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+{"text":"\\section{Introduction}\nAs data acquisition methods become more pervasive, sports analytics has received increased interest in contemporary sports, like soccer, basketball and baseball~\\cite{DBLP:journals\/bigdata\/AssuncaoP18}. One common application in sports analytics is valuing player actions and decision-making. For example, Decroos~et~al.~introduce a framework to value soccer players according to how their actions change their team's chance of scoring~\\cite{DBLP:conf\/kdd\/DecroosBHD19}.\n\nEsports, also known as professional video gaming, is one of the fastest growing sports markets in the world. Yet esports has attracted little sports analytics interest. Most analytical work in esports covers massively online battle arena (MOBA) games, such as League of Legends or Defense of the Ancients 2 (``DOTA2\"). Accordingly, there exists a dearth of work on Counter-Strike: Global Offensive (CSGO), one of the oldest yet most popular esports. A picking and banning process is a common process in many esports, where some entities are banned from being played in a particular game. For example, in League of Legends, teams ban a set of characters, and their players each pick a character to play before the game starts. In CSGO, teams typically perform a map selection process where each team takes turns picking and banning maps to play. However, map selection routines are often not based on analytics and data, but rather on players' inclinations at selection time.\n\nContextual bandits are statistical models that take a context $x$ and return a probability distribution over possible actions $a$, with the objective of maximizing the reward $r$ returned by the action taken. In this paper, we apply a contextual bandit framework to the domain of map selection in CSGO. We use a novel data set of over 25,000 map pick and ban decisions from over 3,500 professional CSGO matches to train three different bandit framings to the problem. We find that teams' choices in the map selection process are suboptimal and do not yield the highest expected win probability.\n\nThe paper is structured accordingly. In section~\\ref{section:RelatedWork}, we review relevant esports and contextual bandit works. In section~\\ref{section:CSMapSelection}, we cover CSGO's map selection process. In section~\\ref{section:Modeling}, we introduce our contextual bandit model. In section~\\ref{section:Experiments}, we describe our dataset and our evaluation methodology. Section~\\ref{section:Results} contains our results. We discuss the benefits of our model, the choices of evaluation metrics and suggest possible areas of future work in section~\\ref{section:Discussion} and conclude the paper in section~\\ref{section:Conclusion}.\n \n\n\n\\begin{figure*}\n    \\center{\\includegraphics[width=\\linewidth]\n    {Figures\/map_picking.png}}\n    \\caption{\\label{fig:mappicking} Example map selection process for a best-of-three match. The available map pool is shown above each pick\/ban decision. The first team, usually decided by tournament rules, bans a map. The second team then does the same. The two teams then both pick a map, and then both ban a map. In total, there are six decisions, four of which are bans, and two are picks.}\n\\end{figure*}\n\n\\section{Related Work} \\label{section:RelatedWork}\n\nReinforcement learning (RL) techniques are increasingly being applied to sports analytics problems. Liu~et~al.~first used RL in sports to estimate an action-value Q function from millions of NHL plays~\\cite{DBLP:conf\/ijcai\/LiuS18}. They used the learned Q function to value players based on the aggregate value of their actions. Liu~et~al.~also apply mimic learning to make their models more interpretable~\\cite{DBLP:conf\/pkdd\/LiuZS18a}. Sun~et~al.~extend this work by considering a linear model tree~\\cite{DBLP:conf\/kdd\/SunDSL20}. While the previous works heavily focused on ice hockey, Liu~et~al.~also learn an action-value Q function for soccer~\\cite{DBLP:journals\/datamine\/LiuLSK20}. Despite the heavy use of other RL approaches such as Q-learning, contextual bandits have not been as heavily utilized in sports analytics.\n\nThis paper applies contextual bandits to the multi-arm map selection process in esports matches for the game CSGO. Contextual bandits are a simplified case of reinforcement learning. In reinforcement learning, an action is chosen based on the context (or state) and a reward is observed, and this process is repeated for many rounds. Rewards are not observed for actions not chosen. In the contextual bandit case, the contexts of different rounds are independent. \\cite{tewari:context_bandit} provides a thorough review of contextual bandits, tracing the concept back to \\cite{woodroofe:context_bandit} and the term back to \\cite{langford:context_bandit}. Many approaches have been explored for learning policies in the contextual bandit setting. \\cite{williams:rl} introduced gradient approaches in the reinforcement learning setting, and \\cite{sutton_bartow:rl} applied the approach to the specific case of contextual bandits. Comparing proposed policies often requires off-policy evaluation: estimating the value of a policy from data that was generated by a different policy (the ``logging policy\"). This paper utilizes two off-policy evaluation approaches: the self-normalized importance-weighted estimator \\cite{swaminathan:sn-iw} and the direct method of regression imputation \\cite{Dud_k_2014}. To our knowledge, ban actions have never been modeled in the bandit setting. \n\nEsports have mostly attracted sports analytics interest in the form of win prediction and player valuation. Numerous efforts have been made to predict win probabilities in popular esports games such as CSGO and DOTA2. \\cite{yang:match_pred} and \\cite{hodge:match_pred} first use logistic regression and ensemble methods to predict win probabilities in DOTA2, a popular MOBA game. \\cite{makarov:csgo} first predicted CSGO win probabilities using logistic regression, however their data only included less than 200 games. \\cite{xenopoulos:csgo} expand on previous CSGO work by introducing a data parser and an XGBoost based win probability model for CSGO. They also value players based on how their actions change their team's chance of winning a round. \\cite{bednarek:csgo} value players by clustering death locations. \n\nMap selection is a process largely unique to CSGO and has not been well studied, but is loosely related to another esports process unique to MOBA games: hero selection. In DOTA2, for example, players from opposing teams alternate choosing from over one hundred heroes, with full knowledge of previous hero selections. \\cite{yang:match_pred} and \\cite{song:dota} use the selected heroes as features to predict win probability, but do not recommend hero selections or explicitly model the selection process.  More relevant is the hero selection recommendation engine of \\cite{conley:dota}, which uses logistic regression and K-nearest neighbors to rank available heroes based on estimated win probability; they do not, however, consider historical team or player context.\n\n\\section{Counter-Strike Map Selection}\\label{section:CSMapSelection}\nCounter-Strike is a popular esport that first came out in 2000, and CSGO is the latest version. The game mechanics have largely stayed the same since the first version of the game. Before a CSGO match starts, two teams go through the map selection process to decide which maps the teams will play for that match. A map is a virtual world where CSGO takes place. Typically, matches are structured as a best-of-three, meaning the team that wins two out of three maps wins the match. A team wins a map by winning rounds, which are won by completing objectives.\n\n\nThe collection of available maps in the map selection process is called the \\textit{map pool}. Typically, there are seven maps to choose from in the map pool. Although the maps rarely change, a new map may be introduced and replace an existing map. Our data contains map selections using the following map pool: \\texttt{dust2}, \\texttt{train}, \\texttt{mirage}, \\texttt{inferno}, \\texttt{nuke}, \\texttt{overpass}, \\texttt{vertigo}. The map selection process is exemplified in Figure~\\ref{fig:mappicking}. First, team A \\textit{bans} a map. This means that the teams will not play the map in the match. The team that goes first in the map selection process is usually higher seeded, or determined through tournament rules. Next, team B will ban a map. The teams then will each \\textit{pick} a map that they wish to play in the match. Next, team A will ban one more map. At this point, team B will ban one of the two remaining maps, and the map not yet picked or banned is called the \\textit{decider}. \n\n\nProfessional teams may sometimes have what is referred to as a \\textit{permaban} -- a map that they will always ban with their first ban. For example, some teams may choose to ban the same map in over 75\\% of their matches. From interviews with four CSGO teams ranked in the top 30, two of which are in the top 10, teams choose their maps from a variety of factors. Some teams try to choose maps they have a historically high win percentage, or maps where their opponents have low win percentages. Other teams may also choose maps purely based on how their recent practice matches performances.\n\n\n\\section{Bandit Model for CSGO Map Selection} \\label{section:Modeling}\n\nIn order to model the map selection process, we elected to use a \\textit{k}-armed contextual bandit. This was a clear choice: the actions taken by teams only yield a single shared reality, where we cannot observe the counterfactual of different choices. The bandit model enables us to approximate the counterfactual reality and frame this problem as a counterfactual learning problem.\n\nIn particular, we used the context from teams' previous matches, as well as information available at the time of selection, such as which maps were still in the selection pool. There are two kinds of actions: picks and bans, which must be manipulated differently. The reward is the map being won by the choosing team or not, as well as more granular version of this in which we include margin of victory.\n\n\\subsection{Context and Actions}\n\nOur initial choice for the context given a particular round $t$ in the map-picking process was a one-hot encoding for the available maps in that particular round, such that the bandit would learn to not pick the map if it was not available. To give the bandit more information about the teams that were deciding for that particular match, we implemented two historical win percentages, the first being the team's historical match win percentage, and the second being the team's historical map win percentage for each map. The first percentage is utilized to indicate team strength compared to other teams, and the second the team's overall ability to play well on each map. We applied Laplace smoothing to the initial percentages for numerical stability, using the formula\n\n\\begin{equation}\n    \\text{Win\\%} = \\dfrac{\\text{Wins} + 5}{\\text{Matches} + 10}.\n\\end{equation}\n\nBoth win percentages were stored in the context vector for both the deciding team and the opponent team alongside the available maps. For both picks and bans, the given \\textit{context} is the same as described above, and the corresponding \\textit{action} would be the map picked or banned by the deciding team.\n\\subsection{Rewards}\n\\subsubsection{Picks}\n\nDue to the nature of the map-picking process, where the decider is a forced pick, we chose to remove the rewards from all final map picks, as it would not make sense to reward either team for a forced choice. As a result, only the first two picks from each map selection process were given a reward. Rewards for map-picking were implemented with two different methods. Our first method utilized a simple 0-1 reward (``0\/1\"), where if the deciding team won on the map they had picked, they would be rewarded with an overall reward of $1$ for that action, or $0$ otherwise. Our second method rewarded the deciding team based on the margin of rounds won (``MoR\") in the best-of-30 rounds on the decided map. The reward function for deciding team $i$ and an opponent team $j$ is given below:\n\n\\begin{equation}\n    R_{i,j}= \\frac{\\text{Rounds won by  $i$} - \\text{Rounds won by $j$}}{\\text{Total number of Rounds on map}} \n\\end{equation}\n\nThe round proportion rewards were implemented as a more granular method to compare team performance on each map.\n\n\\subsubsection{Bans}\n\nSince there is no data on how any deciding team would perform on a banned map, we chose to reward bans based on the deciding team's overall performance in the match, where if the deciding team won the match, they would be rewarded for choosing to not play on the banned map with an overall reward of $1$, or, if they lost, a reward of $-1$. In addition, we implemented a exponentially decreasing reward over the ban process, where earlier bans would have higher rewards. Later map picks have fewer available choices: restricting the action space means a team may be forced to make a choice they do not want, and so we de-emphasize the later choices. The ban reward function for team $i$ playing in match $t$ is given below:\n\\begin{equation}\n    R_{i,t}(n) = \n    \\begin{cases}\n         1 \\cdot \\frac{1}{2^n} & \\text{if team $i$ won match $t$} \\\\\n         -1 \\cdot \\frac{1}{2^n}& \\text{if team $i$ lost match $t$} \\\\\n  \\end{cases}\n\\end{equation}\nwhere $n$ is the $n$th ban in the map picking process. In our case, $n \\in \\{1,2,3,4\\}$, as there are always four bans in the map picking process for CSGO. \n\n\\subsection{Policy Gradient Learning}\n\nThe most straightforward way to train a bandit is via policy gradient learning \\cite{sutton_bartow:rl}. For our policy class, we use a multinomial logistic regression parameterized by weights $\\theta$ and an action-context mapping function $\\phi(x, a)$, with the softmax function to transform the affinity of the bandit for each action into a probability:\n\n\\begin{equation}\n\\pi(a|x) = \\dfrac{\\exp(\\theta^{T} \\phi(x, a))}{\\sum_{i=1}^k \\exp(\\theta^{T} \\phi(x, i))}\n\\end{equation}\n\nThe policy gradient approach trains the model via SGD \\cite{sutton_bartow:rl}, enabling both online and episodic learning. In particular, the optimization maximizes the expected reward for the bandit, using the update function\n\n\\begin{table*}[]\n\\centering\n\\begin{tabular}{@{}lrrrr@{}}\n\\toprule\n                       & \\multicolumn{1}{c}{Picks (0\/1)} & \\multicolumn{1}{c}{Picks (MoR)} & \\multicolumn{1}{c}{Bans (0\/1)} & \n                       \\multicolumn{1}{c}{Bans (MoR)} \\\\ \\midrule\nUniform policy (split) & 0.568\/0.541                    & 0.568\/0.541                          & -0.018\/-0.003                    & -0.018\/-0.003                         \\\\\nLogging policy         & 0.549\/0.549                       & 0.549\/0.549                          & -0.014\/-0.014                 & -0.014\/-0.014                         \\\\\nSplitBandit            & 0.587\/0.554                      & \\textbf{0.659\/0.528}                 & -0.016\/0.004                       & -0.016\/0.004                         \\\\\nComboBandit             & \\textbf{0.640\/0.528}              & 0.613\/0.573                          & \\textbf{0.021\/0.003}               & \\textbf{0.036\/-0.015}                 \\\\\nEpisodicBandit         & 0.568\/0.551                       & 0.561\/0.547                          & 0.013\/0.006                    & 0.013\/0.006                           \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Expected reward for each policy type under four different evaluations. The best policy parameters were found via grid search and the policy was optimized with policy gradient. Both the SN-IW (left) and DM (right) evaluation methods are presented, except for Logging policy where the on-policy value is presented. Every model tested outperforms or matches the baseline uniform policy, with the best overall model being the bandit trained on both picks and bans. Comparisons between the uniform and logging policy indicate teams choose their bans well, but their picks poorly.}\n\\label{table:mainresults}\n\\end{table*}\n\n\\begin{equation}\n\\theta_{t+1} \\leftarrow \\theta + \\eta R_t(A_t) \\nabla_{\\theta} \\log \\pi_{\\theta_t}(A_t|X_t)\n\\end{equation}\n\nwith $\\pi$ defined above and the gradient\n\n\\begin{equation}\n\\resizebox{.91\\linewidth}{!}{$\\nabla_{\\theta} \\log \\pi(a|x) = \\phi(x,a) - \\dfrac{\\sum_{i=1}^k \\phi(x, i)  \\exp(\\theta^T \\phi(x, i))}{\\sum_{i=1}^k \\exp(\\theta^T \\phi(x, i))}.$}\n\\end{equation}\n\nIn the context of picks, we can use online learning to iteratively update the parameters $\\theta$. For bans, however, we do not observe a reward at the time the choices are made; as a result, we used episodic learning, where an episode is an entire match.\n\n\\section{Experiments}\\label{section:Experiments}\n\\subsection{Data}\nWe obtained our data from HLTV.org, a popular CSGO fan site. The site contains scores, statistics and map selections for most professional matches. We use matches from April 2020 to March 2021. In total, this consisted 628 teams that played a total of 6283 matches, summing to 13154 games. We only consider best-of-three matches, which are by far the most popular match format. We focus on the games where the most common set of seven maps is selected from the map pool of \\texttt{dust2}, \\texttt{inferno}, \\texttt{mirage}, \\texttt{nuke}, \\texttt{overpass}, \\texttt{train}, \\texttt{vertigo}. In addition, we also remove teams such that in the final dataset, each team has played at least 25 games, or approximately 10 matches, with another team in the dataset. This leaves us with 165 teams, playing a total of 3595 matches, summing to 8753 games. The resulting dataset was split into an 80-20 train-test split by matches for bandit learning and evaluation.\n\n\\subsection{Evaluation}\nWe use two typical off-policy evaluation methods, the \\textit{direct method}  (``DM\") \\cite{Dud_k_2014} and the self-normalized importance-weighted estimator (``SN-IW\") \\cite{swaminathan:sn-iw}. We also present the mean reward observed as a baseline. \n\nThe goal of the direct method is to estimate the reward function $r(x, a)$ that returns the reward for any given action $a$ for the context $x$. We estimate the reward function by using an importance-weighted ridge regression for each action. We use the self-normalized importance-weighted estimator with no modifications.\n\nValue estimates are presented for four different reward and model training settings:\n\\begin{itemize}\n    \\item Picks(0\/1): Expected pick reward for models trained with 0\/1 rewards\n    \\item Picks(MoR): Expected pick reward for models trained with MoR rewards\n    \\item Bans(0\/1): Expected ban reward for a models trained with 0\/1 rewards\n    \\item Bans(MoR): Expected ban reward for models trained with MoR rewards\n\\end{itemize}\n\n\\subsection{Variety of Policies}\n\nWe experimented with three different varieties of contextual bandits: \\texttt{SplitBandit}, \\texttt{ComboBandit}, and \\texttt{EpisodicBandit}. \n\n\\texttt{SplitBandit} is composed of two individual, simple contextual bandits, each with a $\\theta$ parameter size of $(\\texttt{n\\_features} \\cdot \\texttt{n\\_arms})$. The first contextual bandit is trained on the picks via online learning. The second contextual bandit is trained on the bans in an episodic fashion.\n\n\\texttt{ComboBandit} is a single model also trained on the picks via online learning and on the bans via episodic learning with a $\\theta$ parameter size of $(\\texttt{n\\_features} \\cdot \\texttt{n\\_arms})$. \\texttt{ComboBandit} learns a single set of parameters that define a policy for both picks and bans. The ban policy is derived from the pick policy:\n\\begin{equation}\n    \\pi_B(a|X) = \\frac{1-\\pi_P(a|X)}{\\sum_{\\alpha \\in A}1-\\pi_P(\\alpha|X)}\n\\end{equation}\nfor pick policy $\\pi_P$ and ban policy $\\pi_B$ over actions $A$ and context $X$.  \n\n\n\\texttt{EpisodicBandit} is similarly a single model, but it is trained on both the picks and bans simultaneously via episodic learning with a $\\theta$ parameter size of $(2 \\cdot \\texttt{n\\_features} \\cdot \\texttt{n\\_arms})$. We expected this model to perform similarly to \\texttt{SplitBandit}, since its gradient estimates are better estimates than the estimates derived from individual datapoints, offsetting the quicker adaptability of the online gradient calculation with less noise.\n\n\\section{Results} \\label{section:Results}\n\nOur main results are summarized in table~\\ref{table:mainresults}. Considering the self-normalized estimator, the best model for picks was \\texttt{SplitBandit} trained on proportional rewards, while the best model for bans was \\texttt{ComboBandit} trained on proportional rewards. The uniform policy performs better than the logging policy for the picks in our dataset but worse for bans, which indicates teams' picks might be overconfident, whereas their bans are chosen more carefully.\n\n\\begin{figure}[t]\n    {\\includegraphics[width=9cm]\n    {Figures\/pick_value_over_time.png}}\n    \\caption{\\label{fig:value_over_time} Picks(0\/1) value on the test set for \\texttt{ComboBandit} and Uniform policy, evaluated every 100 rounds over 3 epochs of training. The bandit quickly surpasses the uniform policy's performance and plateaus around an expected reward value of approximately $0.64$.}\n\\end{figure}\n\n\\texttt{ComboBandit} substantially outperforms all other policies. We believe this is due to its training including the additional data from both picks and bans instead of selecting only one of the two categories for training a given parameter. This yields a better optimization through better gradient estimates. \\texttt{EpisodicBandit} is trained on both picks and bans, but its parameters do not depend on both subsets of data, which does not provide that optimization advantage. The learning curve in Figure \\ref{fig:value_over_time} shows that ComboBandit surpasses the uniform policy benchmark after only a few training rounds, continuing to improve over 3 epochs of training.\n\n\\begin{figure}[t]\n    {\\includegraphics[width=9cm]\n    {Figures\/policy_ex.png}}\n    \\caption{\\label{fig:policy_comparison} The best model's probability distribution for pick 4 in a match between \\textit{TIGER} and \\textit{Beyond}. \\textit{TIGER}, the deciding team, chose \\texttt{Nuke} and lost the map, later going on to win map \\texttt{Overpass}, which was \\texttt{ComboBandit}'s suggestion.}\n\\end{figure}\n\nFigure~\\ref{fig:policy_comparison} shows an example of \\texttt{ComboBandit}'s policy. In this match, team \\textit{TIGER} chose to play on the map \\texttt{Nuke}, which they later lost. \\texttt{ComboBandit} suggested instead to play on \\texttt{Overpass}, with 71\\% probability. In the same match, \\texttt{Overpass} was chosen as the decider and \\textit{TIGER} won that map, indicating that the bandit model's policy distribution was more valuable than the team's intuition on map choice.\n\n\n\\section{Discussion} \\label{section:Discussion}\n\nThe results indicate that teams using our chosen policy instead of their traditional map-picking process can increase their expected win probability by 9 to 11 percentage points, depending on the policy used. This is a substantial advantage for a best-of-3 match, since the model could confer that added win probability to all three map choices. The ban choice can be improved as well by using our model. The logging policy yields an expected reward of approximately $-0.014$, which indicates that bans have a slight negative effect on match win probability. However, our best model's expected reward for bans is $0.036$, thus increasing match win probability by approximately 5\\% after a ban choice. For two teams that are evenly matched, using our bandit for both pick and ban decisions translates to the team that uses the model having an expected overall match win probability of 69.8\\% instead of 50\\%, a substantial advantage for a team.\n\nThe choice of evaluation metric is particularly important in examining the results. Using the direct method instead of the self-normalized estimator, we reach drastically different conclusions about which model to use, with the best overall model being \\texttt{EpisodicBandit}. In our experiments, we used ridge regressions for our regression imputation. This is clearly a suboptimal model for this estimation, since the context features of win probabilities are bounded: there is a non-linear relationship between the context and the rewards. This is a big limitation of our experiments: we instead relied on the importance-weighted estimator, which is known to be imprecise in estimating policies far from the logging policy.\n\nFuture work in this area will be concentrated on examining better choices for evaluation metrics, as well as expanding the contextual features further by adding, for example, player turnover, team-based Elo metrics or rankings, or examining recent performances, such as win percentage in the last 10 matches. The rewards can also be expanded by using not only margin of rounds won per map, but also the margin of players alive per map at the end of a round. Additionally, different framings for the bandit can be considered, such as creating a ranking of which maps are best to choose instead of the model selecting a single map for the user. \n\n\n\\section{Conclusion} \\label{section:Conclusion}\n\nWe modeled the map selection process in Counter-Strike: Global Offensive as a bandit, framing the problem in several different ways. Our key contributions are (1) the introduction of bandits and simple reinforcement learning models to esports and CSGO in particular, and (2) novel ways of implementing negative choices in bandits, for which we explicitly choose not to observe their rewards. We find that our model shows that teams are making sub-optimal map selections. \n\n\n\\section*{Acknowledgments}\nThis work was partially supported by: NSF awards CNS-1229185, CCF-1533564, CNS-1544753, CNS-1730396, CNS-1828576, CNS-1626098. We additionally thank David Rosenberg.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNowadays, a lot of information is daily generated. It is necessary to have available memory storage because each datum must be processed and the information contained therein analyzed. The manual analysis is impossible because it is necessary a huge number of persons to analyze this information in an available time. The summary is a short text with main ideas of original text \\cite{torres2014automatic} and reduces the read time to analyze these data.\n\nAudio is widely used in daily life on the radio and on the internet, in news, interviews and conversations. A Call Centre Conversation creates a lot of conversations every day. These centers has issues and tasks. It is essential the control of the discussed topics and the results obtained by customers in these calls. One way to analyze and accelerate the data processing is speech summarization, that is different from traditional text summarization because there are other problems in these texts as speech errors, sentences of different sizes and colloquialisms.\n\n\n\\textsl{``Multiling is a community-driven initiative for benchmarking multilingual summarization systems, nurturing further research, and pushing the state-of-the-art in the area''}\\footnote{\\url{http:\/\/multiling.iit.demokritos.gr\/pages\/view\/1517\/multiling-2015-call-for-participation}}. The MultiLing 2015 initiative features the following tasks: Multilingual Multi-document Summarization, Multilingual Single-document Summarization, Online Forum Summarization and Call Centre Conversation Summarization (CCCS). The CCCS pilot task consists in \\textsl{``creating systems that can analyze call centres conversations and generate written summaries reflecting why the customer is calling, how the agent answers that query, what are the steps to solve the problem and what is the resolution status of the problem\"} \\cite{benoit}.\n\nWe developed the LIA-RAG summarization system based on the RAG system \\cite{rag:15}, coupled with some post-processing rules\nin order to generate a final summary. LIA-RAG uses a graph model to analyze and verify a set of documents (e.g., the conversation transcription) for MultiLing'15 CCCS pilot task. \nLIA-RAG creates a summary computing the relevance of the words and the similarity among the sentences. The system uses a simple post-processing to improve the quality of the final summary. \n\nThe rest of the paper is organized as follows: \nsection \\ref{sc:rt} describes related work on automatic summarization of texts and conversations. Sections \\ref{sc:mod} and \\ref{sc:rag} analyze the graph model and the system used in this work. Section \\ref{sc:results} describes the results obtained for Multiling\/DECODA French corpus and section \\ref{sc:conc} concludes this work.\n\n\\section{Related Works}\n\\label{sc:rt}\n\n\\ac{ATS} aims to creates a summary containing the main ideas of a textual document \\cite{mani:mayburi:99,mani:01,torres2014automatic}. The summary can be an extraction or abstraction of a single document or multi-document. The extraction process identifies the most informative sentences of a document and creates a summary by assembling of these sentences \\cite{Luhn,torres2014automatic}. Extraction may be guided (by a query). In this case, the algorithm selects the most relevant information follow a particular topic. The abstraction algorithms create new (or reformulate) sentences from original texts \\cite{seno1,seno2} and the extraction methods use the key sentences of texts \\cite{Barzilay,torres2014automatic}.\n\nWorks about abstraction usually uses syntactic and semantic knowledge of a language to create the summary. This procedure verifies the best construction of a sentence \\cite{Barzilay2}. This type of summarization uses fusion to help the review of information. \\cite{seno1} proposed a method to fusion similar sentences in Brazilian Portuguese based on a symbolic and domain-independent approach. This method allows the fusion by union and by intersection of a document cluster. Fusion by union preserves the overall message of the cluster while fusion by intersection analyses the redundant information considered most important in the cluster. \\cite{seno2} described how to identify common information between sentences in Brazilian Portuguese using lexical \nknowledge, syntactic and semantic rules of paraphrasing.\n\n\\cite{cstsumm} developed a summarizer system based on the CST model (Cross-document Structure Theory). The system proposed analyses redundancy and contradiction among different information sources in Brazilian Portuguese.\n\n\\cite{Barzilay2} developed a method to generate automatic summaries by identifying and synthesizing similar elements in a cluster of documents. This method creates the summary based on similarity between the sentences and topic. \\cite{Barzilay} described an approach to fusion sentences through the text-to-text technique, to synthesize repeated information from multiple documents. This method uses a syntactic alignment in sentences to identify common information. After the identification step, sentences are processed and a new text is generated with the same content.\n\nA way to calculate the similarity between sentences is to use co-occurrence of words. \\cite{He} proposed a fusion method using similarity metrics, co-occurrence skip-bigram and information density to evaluate sentences and to select the most relevant ones. \\cite{Hennig} developed a multi-document model to summarize by analyzing the co-occurrence of sentence-term and sentence-bigram using the \\ac{JS} divergence.\n\nAnother method to obtain relevant sentences uses compression, as reported in \\cite{Pitler}. Pitler uses approaches based on syntactic trees, sentences and \ndiscourse. \\cite{Filippova} describes a multi-sentence compression method using a word-based graph.\n \nThe summarization by extraction does not have the same quality as the summaries produced by abstraction because it uses surface methods based on statistical calculations to verify the sentence relevance. However, the extraction is general and do not require deep analysis of the language \\cite{Barzilay,sasi}.\n\n\\cite{sasi} use Graph theory concomitant with \\ac{JS} divergence to create multi-document summaries by extraction. Their system describes a text model as a graph where the sentences are represented by vertices and the edges connect two similar sentences. Their approach calculates the stable set of the graph aiming creating the summary containing sentences with general information of the cluster and without redundancy. \\cite{glouton} model the text as graph model and use a heuristic (greedy algorithm) to obtain the relevant sentences in the text.\n\nThe speech summarization task is more complex and it involves other problems. It is more difficult to identify utterance boundaries because it may be fragmented,  contain disfluencies and also because speech recognition introduces errors. Meetings involve multi-party conversation with overlapping speakers. The language used is informal and utterances tend to be partial, fragmentary, ungrammatical and include many ellipses and pronouns. However, the speech signal may provides additional information that emphasizes a piece of text as prosody \\cite{Murray}.\n\n\\cite{Mckeown} described some ways to use a text summarization as a speech summarization. They described some work about summarization of broadcast news and meetings. \\cite{Murray} analyzed extractive summarization of multiparty meetings. They described Maximal Marginal Relevance and Latent Semantic Analysis to create the summary based on prosodic and lexical features.\n\n\n\\section{Modeling the problem}\n\\label{sc:mod}\n\nThis paper aims to design a system to summarize several documents by extraction its most important sentences. Statistical techniques were used to build a language independent system. The proposed methods are based on a specific preprocessing of words, a weighting function of sentences and a bag-of-words model to represent the text content.\n\nThis model uses $K$ matrices represented by $S^{K}_{[m \\times n]}$ and constructed from $K$ documents, where $m_a$ is the number of sentences and $n_a$ is the number of distinct words in the document $a$ ($a \\in K$). The cell $s^{a}_{ij}$ of the matrix represents the frequency of word $j$ in the sentence $i$ ($FP_{ij}$) of the document $a$.\nThis stage was constructed using the libraries and algorithms from Cortex summarization system \\cite{torres2002condenses,Torres-Moreno2001}.\n\n\n\\begin{equation}\n\\begin{split}\n S^{a}=\\left(\n\\begin{array}{cccc}\ns^{a}_{11} & s^{a}_{12} & \\ldots & s^{a}_{1n}\\\\\ns^{a}_{21} & s^{a}_{22} & \\ldots & s^{a}_{2n}\\\\\n\\vdots & \\vdots &  & \\vdots \\\\\ns^{a}_{m1} & s^{a}_{m2} & \\ldots & s^{a}_{mn}\\\\\n\\end{array}\n\\right), a \\in K\\\\\ns^{a}_{ij}=\\left\\{\n\\begin{array}{cc}\nFP_{ij}, & \\textrm{if}\\ \\exists\\ \\textrm{word\\ j\\ in\\ sentence\\ i}\\\\\n0, & \\textrm{otherwise}\\\\\n\\end{array}\n\\right.\n\\end{split}\n\\end{equation}\n\n\\subsection{Jensen-Shannon divergence}\n\\label{ss_djs}\n\nWe use Jensen-Shannon (JS) divergence to measure the similarity between sentences. Let $w$ be a words' set in P and Q. P and Q represent\nthe probability distribution between two objects:\ntwo individuals sentences or a sentence and a set of sentences.\nThe divergence will then calculated among these two objects. The \\ac{JS} divergence is symmetric and provides a stable way to measure the difference between two distributions (equation \\ref{DJS}).\n\n\\begin{eqnarray}\n\\label{DJS}\nD_{JS}(P||Q) &=& \\frac{1}{2}\\sum_{w \\in W} \\Bigg[ P_w\\log{\\Bigg( \\frac{2 \\times P_w}{P_w+Q_w} \\Bigg)} \\nonumber \\\\\n  &+& Q_w\\log{\\Bigg( \\frac{2 \\times Q_w}{P_w+Q_w} \\Bigg)} \\Bigg]\\\\ \\nonumber\n\\end{eqnarray}\n\nThe \\ac{JS} divergence value ranges from $[0,\\infty+)$. It is closer to zero when the distributions are similar and they differ in another case. \n\nIn the case there is a word in a sentence that is missing in another one, a smooth (different weighting) will be used to avoid null values and have a smoother distribution \\cite{smoothBook}. If a word $w$ is not present in the sentence $Q$, then the smooth is calculated by the equation \\ref{smooth}, where $ \\beta = 1.5 \\times voc$, which $voc$ is the number of distinct words in $R$, $\\gamma$ is the variable that controls the relevance of the missing word in the sentence and $N$ is the number of words in $R$ \n\\cite{nenkova}.\n\n\\begin{equation}\n \\label{smooth}\n  Q_w = \\left( \\frac{P_w + \\gamma}{N + \\gamma \\times \\beta} \\right)\n\\end{equation}\n\n\\subsection{Term Frequency-Inverse Sentence Frequency (TF-ISF)}\n\\label{sc_tfisf}\n\nOne way to verify the initial relevance of a word and a sentence to the text is through the TF-ISF. This metric is based on term frequency in the text and it is calculated by the equation \\ref{tfisf}.\n\n\\begin{equation}\n \\label{tfisf}\n  tf\\_isf(w) = tf(w) \\times \\log \\left( \\frac{n}{n_{w}} \\right)\n\\end{equation}\n\n\\noindent where $tf(w)$ is frequency of term $w$, $n$ is total number of documents and $n_w$ is number of documents that contain the term $w$.\n\n\\section{The LIA-RAG system}\n\\label{sc:rag}\n\nIn general lines, a text consists of several sentences with different topics. The text can be divided into several groups and each of them describes one step\/idea in the text. If a group is large, then it is relevant to the text. It is possible to choose the sentences of the largest group and obtains the most relevant content.\n\nThe main ideas of a text are generally analyzed and discussed several times. The vertices with higher degree have more similar sentences and then, are important to the text. However, it is not necessary to have a lot of similar sentences to be a relevant one.\n\n\\ac{RAG} is a summarizer system by sentence extraction, which selects the main sentences of a text and uses a post-processing to remove some errors and make the text more concise and compact.\n\n\\subsection{The RAG algorithm}\n\\label{ssc:rag_desc}\n\n\\ac{RAG} uses Graph theory and divergence metrics to calculate the similarity and to group the sentences. Initially, the system performs a filtering process to remove the brackets. Then, it performs a segmentation, filtering and stemming processes to remove stopwords and reduce the words to their roots. RAG accomplished this preprocessing and matrix transformation based on \\cite{Torres-Moreno2001}. It calculates the relevance of each sentence based on TF-ISF metric (equation \\ref{tfisf}) and removes the less relevant sentences.\n\nThe system creates a graph $G$ which each vertex represents a sentence previously selected. The text is analyzed and modeled as a sentence graph (vertices). Based on equation \\ref{tfisf}, it calculates the similarity between sentences. If the similarity between two sentences is less than 0.16 (threshold obtained by empirical testing), then the system creates an edge between them. So, the vertices with higher degrees have the most relevant content of the text. However, some sentences may have a small degree, but they may contain important information.\n\nRAG combines the TF-ISF and degree sentences to analyze the relevance of them. The relevance of the sentence $i$ is defined by:\n\n\\begin{equation}\n\\label{eq:score}\n  rel(i) = degree(i) \\times tf\\_isf(i)\n\\end{equation}\n\n\\noindent where $degree(i)$ is the degree of vertex $i$ and $rel(i)$ is the relevance of the sentence $i$.\nAfter, the system creates a summary with the higher score sentences, excluding similar (or redundant) sentences based on Dice's coefficient \\cite{dice}. \n\nThe figure \\ref{fig:rag} describes the RAG system.\n\n\\begin{figure}[!h] \n\\begin{center} \n\\includegraphics[scale=0.42]{rag.pdf}\n\\end{center} \n\\caption{Architecture of the RAG system.} \n\\label{fig:rag} \n\\end{figure}\n\n\\subsection{LIA-RAG: RAG with a specific speech post-processing}\n\\label{ssc:rag_post}\n\nThe speech recognition process produces a text that contains several grammatical problems (slang, colloquialisms, expressions and speech recognition errors). An extraction summary algorithm selects the relevant sentences, however the sentences may have some grammatical problems. So, it is necessary to perform a treatment of this summary.\n\nThe main analyzed aspects in this process are: \n\\begin{itemize}\n\\item Colloquialisms, \n\\item Speech expressions and \n\\item Dates. \n\\end{itemize}\n\nLIA-RAG system receives the summary as an input.\nIn this input, some speech expressions are used to connect ideas or concepts in oral conversations.\nLIA-RAG removes these expressions, because often they are incorrectly transcripted (a noise source). Also, the system eliminates several colloquialisms and the duplicated words. The system replaces some mistaken words by its correct form. The figure \\ref{fig:lia-rag} shows the architecture of the LIA-RAG system.\n\n\\begin{figure}[!h] \n\\begin{center} \n\\includegraphics[scale=0.38]{lia-rag.pdf}\n\\end{center} \n\\caption{Architecture of the LIA-RAG system.} \n\\label{fig:lia-rag} \n\\end{figure}\n\n\\section{Results}\n\\label{sc:results}\n\nThe tests were carried on a computer with i5@2.6 GHz processor and 4 GB of RAM on GNU\/Linux Debian 64-bit operating system. The algorithms of RAG were implemented using the Perl language.\n\nWe used the French DECODA corpus \\cite{BECHET12.684}. The systems have to generate textual summaries with the main idea of each conversation belonging to the corpus. \\textsl{``The conversations topics range\nfrom itinerary and schedule requests, to lost and found, to complaints (the calls were recorded during strikes)''}\\cite{benoit}. Each summary has 7\\% of the number of words of each conversation transcription. We compared LIA-RAG and RAG systems with two baseline systems (random and  first lead base). \n\n\nIn order to evaluate the quality of the summaries, Multiling CCCS used the system \\ac{ROUGE}\\footnote{The options for running ROUGE 1.5.5 are -a -l 10000 -n 4 -x -2 4 -u -c 95 -r 1000 -f A -p 0.5 -t 0}, which determines the quality of an automatic summary based on the intersection of the $n$-grams of a candidate summary and the $n$-grams of a set of reference summaries. More specifically, we used ROUGE-N and ROUGE-SU measures. ROUGE-N, N $\\in\\ [1, 2]$. ROUGE is an $n$-gram recall measure \\cite{rouge}\\footnote{\\url{http:\/\/www.berouge.com\/Pages\/default.aspx}}. The values of these metrics belongs to $[0, 1]$, 1 for the best result.\n\nThe table \\ref{tb:rouge_training} shows the results obtained using the systems over the training corpus.\nThis corpus contains 50 conversations transcription with 23,363 words and 115 summaries. Both versions of RAG provided the best results. The RAG system identified the main sentences discussed in conversations. \nHowever, the errors and speech expressions decreased the informativeness. \nThe post-processing of LIA-RAG allowed to improve the results. \nThis process reduces errors and generates a more informative and concise summary.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|ccc|}\n\\hline\n \\textbf{Systems} \t& \\small \n\\textbf{ROUGE-1}    \t& \\small \\textbf{ROUGE-2}    \t& \\small \\textbf{ROUGE-4}     \t\\\\ \\hline\n \\textbf{LIA-RAG:1}\t\t\t&   \\textbf{0.1893}  \t&   \\textbf{0.0628}  \t& \\textbf{0.0683} \\\\ \n \\textit{RAG}\t\t\t&   \\textit{0.1833}  \t&   \\textit{0.0614}  \t& \n \\textit{0.0654} \\\\\n Base-first \t&   0.1578  \t&   0.0556  \t& 0.0583 \\\\\n Base-rand \t\t&   0.1170  \t&   0.0310  \t& 0.0371 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tb:rouge_training} Evaluation of training corpus.}\n\\end{table}\n\nThe French test corpus has 100 conversations transcription with 42,130 words and 212 summaries. The ROUGE-2 official performance for the systems participating to CCCS pilot task is showed in table \\ref{tb:rouge_test} \\cite{benoit}. The LIA-RAG system  obtained the best results.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\n \\textbf{Systems} \t& \\textbf{ROUGE-2} \t\\\\ \\hline\n \\textbf{LIA-RAG:1}\t\t&  \\textbf{0.037} \\\\\n NTNU:1 \t\t&  0.035 \\\\\n NTNU:3\t\t\t&  0.034 \\\\\n NTNU:2 \t\t&  0.027 \\\\\n \\hline\n\\end{tabular}\n\\caption{\\label{tb:rouge_test} Evaluation of test corpus.}\n\\end{table}\n\n\\section{Conclusion and perspectives}\n\\label{sc:conc}\n \nDivergence of probabilities in a graph model to extract key sentences in French speech-to-text summarization was very interesting. LIA-RAG system uses very few language resources (stopwords and stemming) and has achieved good results. \nNevertheless, the system is easily adaptable to other languages with only some modifications in the preprocessing stage.\n\nAn interesting perspective of this work consists in the utilization of the speech TAGs markers to improve the computation of the sentences score. \nIn addition, it is necessary to improve the post-processing in order to increase the quality of the final summary.\nFinally, the verification of the grammaticality and readability of the extracted key sentences can help to produce more realistic abstracts.\n\n\n\\section*{Acknowledgments}\nThis project was partially founded by a scholarship from FUNCAP-CE (Brazil).\n\n\\newpage\n\n\\bibliographystyle{acl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\n\\input{module_intro}\n\n\\section{model} \\label{model}\n\\input{module_YSK_normal}\n\n\\section{Inverse proximity effect} \\label{inverseproximity}\n\\input{module_result_normal}\n\n\\section{Numerical Calculation of the ground state energies} \\label{numerics}\n\\subsection{Single-layer limit of the superconductor}\n\\input{module_YSK_SC}\n\\input{module_result_SC}\n\n\\subsection{Multiple-layers of the superconductor}\n\n\\input{module_result_multilayer}\n\\section{transport measurement} \\label{transport}\n\n\\input{module_result_transport}\n\\input{module_result_transport2}\n\\section{conclusion} \\label{conclusion}\nIn conclusion, we have studied the stability of the FF phase in magnetically doped TI-BCS superconductor heterostructures. We find that the FF state can be more energetically favorable than the traditional BCS pairing. This is due to the anisotropy of the Fermi surface in the superconductor that arises from the IPE where the normal band of the superconductor near the interface has the effective spin-orbit coupling and the Zeeman field. We find that the IPE quickly decays as the coupled state moves farther away from the interface into the bulk of the superconductor. As a consequence, the FF state gains more energy as the thickness of the superconductor increases and the stability of the FF state quickly decays. Nevertheless, in the thick superconductor limit, we find the FF phase can survive at the interface of the proximity structure.  We expect the FF pairing in our proposal can be experimentally measured through the four probe transport experiment utilizing a Josephson junction or through the Y-junction method.\n\n\n\n\n\n\\section{Acknowledgement}\nM.J.P. appreciates David ChangMo Yang for helpful discussions. This work is supported by NSF CAREER ECCS-1351871.\n\n\n\n\n\n\n\\subsection{Y junction}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.8]{experiment_config2}\n\\break\n\\includegraphics[scale=0.8]{schematic2}\n\\break\n\\includegraphics[scale=0.35]{G_angle}\n\\caption{(a) A schematic of the Andreev interferometer using Y-junction method to measure the unconventional superconductivity. (b) A plot of conductance as a function of finite momentum angle, $\\theta$, of the FF phase. From the outermost to innermost, we plot the calculation results with different $\\vec{q}$ where $|q|=0,\\;\\pi\/8,\\;\\pi\/6, and \\;\\pi\/4$, respectively. We use a value of $v_F=6.61\\times 10^{5}$ m\/s for the Fermi velocity, that has been extracted from the metal Hamiltonian parameter, and plot the conductance at the incident electron the energy of $E=0.5\\Delta$, where the $\\Delta$ is superconducting gap. We set the barrier height at the interface of the metal arms and superconductor to be transparent.\n} \\label{fig:exp2}\n\\end{figure}\n\nAnother useful experimental method to detect FF phase is Andreev interferometer\\cite{2014Xing}. Fig. \\ref{fig:exp2} (a) shows the Andreev interferometer with the Y-junction with two arms separated by $L_x$ in $\\hat{x}$ direction placed on the top of the superconductor. In the presence of the magnetization vector, $\\vec{m}=|m|(sin\\theta,cos\\theta)$ on the TI surface, our analysis shows that the FF phase with the momentum vector, $\\vec{q}=|q|(cos\\theta,sin\\theta)\\perp \\vec{m}$, is induced. In this case, the superconducting order parameter at each contact has different phases due to the phase modulation resulting from the finite longitudinal separation with respect to the momentum of the FF phase, $q$. We parameterize the different phases by assigning the order parameters $|\\Delta| e^{-i\\vec{q} x_1}$  and $|\\Delta| e^{-i\\vec{q} x_2}$ at the upper and the lower contacts respectively, where $x_1$ and $x_2$ are the coordinates of the upper and lower contacts. The phase difference between the two contacts is given as $\\Delta \\phi=q(x_1-x_2)cos\\theta=|q|L_x cos\\theta$. When the current flows through the Y-junction, the electrons injected from lower and the upper contacts undergo Andreev reflection process and reflected as holes. Due to the presence of the FF order parameter, the holes gain additional phases of either $\\Delta e^{-iq_x x_1}$ or $\\Delta e^{-iq_x x_2}$ depending on whether it is reflected from the upper or the lower contacts that comprise the Y-junction. The generation of this additional phase can be understood from an examination of the pairing Hamiltonian, $H_{pairing}(x)=\\Delta e^{-iq_x x}c_x i\\sigma_y c_x+h.c.$, at the interface between the contact, which annihilates a electron and create a hole with an additional phase of $\\Delta e^{-iq_x x}$. Eventually, when the holes are collected to the central branch of the Y junction, the phase difference between different contacts generates an interference pattern as a function of $\\Delta \\phi\\approx cos\\theta$ and, most importantly, when $|q| L_xcos\\theta=\\pi$, destructive interference occurs and the conductance vanishes.\n\nTo illustrate the qualitative behavior of the Y-junction Andreev interferometer, we use the metallic Hamiltonian in Eq. (\\ref{eq:Hm}) with assumed FF superconducting order. The conductance is obtained from Blonder-Tinkham-Klapwijk theory\\cite{1982BTK} with an assumed interface barrier height that is transparent \\cite{2014Xing}.The outermost line (Green solid line) in Fig. \\ref{fig:exp2}b shows the conductance with no finite momentum in the superconducting system, or $\\vec{q}=0$. The conductance shows a uniform distribution whereas we observe non-uniform conductance oscillation for $Q>0$. The innermost line (Red solid line) in Fig. \\ref{fig:exp2} (b) shows $q_x=\\pi\/L_x$ where the phase difference between two arms is $q_x L_x=\\pi\\cos\\theta$, and the conductance shows a destructive interference at $\\theta=0$ and $ \\pi$. Consequently, the signature of the conductance oscillation in Y-junction is a direct result of spatially varying nature of the order parameter. In addition, the Andreev interferometer is an optimal scheme for our proposal as one can adjust the angle of the finite momentum ($\\theta$) before each transport measurements by applying in-plane magnetic field to adjust the orientation of the magnetic dopants rather than needing to fabricate different devices or multiple Y-junctions.\nHowever, it is important to note that that the minimum momentum shift required to observe a clear destructive interference pattern is either $q=\\pi\/L_x$  or $L_x$ and that this quantity needs to be chosen within the scope of the maximum $Q$ that can be realized by the magnetic doping on the TI surface.\n\n\n\n\\subsection{Four terminal Josephson junction}\n\nIn the previous sections, we analyzed the stability of the FF pairing. In this section, we now propose a Josephson junction transport and compare the transport signatures of the three different pairing scenarios: the conventional BCS phase, the homogeneous FF phase, and the interface FF phase. Fig \\ref{figtrans} (a) shows the schematic figure of the transport configuration which consists of a Josephson junction between the TI-SC heterostructure and the conventional BCS superconductor separated by normal insulator. On the top of the superconductors we attach the four transport terminals. The two terminals are attached on the top of the two different superconductors so that the two junction can have a different phase of the superconducting order parameter by either applying the voltage bias or external current, $I_{J}$. The other two contacts are attached on the top of the BCS superconductor to drive the current in the perpendicular direction($I_{per}$) of the Josephson junction.\n\nAfter establishing the setup of the Josephson junction, we now explain the manner in which current flows in this Josephson junction. Our setup utilizes the mismatch of the order parameter wave function on the interface between the BCS pairing and the FF pairing, this method has been similarly proposed to measure the LO state in the bulk doped inversion symmetric Weyl semi-metal\\cite{PhysRevB.93.214511}. We first consider the weak coupling regime of the junction where the normal insulator is thick enough so that the Josephson current between the BCS and the FF superconductor can be approximated as,\n\\begin{gather}\n\\label{transporteq}\nmax(I_J)\\approx t_{j} \\int d^2 x \\Delta_{top}(x)^* \\Delta_{bottom}(x),\n\\\\\n\\nonumber\nI_J(\\phi)\\approx max(I_J)sin(\\phi).\n\\end{gather}\nwhere $t_j$ is the coupling strength between the junction. $\\Delta_{top}$ and $\\Delta_{bottom}$ is the order parameter wave function of the top and bottom superconductor respectively. The integration indicates the sum over the two dimensional junction region. As it can be seen from Eq. (\\ref{transporteq}), the Josephson current is strongly suppressed when there exists a spatial interference pattern in the inner product of the order parameters of the two superconductor. As a result, the intrinsic spatial oscillations of the FF order parameter(i.e. $\\Delta_{bottom}\\approx |\\Delta| e^{iqx}$) strongly suppress $I_{J}$ when it is coupled to BCS superconductor(i.e. $\\Delta_{top}\\approx |\\Delta|$) in equilibrium. However, when $I_{per}$ is applied to the BCS superconductor, the BCS Cooper pairs possess the finite net momentum, resulting in the form of the order parameter, $\\Delta_{top}=|\\Delta| e^{iq_{per}x}$. The current induced spatial oscillations of the order parameter can cancel the oscillatory component of the FF order parameter in Eq. (\\ref{transporteq}) when $q_{per}=q$, and recover $I_J$. Due to this momentum mismatch between the two superconductors, the Josephson junction between the FF state and the BCS state have a maximum of $max(I_J)$ under the non-zero parallel current, $I_{per}$, while the junction made with the two BCS superconductor always have a maximum in the absence of the parallel current.\n\n\\begin{figure\n\\centering\n\\includegraphics[width=.45\\textwidth]{fig55}\n\\caption{\\label{figtrans}(a) The schematic figure of the Josephson junction setup. On the top of the magnetically doped TI-superconductor junction, the normal insulator barrier is deposited, and the another superconductor is placed on the top of the normal insulator. The four terminal current is placed on the top of the superconductors. The two are attached on the different superconductors to drive the Josephson current. The other two are attached on the top BCS superconductor to drive the current in a direction parallel to the FF momentum. (b) Numerically calculated Josephson current as a function of the transverse momentum $q_{per}$. The blue, the black, and the red lines represent the Josephson current in the BCS pairing, the homogeneous FF pairing and the interface FF pairing respectively. We find that the blue(BCS) line has the maximum located at the $q_{per}=0$ and the black(homogenous FF) line has the maximum located at the $q_{per}=q=0.3$. The red(interface FF) line which has the peak at the $q_{per}=0$ are the interface FF phase with $N_{Layer}=2$.}\n\\end{figure}\n\nWe now illustrate this idea discussed above by numerically calculating the Josephson current. In this calculation, we model the normal insulator barrier using a small coupling strength $t_J$ between the superconductors. We also model $I_{per}$ by adding the finite momentum ,$q_{per}$, in the order parameter of the BCS superconductor. The Josephson current can be calculated from the full energy spectrum of the bound state in the junction by using the following formula\\cite{1966Gennes}\n\\begin{eqnarray}\nJ(\\phi)=\\frac{\\partial E_{ground}(\\phi)}{\\partial \\phi}\n\\end{eqnarray}\nwhere $\\phi$ is the phase difference between the two superconductors. $E_{ground}$ is the ground state energy. By explicitly sweeping $\\phi$ from $0$ to $2\\pi$, we derive the amplitude of the Josephson current as given in Eq. (\\ref{transporteq}).\nFig. \\ref{figtrans} (b) shows the amplitude of the numerically calculated Josephson current as a function of $I_{per}$ in the case of the three different scenarios of the superconducting order parameter. First of all, the blue curve shows the current in the case of the BCS pairing. As explained above, we find that the maximum of the current occurs in equilibrium when $q_{per}=0$ and the addition of the transverse current strongly suppresses the Josephson current as it introduces an additional spatial variation in the order parameter products. Unlike the case of the BCS superconductor, the black lines, which shows the Josephson current in the BCS-FF case, have maximum in the presence of non-zero parallel current which cancel the intrinsic spatial variation of the FF superconducting order parameter. As long as the FF state persists we find that this non-trivial Josephson current serves as an important signature which is distinguished from the conventional BCS pairing. Further, the red lines shows the transport of the interface FF pairing. Unlike the BCS and FF order parameter, we now find a crossover in the location of the maximum current layer increases. In the single layer limit, we find the maximum of the current occur in the same position as FF phase. However, as the $N_{Layer}$ increases more than two, we find that the current patter resembles the BCS phase, since the interface FF has identical order parameter to the BCS order parameter on the top. This shows that the Josephson current is only sensitive to the form of the order parameter near the junction region and the interface FF shows the distinct signature of the FF phase only in the thin superconductor limit.\n\n\n\\subsection{Multi-layered metallic Hamiltonian}\nUsing the metal-TI Hamiltonian we constructed in Eq. (\\ref{eq:HMTI}), we introduce the Hamiltonian that connects two adjacent metallic systems, $\\hat H_{m}=t_mI_2$, where $t_m$ is inter-layer hopping parameter. Then we construct the multi-layer metallic Hamiltonian from the following Hamiltonian construction:\n\\begin{equation}\n\\begin{split}\n\\hat{H}_{3D} =\n\\left(\n\\begin{matrix}\n{\\hat H_{M}} & {\\hat H_m} & 0 & \\cdots & \\cdots & 0 \\\\\n{\\hat H^\\dagger_m} & {\\hat H_M} & {\\hat H_m} & \\cdots & \\cdots & 0 \\\\\n0 & {\\hat H^\\dagger_m} & {\\hat H_M} & \\ddots & & \\vdots \\\\\n\\vdots & \\vdots & \\ddots & \\ddots & {\\hat H_m} & \\vdots \\\\\n\\vdots & \\vdots & & \\hat H^\\dagger_m & {\\hat H_M} & {\\hat H_{couple}} \\\\\n0 & 0 & \\cdots & \\cdots & {\\hat H^\\dagger_{couple}} & {\\hat H_{TI}}\n\\end{matrix}\n\\right).\n\\end{split}\n\\end{equation}\nwhere $\\hat H_{couple}=t_c I_2$ and $\\hat H_M=t_mI_2$\n\n\\subsection{System Hamiltonian}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{schematic}\n\\caption{\\label{schematic}\nThe schematic figure of the magnetically doped topological insulator superconductor hetero-structure. On the top of the topological insulator surface, a thin film of the BCS superconductor is deposited. The magnetization points out the parallel direction to the surface of the topological insulator to shift the location of the Dirac cone.}\n\\end{figure}\n\nIn Fig. \\ref{schematic}, we show the system comprised of a metallic superconductor grown on top of a magnetically doped 3D TI. We choose the NbSe2 as our metallic superconductor as it has been widely used for the TI-superconductor heterostructure due to its lattice matching with the $Bi_2Se_3$. We begin our discussion by writing down the metallic Hamiltonian which describes the parent superconductor of $NbSe_2$: \\cite{2008Inosov,2012Rahn}\n\\begin{equation} \\label{eq:Hm}\n\\hat{H}_{M}(\\vec{k})=t_M k^2-\\mu_M\n\\end{equation}\nwhere  $k=\\sqrt{k_x^2+k_y^2}$ is the magnitude of the in-plane momentum, $t_M$ is the material parameter that determines the slope of the parabolic band. $a$ is the lattice constant, and $\\mu_M$ is the chemical potential. We choose the value of these parameters to be $a=0.344nm$,$t_M=-0.5eV\/a^2$, and $\\mu_M=0.8eV$. $\\mu_M$ is obtained by fitting the tight-binding Hamiltonian\\cite{2008Inosov,2012Rahn} of the 2H-NbSe$_2$ to the quadratic band near the chosen chemical potential.\nWe now consider the surface state Hamiltonian of the magnetically doped 3D TI:\n\\begin{eqnarray} \\label{eq:HTI}\n\\hat{H}_{TI}(\\vec{k})= v_F( k_x\\sigma_y-k_y\\sigma_x)+m\\sigma_x -\\mu_{TI} I_2,\n\\end{eqnarray}\nwhere, without loss of generality, we set $\\hbar=1$.\n$v_F$ is the Fermi velocity of the TI surface state, $m$ is the exchange field Zeeman term, $\\mu_{TI}$ is the chemical potential of the topological insulator, $I_2$ is $2\\times2$ identity matrix, and $\\sigma_i$ is the $i$-th Pauli matrix for spin. The choice of the above parameters is taken from ARPES experiments\\cite{PhysRevB.82.045122} of the surface bands to derive the values of the parameters: $v_F=1.19eV$ and $\\mu_{TI}=0.26$. From Eqs. (\\ref{eq:Hm}) and (\\ref{eq:HTI}), our system is described by the total Hamiltonian written as\n\\begin{eqnarray} \\label{eq:Htot}\nH_{M-TI}=H_{M}+H_{TI}+H_{coupling}.\n\\end{eqnarray}\nIn Eq. (\\ref{eq:Htot}), the metallic Hamiltonian is $H_{M}=\\sum_\\vec{k}\\psi_{M,\\vec{k}}^\\dagger \\hat{H}_M(\\vec{k})\\psi_{M,\\vec{k}}$ where we define the 2 component spinor $\\psi_{M,\\vec{k}}=[d_{\\vec{k}\\uparrow}, d_{\\vec{k}\\downarrow}]^T$, and $d_{\\mathbf{k}\\uparrow}^\\dagger$ ($d_{\\mathbf{k}\\downarrow}$) is up-spin (down-spin) electron creation (annihilation) operator of the metal. Likewise, the TI Hamiltonian is $H_{TI}=\\sum_\\vec{k}\\psi_{TI,\\vec{k}}^\\dagger \\hat{H}_{TI}(\\vec{k})\\psi_{TI,\\vec{k}}$ where we define $\\psi_{TI,\\vec{k}}=[c_{\\mathbf{k}\\uparrow}, c_{\\mathbf{k}\\downarrow}]^T$ where $c_{k\\uparrow}^\\dagger$ ($c_{\\mathbf{k}\\downarrow}$) is up-spin (down-spin) electron creation (annihilation) operator.\nIn Eq. (\\ref{eq:Htot}), we introduce $H_{coupling}$ which couples the TI and the metallic system as\n\\begin{eqnarray} \\label{eq:Htc}\nH_{coupling}=\\sum_{\\mathbf{k},s=\\mathbin\\uparrow\\hspace{-.0em}\\downarrow} t_c( c^\\dagger_{\\mathbf{k}s} d_{\\mathbf{k}s} + d^\\dagger_{\\mathbf{k}s} c_{\\mathbf{k}s} )\n\\end{eqnarray}\nwhere $t_c$ is a coupling constant.\nFrom the Hamiltonian in Eqs. (\\ref{eq:Hm}, \\ref{eq:HTI}, \\ref{eq:Htc}), we construct the matrix form of the metal-TI Hamiltonian, $H_{M-TI}=\\sum_\\vec{k}\\Psi_\\vec{k}^\\dagger \\hat{H}_{M-TI}(\\vec{k}) \\Psi_\\vec{k}$, where\n\\begin{eqnarray} \\label{eq:HMTI}\n\\hat{H}_{M-TI}(\\vec{k})=\n\\begin{pmatrix}\n\\hat{H}_{M}(\\vec{k}) &\\hat{H}_{coupling} \\\\\n\\hat{H}^\\dagger_{coupling} & \\hat{H}_{TI}(\\vec{k})\n\\end{pmatrix},\n\\end{eqnarray}\nwith the operator $\\Psi_\\vec{k}=[\\psi_{TI,\\vec{k}},\\psi_{M,\\vec{k}}]^T$, and the coupling Hamiltonian $\\hat{H}_{coupling}=t_cI_2$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $C$ be a genus $2$ curve over a number field $K$ and let $J$ be its Jacobian. We assume that $C$ is given by an affine equation of the form $y^2 = f(x)$, where $f(x)$ is a monic polynomial of degree $5$ with coefficients in the ring of integers $\\mathcal{O}$ of $K$ and no repeated roots.  Let $\\iota\\colon C\\xhookrightarrow{} J$ be the embedding with respect to the unique point at infinity of $C$, and let $\\Theta$ be the corresponding theta divisor of $J$. Fix a (any) rational prime $p$ and a continuous idele class character $\\chi\\colon \\mathbb{A}_K^{\\times}\/K^{\\times}\\to \\mathbb{Q}_p$. With respect to this data, we explicitly construct, at every non-archimedean place $v$ of $K$, $p$-adic analogues of real-valued N\\'eron functions with respect to $2\\Theta$ (\\S \\ref{subsec:local_Neron}). \n Let \n\\begin{equation*}\n\\lambda_v\\colon J(K_v)\\setminus\\Supp(\\Theta)\\to \\mathbb{Q}_p\n\\end{equation*}\nbe such a $p$-adic N\\'eron function at $v$. We obtain a quadratic form $h_p\\colon J(K)\\to \\mathbb{Q}_p$ by extending quadratically to $J(K)$ the following function on $ J(K)\\setminus\\Supp(\\Theta)$:\n\\begin{equation*}\nh_p(P) =\\frac{1}{[K:\\mathbb{Q}]} \\sum_v n_v\\lambda_v(P), \n\\end{equation*}\nwhere $n_v = [K_v:\\mathbb{Q}_{\\ell}]$ if $v\\mid \\ell$; we call $h_p$ a global $p$-adic height.\n\nTo define $\\lambda_v$, we use the explicit embedding of $J$ into $\\mathbb{P}^8$ due to Grant \\cite{Grant1990} and genus $2$ hyperelliptic division polynomials of Kanayama \\cite{Kanayama, Kanayama_corrections} and Uchida \\cite{Uchida}, a generalisation of the standard division polynomials on elliptic curves (see \\S \\ref{subsec:jac}). \nIf $\\chi$ is unramified at $v$, i.e.\\ the local component $\\chi_v$ is identically zero on $\\mathcal{O}_v^{\\times}$ \\footnote{Note that the places of ramification of $\\chi$ are a subset of the places of $K$ above $p$.}, this is all that is needed for the definition of the N\\'eron function at $v$ (cf.\\ Definition \\ref{def:Neron_fct_away_p}). Moreover, up to a multiplicative constant, this is the same as the real-valued N\\'eron function at $v$, as explicitly constructed by Uchida \\cite{Uchidacanloc}.\n\nAt the places $v$ at which $\\chi$ is ramified, the $p$-adic N\\'eron function $\\lambda_v$ depends on the choice of a subspace $W_v$ of $H^1_{\\dR}(C\/K_v)$ that is complementary to the space of holomorphic forms and isotropic with respect to the cup product pairing. In particular, we attach to $W_v$ a unique $v$-adic analogue $\\sigma_v$ of the complex hyperelliptic sigma function for $C$ (see Section \\ref{sec:sigma_functions}). This extends work of Blakestad. Indeed, suppose that $K_v$ is an unramified extension of $\\mathbb{Q}_p$, that $v$ is of good reduction for $C$, ordinary reduction for $J$ and that $p\\geq 5$. Then the $v$-adic sigma function attached to the unit root subspace of the Frobenius endomorphism of $H^1_{\\dR}(C\/K_v)$ is the ``canonical'' $v$-adic sigma function of \\cite{blakestadsthesis} (see Theorem \\ref{thm:Blakestad_main}, Remark \\ref{rmk:inver_H1} and Proposition \\ref{prop:Blakestad_space_unit_root}). In fact, Blakestad's thesis was the motivation for the work presented in this paper: for elliptic curves, the Mazur--Tate $p$-adic height  \\cite{mazur-tate} can be described explicitly using $v$-adic analogues of the Weierstrass sigma function \\cite{padicsigma,MST} (see \\S \\ref{subsec:ell_curves_analogues}). Our initial goal was to extend this to genus $2$. We will show in forthcoming work with Kaya and M\\\"uller \\cite{BKM22} that our explicit definition of $h_p$ in the canonical case indeed recovers the canonical Mazur--Tate $p$-adic height for $J$.\n\n The complex sigma function is a holomorphic function $\\mathbb{C}^2\\to \\mathbb{C}$, which depends on an analytic isomorphism $J(\\mathbb{C})\\cong \\mathbb{C}^2\/\\Lambda$ (\\S \\ref{subsec:jac}). In general, no uniformisation theorem is at our disposal in the $v$-adic setting; however, the kernel of reduction of $J$ modulo $v$, which we denote by $J_1(K_v)$, is the group associated to a commutative formal group law $F(T_1,T_2)$ of dimension $2$ over $\\mathcal{O}_v$. Upon base-changing to $K_v$, the formal group law $F$ becomes isomorphic to the additive group $\\mathbb{G}_a^2$ (see \\S \\ref{subsec:formalgps_gen},\\ref{subsec:formal}).\n \n Using a suitable isomorphism, we can mimic the complex theory and define the sigma function $\\sigma_v$ as the solution of a  system of differential equations on the formal group\\footnote{This will not be surprising to the reader familiar with $v$-adic sigma functions on elliptic curves (\\S \\ref{subsec:ell_curves_analogues}): see \\cite{Perrin-Riou, bernardi, padicsigma}.}. \nAll this can be made explicit thanks to Grant's description of formal group parameters $T_1$ and $T_2$ \\cite{Grant1990}, and the formal power series $\\sigma_v(T)\\colonequals \\sigma_v(T_1,T_2)$ converges on a finite index subgroup $H_v$ of $J_1(K_v)$, and hence of $J(K_v)$ (Proposition \\ref{prop:finite_index_subgp}).  When it exists, Blakestad's is the unique $v$-adic sigma function that is given as a power series in $T_1$ and $T_2$ with $v$-adically integral coefficients.  \n\nThe connection between the complex sigma function and real-valued N\\'eron functions was proved by Yoshitomi \\cite{Yoshitomi} (see also \\cite{Uchida} for a generalisation to arbitrary genus). By construction, the $v$-adic sigma function $\\sigma_v$ satisfies similar properties to the complex one; therefore, it can be used to define a $p$-adic analogue of a N\\'eron function: for $P\\in J(K_v)\\setminus \\Supp(\\Theta)$, we let\n\\begin{equation*}\nn_v\\lambda_v(P) = -\\frac{2}{ m^2}\\cdot \\chi_{v}\\left(\\frac{\\sigma_v(T(mP))}{\\phi_m(P)}\\right),\n\\end{equation*}\nwhere $m=k[J(K_v):H_v]$ for some $k\\in\\{1,2\\}$ and  $\\phi_m$ is the $m$-th division polynomial (see Definition \\ref{def:Neron_fct_above_p}).\n\n A natural question that arises is how the global height $h_p$ and the $p$-adic N\\'eron functions $\\lambda_v$ compare to well-known global and local $p$-adic height theories that apply to Jacobians of curves in general. We explore this question for the $p$-adic height pairing of Coleman--Gross \\cite{ColemanGross} (and its extension to bad reduction due to Colmez \\cite{Colm96} and Besser \\cite{Besser_pArakelov, BesserPairing}); by \\cite{Besser:CG_and_Nekovar, BesserPairing}, this is equal to the $p$-adic height of Nekov\\'a\\v{r} \\cite{Nekovar} if the curve has semistable reduction at every prime above $p$. \n \n The extended\\footnote{i.e.\\ ``extended to bad reduction''.} Coleman--Gross height for $C$ is defined in a crucially different way from our $h_p$. In particular, the local part of the theory is given by pairings on degree $0$ divisors on $C$ with disjoint support, and it is defined using Coleman--Colmez--Vologodsky integration on $C$ at the places of ramification of $\\chi$ and arithmetic intersection theory at the other places. \n  \n  \nJust for this introduction, we denote by $\\langle D_1,D_2\\rangle_v$ the extended Coleman--Gross local height pairing at $v$, evaluated on the pair of $K_v$-rational degree $0$ divisors $D_1,D_2$ with disjoint support.  As is the case for $h_p$, the local pairings $\\langle D_1,D_2\\rangle_v$ at the primes $v$ of ramification for $\\chi$ depend on a choice of isotropic subspace of $H_{\\dR}^1(C\/K_v)$, complementary to $H^0(C\/K_v,\\Omega^1)$.\n \n We prove that, provided that we make the same choices of subspaces at the primes of ramification, the global height $h_p$ is the same as the extended Coleman--Gross height for $C$ (Corollary \\ref{cor:global_CG_same_as_this}) and that the local Coleman--Gross height pairing at any $v$ is equal to a suitable linear combination of pullbacks of $\\lambda_v$ to $C$: see Corollary \\ref{cor:lambda_eq_CG} for a precise statement. For the reader's convenience, we state here a simplified version of the local and global comparison:\n \\begin{thm}\\label{thm:intro_1}\nLet $P_1,P_2,Q_1,Q_2\\in C(K_v)$ such that $Q_i\\neq P_j$ for all $i,j\\in\\{1,2\\}$. Then there exists a finite set of points $S\\subset C(K_v)$, such that for all $R\\in C(K_v)\\setminus S$, we have:\n \\begin{equation*}\n\\langle P_1-P_2,Q_1-Q_2\\rangle_v = -\\frac{n_v }{2}\\sum_{1\\leq i,j\\leq 2} (-1)^{i+j}\\lambda_v(\\iota(Q_i)  -\\iota(P_j)-\\iota(R)). \n \\end{equation*}\n Moreover,\n \\begin{equation*}\n \\sum_{v}\\langle P_1 - P_2, Q_1 - Q_2 \\rangle_v =\\frac{1}{2}( h_p([P_1+Q_1-P_2-Q_2])-h_p([P_1-P_2])-h_p([Q_1-Q_2])).\n \\end{equation*}\n \\end{thm}\n \\begin{rmk}\n This is essentially a $p$-adic analogue of the real-valued theorem of Faltings \\cite{faltings_calculus_arithm_surfaces} and Hriljac \\cite{hriljac} (specialised to the setting of a genus $2$ odd degree hyperelliptic curve).\n \\end{rmk}\n In order to prove these comparison results, we take a detour into Colmez's theory of $p$-adic integration on curves and symmetric $p$-adic Green functions of theta divisors on their Jacobians \\cite{Colm96}, which we apply to $C$ and to the divisor $\\Theta$ on $J$. The link to the above problem is the following. In the original paper of Coleman--Gross \\cite{ColemanGross}, at a place $v$ of ramification for $\\chi$, the local height paring $\\langle D_1,D_2\\rangle_v$ is given by\n \\begin{equation*}\n\\langle D_1,D_2\\rangle_v =  t_v\\biggl(\\int_{D_2}\\omega_{D_1}\\biggr),\n \\end{equation*}\n where the integral is a $K_v$-valued (Coleman \\cite{Col82, coleman, ColemandeShalit}) integral of a differential on $C$, and $t_v\\colon K_v\\to \\mathbb{Q}_p$ is a trace map. \nThe trace map is uniquely determined by $\\chi$, and so is the branch of the $p$-adic logarithm that the integral depends on (for certain choices of divisors).\n\nThe dependency of $v$-adic integration on the branch of the $p$-adic logarithm is resolved in Colmez's theory (which, unlike Coleman's, makes no assumption on the reduction) in a different manner: by viewing the value $\\mathop{\\mathrm{Log}}\\nolimits{p}$ of the logarithm at $p$ as a variable. In fact, the Colmez integral of $\\omega_{D_1}$ takes values in $\\mathscr{L}(K_v)\\colonequals K_v\\oplus \t\\mathbb{Q}\\mathop{\\mathrm{Log}}\\nolimits{p}$.  With such a convention, Colmez gives a unified theory of (extended) Coleman--Gross local height pairings: in terms of his integration theory, for any $v$ (including the unramified primes), we have\n \\begin{equation}\\label{eq:D1D2v_colmez}\n\\langle D_1,D_2\\rangle_v = T_v\\biggl(\\int_{D_2}\\omega_{D_1}\\biggr),\n \\end{equation}\n for some $\\mathbb{Q}$-linear map $T_v\\colon \\mathscr{L}(K_v)\\to \\mathbb{Q}_p$ uniquely determined by $\\chi$. At an unramified prime $v\\mid q$, this trace map $T_v$ satisfies $T_v(a+b\\mathop{\\mathrm{Log}}\\nolimits{q}) = b\\chi_v(q)$. In other words, the intersection multiplicity appearing in Coleman--Gross's definition of $\\langle D_1,D_2\\rangle_v$ at such a prime is replaced by the coefficient of $\\mathop{\\mathrm{Log}}\\nolimits{q}$ of a $v$-adic integral, constructed analogously to the one appearing at the primes of ramification for $\\chi$.  \n \nMoreover, Colmez constructs $v$-adic analogues of Green functions on abelian varieties. He shows that the integral appearing in \\eqref{eq:D1D2v_colmez} can be expressed in terms of pullbacks to $C$ of a suitable symmetric\\footnote{In our case, ``symmetric'' is a synonym of ``even'' (cf.\\ \\S\\ref{subsec:comparison})} $v$-adic Green function $G_{\\Theta}$ associated to the divisor $\\Theta$ on $J$. The choice of $G_{\\Theta}$ at a ramified prime $v$ depends on $W_v$. We prove (see Theorem \\ref{thm:comparison} for a more precise statement): \n \\begin{thm}\nThere exists a symmetric $v$-adic Green function $G_{\\Theta}$ of divisor $\\Theta$ such that\n \\begin{equation*}\n n_v\\lambda_{v} = -2T_{v}(G_{\\Theta}).\n \\end{equation*}\n Moreover, if $\\chi$ is ramified at $v$ and $\\lambda_v$ corresponds to the choice of $W_v\\subset H^1_{\\dR}(C\/K_v)$, so does $G_{\\Theta}$.\n\\end{thm} \n  From this, Theorem \\ref{thm:intro_1} and the more general Corollaries \\ref{cor:lambda_eq_CG} and \\ref{cor:global_CG_same_as_this} follow easily. \n\nWe now turn to the computational aspects of the paper (Section \\ref{sec:implementation}). Our \\texttt{SageMath} \\cite{sage} implementation is available at \\cite{github_padic_g2}. In recent years, algorithms for $p$-adic heights have attracted considerable interest due to their crucial role in explicit versions of the Chabauty--Kim method \\cite{KimP1, Kimunipotent} for determining rational points on curves: see especially the groundbreaking \\cite{BDQCI} and \\cite{SplitCartan} on quadratic Chabauty. We therefore believe that an implementation of the local $p$-adic N\\'eron functions could have some applications beyond the ones that we already present in this paper.\n\nGiven a subspace $W_v$, computing the $v$-adic sigma function associated to it requires working explicitly with the formal group law of $J$. In particular, one needs to compute expansions of functions in the formal group parameters and a formal group isomorphism of the base-change of $J_1$ to $K$ (or $K_v$) with the formal additive group of dimension $2$. All this can be achieved essentially by implementing the explicit results of \\cite{Grant1990}.\n\nFor instance, we can compute a ``naive'' $v$-adic sigma function (\\S \\ref{subsec:naive_sigma}), a genus $2$ analogue of Bernardi's sigma function for elliptic curves \\cite{bernardi}. This sigma function is quite useful for computational purposes, because its coefficients as a series in $T_1,T_2$ belong to $K$ and are independent of the prime $v$, which allows us to consider large primes. \n\nHowever, if $J$ has good ordinary reduction at all places $v$ of ramification of $\\chi$, there are instances where one might want to instead compute Blakestad's canonical $v$-adic sigma function. \nIndeed, as mentioned above, we prove that Blakestad's $v$-adic sigma function corresponds to the unit root subspace of Frobenius, and by the comparison result with the extended Coleman--Gross height (Corollary \\ref{cor:global_CG_same_as_this}), the global height $h_p$ with respect to this choice is related to a $p$-adic analogue of the Birch and Swinnerton-Dyer conjecture \\cite{BaMuSt12}. In addition, in light of its integrality, Blakestad's $v$-adic sigma function has better convergence properties than any other and this might make it a preferable choice in various computational situations. In \\S \\ref{subsec:implementation_canonical_sigma} we discuss how we can compute it when $K_v\\cong \\mathbb{Q}_p$ using Kedlaya's algorithm \\cite{kedlaya}.\n\nThe final ingredients for the computation of $p$-adic N\\'eron functions are division polynomials, continuous idele class characters and the index $[J(K_v):J_1(K_v)]$. For division polynomials, we improve an implementation of de Jong--M\\\"uller \\cite{muller_de_jong} (\\S \\ref{subsec:division_polynomials}); for the characters we could use \\cite{QCnfs} to address the general case (but have currently restricted the implementation to the cyclotomic character), and for the index we use work of Bruin--Stoll \\cite{Bruin-Stoll:MWSieve} (see \\S \\ref{subsec:implementation_Neron_fcts} for details). \n\nTo illustrate our implementation, we present two examples. In \\S \\ref{subsec:eg_large_p}, we consider the curve $C\\colon y^2 = x^5-1$ over $\\mathbb{Q}$  and we compute the canonical cyclotomic $p$-adic height of a non-torsion point in $J(\\mathbb{Q})$, for the good ordinary prime $p= 10^6 + 81$. What allows us to consider such a large prime and yet compute the canonical $p$-adic height (to $p$-adic precision 20, in approximately 15 seconds) \n is that, in this case, we can find the unit root subspace without appealing to Kedlaya's algorithm, as a consequence of the fact that $C$ has extra automorphisms over $\\mathbb{Q}(\\zeta_5)$.\n\nIn \\S \\ref{subsec:example_hts_Neron_fcts}, we consider a curve for which some computations of canonical Coleman--Gross heights were carried out in \\cite{BBM0}. We use our comparison results to replicate these computations using $p$-adic N\\'eron functions in place of Coleman integrals and intersection multiplicities. \n\nThe two examples that we have chosen to present are of curves over $\\mathbb{Q}$ that have good ordinary reduction at $p$. The good and ordinary assumptions can both (or either one) be removed in our implementation if we replace the unit root subspace of Frobenius with any explicitly given isotropic complementary subspace. In forthcoming work with Kaya and M\\\"uller \\cite{BKM22} we address the problem of computing a canonical sigma function also in the semistable ordinary case. Finally, the assumption that the curve is defined over $\\mathbb{Q}$ is also not essential for the algorithms (however, if our choices require computing unit root eigenspaces of Frobenius using Kedlaya's algorithm, we need to assume that $K_v\\cong \\mathbb{Q}_p$ at every prime $v$ of ramification of $\\chi$, due to current limitations in the \\texttt{SageMath} \\cite{sage} implementation). \n\nCompared to other explicit approaches to $p$-adic heights, such as the algorithms of \\cite{Bes-Bal10, BaMuSt12} for Coleman--Gross heights, ours has the clear disadvantage that it assumes that $C$ is of genus $2$. On the other hand, we believe there are features that might make our approach preferable in the genus $2$ case. For example, its essential insensitivity to the reduction type at the primes $v\\mid p$. \nThis is inherited from the N\\'eron functions definition. Namely, the N\\'eron function at $v$ is first defined on the model-dependent kernel of reduction $J_1(K_v)$, and then extended to $J(K_v)$ by combining the facts that $[J(K_v):J_1(K_v)]$ is finite and that $\\lambda_v$ is ``almost'' a quadratic function (Proposition \\ref{prop:properties_neron_fcts}).\n\n By Theorem \\ref{thm:intro_1} (and, more generally, Corollary \\ref{cor:lambda_eq_CG}), we can then compute local Coleman--Gross heights for arbitrary reduction. In \\S \\ref{subsec:diff_first_2nd_third} we explain how, by replacing quasi-quadraticity with a suitable transformation property under multiplication on $J$, we may also express integrals of differentials of the first and second kind on $J$ (and hence on $C$, by pullback) in terms of integrals on $J_1(K_v)$. In summary, the same or similar techniques to the ones for heights can be applied to compute integrals of differentials of the first, second and third kind on $C$, without any assumptions on the reduction. \n\nThe final section (Section \\ref{sec:application_bihyper}) is devoted to an elementary quadratic Chabauty-type criterion for the rational points on certain genus $4$ hyperelliptic curves that admit degree two maps to two genus $2$ curves (as $C$), each with a rank $2$ Jacobian. See Proposition \\ref{prop:QC_bihyper}.\n\n\\subsection{Elliptic curve analogues}\\label{subsec:ell_curves_analogues}\nEssentially all of the results of this paper admit an elliptic curve analogue, which either already appears in the literature or could be deduced in a similar way to the genus $2$ case.  The construction of the genus $2$ naive sigma function and the analysis of its convergence properties that we present in \\S \\ref{subsec:naive_sigma} are based on the work of Bernardi for elliptic curves \\cite{bernardi}; Blakestad's sigma function (\\S \\ref{subsec:infty_sigma}) is an analogue of the canonical sigma function of Mazur--Tate \\cite{padicsigma}. That the elliptic curve canonical sigma function for $E\/K_v$ is related to the unit root eigenspace of $H^1_{\\dR}(E\/K_v)$ follows from \\cite[Proposition 3]{padicsigma} and \\cite{Katz, Katzinterpolation}\nand is used in the algorithms of Mazur--Stein--Tate \\cite[\\S 3.2]{MST} and Harvey \\cite[Sections 3,4]{harvey}. \n\n$p$-Adic heights on elliptic curves constructed using $p$-adic sigma functions are again subject of \\cite{bernardi, padicsigma, MST,harvey} and they are $p$-adic heights in the sense of \\cite{mazur-tate}. See also \\cite{Perrin-Riou-comptes, Perrin-Riou, wuthrichheights}. Typically, such sigma functions are used to  define a global $p$-adic height. The point of view of decomposing this as a sum of local $p$-adic N\\'eron functions with respect to the divisor given by twice the point at infinity is often not pursued. Perhaps this is because the natural domain of a $v$-adic sigma function (for $v\\mid p$) is a subgroup of the formal group. Therefore, if we are only interested in values of the global $p$-adic height, it is more convenient to extend the latter using quadraticity, rather than extending each local N\\'eron function using formulae involving division polynomials. \n\nHowever, the point of view of $p$-adic N\\'eron functions is natural because it is a direct analogue of the real-valued setting (for which, see for instance \\cite[Chapter VI]{silvermanadvancedtopics}), and it is convenient in the context of the Chabauty--Kim method. It is thus taken in \\cite{nonabelianconjecture} at the primes away from $p$, and in \\cite[\\S\\S 2.A, 4.A]{Bianchi20} at all primes. \n\nDirect analogues of the local comparison results of \\S \\ref{subsec:comparison} are,  to our knowledge, not explicitly stated in the literature, but they could be derived in an analogous manner. Some remarks are however in order. A relation between Colmez Green functions and the work of Mazur--Tate \\cite{padicsigma} is hinted by Colmez himself before \\cite[Proposition II.2.20]{Colm96}. The equality of the global Coleman--Gross and Mazur--Tate height is known, at least in the good ordinary reduction case \\cite{Coleman:universal, Besser:CG_and_Nekovar} (see also the introduction to \\cite{ColemanGrosspAdicSigma}). A different kind of local comparison in the good ordinary case is provided by \\cite[Corollary 4.2]{ColemanGrosspAdicSigma}.\n\n\\subsection{Notation} $\\mathbb{N}$ includes $0$. Given a number field $K$ and a non-archimedean place $v$, the notation $\\mathop{\\mathrm{ord}}\\nolimits_v$ is used for the $v$-adic valuation on $K_v^{\\times}$, normalised so that $\\mathop{\\mathrm{ord}}\\nolimits_v(K_v^{\\times}) = \\mathbb{Z}$. The $v$-adic absolute value $|\\cdot |_v$ on $K_v$ extends the standard $\\ell$-adic absolute value, if $v$ lies above $\\ell$.  Various ``logarithms'' appear in the paper. In order to help the reader overcome potential confusion, we compiled a list here:\n\\begin{itemize}\n\\item We use the symbol $\\log$ in logarithmic derivatives. That is, if $f$ is a power series over some field and $D$ is a differential operator, by $D(\\log(f))$ we mean $\\frac{Df}{f}$. We also use $\\log$ for the real logarithm.\n\\item $\\mathop{\\mathrm{Log}}\\nolimits$ and $\\mathop{\\mathrm{Log}}\\nolimits_v$ are logarithms in the sense of Colmez (Section \\ref{sec:Colmez}, Equation \\eqref{eq:Log_colmez} and \\S\\ref{subsubsec:notation_colm_heights}). \n\\item $\\log_p$ is the branch of the $p$-adic logarithm vanishing at $p$ (\\S\\ref{subsec:implementation_Neron_fcts}).\n\\item $\\mathcal{L} = (\\mathcal{L}_1,\\dots, \\mathcal{L}_n)$ is a formal group logarithm and, usually, the strict one (Definition \\ref{def:strict_log}).\n\\item A \\emph{one-logarithm} of an abelian variety $X$ over a $p$-adic field $K$ is a locally analytic group homomorphism $X(K)\\to K$ (\\S \\ref{subsec:Colm_int}).\n\\item Given a smooth, projective, geometrically irreducible curve $X$ over a $p$-adic field $K$ with Jacobian $J$, the logarithm $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}$ is the canonical homomorphism $T(K)\/T_{\\ell}(K)\\to H^1_{\\dR}(X)$ from differentials of the third kind on $X$ modulo logarithmic differentials to the first de Rham cohomology of $X$ (Definition \\ref{def:LogJtilde}, Remark \\ref{rmk:logJtilde}).\n\\end{itemize}\n\n\\subsection*{Acknowledgements}\nFirst and foremost, I am extremely grateful to Steffen M\\\"uller for suggesting that Blakestad's work on $p$-adic sigma functions could be used in the context of heights, as well as for helpful discussions and generous feedback at various stages of this project. I would also like to thank Clifford Blakestad for sharing insights about his work,  Stevan Gajovi\\'c for coming up with Lemma \\ref{lemma:stevan} and for helpful comments, Enis Kaya for useful feedback on an earlier draft and for inspiring conversations on $p$-adic heights, and Remco Wouts for assistance with the computations. This work was supported by an NWO Vidi grant.\n\n\\section{Preliminaries}\n\\subsection{Formal group laws}\\label{subsec:formalgps_gen}\nIn this subsection we recall some definitions and properties of formal group laws. We are mainly interested in an explicit description of the isomorphisms of certain formal group laws with the formal additive group of a suitable dimension: see Theorem \\ref{thm:log_iso_Ga}, Proposition \\ref{prop:exp_exp_log} and Lemma \\ref{lemma:log_and_inv_derivations} below. The main references are \\cite{Freije, zink, cassels_flynn}. In particular, up to and including Definition \\ref{def:strict_log} we follow \\cite{Freije} closely; Proposition \\ref{prop:exp_exp_log} is then a stronger version of results in \\cite{cassels_flynn}; finally the discussion on invariant derivations is based on \\cite{zink}.\n\nLet $R$ be a commutative ring with identity, let $n\\in \\mathbb{Z}_{\\geq 1}$, let $X = (x_1,\\dots, x_n)$ and let $R[[X]]$ be the power series ring in $x_1,\\dots,x_n$ over $R$. \n\n\\begin{mydef}\nA \\emph{formal group law} $F(X,Y)$ of dimension $n$ over $R$ is an $n$-tuple of power series $F_i(X,Y)\\in R[[X,Y]]$ such that\n\\begin{enumerate}[label = (\\roman*)]\n\\item\\label{it:formal_gp_1} $F(X,Y) = X+Y + O(X,Y)^2$, where $O(X,Y)^2$ means terms of total degree at least $2$;\n\\item\\label{it:formal_gp_2} $F(X,F(Y,Z)) = F(F(X,Y),Z)$. \n\\end{enumerate}\n\\end{mydef}\nThe formal group law $F(X,Y)$ is said to be \\emph{commutative} if $F(Y,X) = F(X,Y)$. \n\nIt follows from the formal group axioms that there exists a unique \\emph{formal inverse}, that is an $n$-tuple of power series $i(X)\\in R[[X]]^n$ such that $F(X,i(X)) = F(i(X),X) = 0$ and $i(X) =  -X +O(X)^2$. Moreover, $F(X,0) = F(0,X) = X$.\n\n\\begin{mydef}\nGiven formal group laws $F(X,Y)$ and $G(X,Y)$ of dimensions $n$ and $m$ over $R$, a \\emph{formal group homomorphism} $\\alpha\\colon F\\to G$ is an $m$-tuple of power series $\\alpha_i(X)\\in XR[[X]]$ such that $\\alpha(F(X,Y)) = G(\\alpha(X),\\alpha(Y))$. The homomorphism $\\alpha$ is an \\emph{isomorphism} if there exists a formal group homomorphism $\\beta\\colon G\\to F$ such that $\\alpha(\\beta(Y)) = Y$ and $\\beta(\\alpha(X)) = X$.\n\\end{mydef} \nAn example of a commutative formal group law of dimension $n$ is the additive group $\\mathbb{G}_a^{n}(X,Y) = X+Y$. If $R$ is a $\\mathbb{Q}$-algebra, any commutative formal group law of dimension $n$  over $R$ is isomorphic to $\\mathbb{G}_a^{n}$:  see Theorem \\ref{thm:log_iso_Ga} below. \n\nIn order to describe the isomorphism explicitly, we first recall the notion of invariant differentials on a commutative formal group law $F(X,Y)$ of dimension $n$ over an arbitrary commutative ring $R$ with identity.\n\nWe define the $R[[X]]$-module of \\emph{differential $1$-forms} as $\\Omega = \\sum_{i=1}^n R[[X]] dx_i$, and a corresponding total derivative map $d\\colon R[[X]]\\to \\Omega$, $d(f) = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i} dx_i$. A differential in $\\Omega$ is \\emph{exact} if it is in the image of $d$. Given $\\omega = \\sum_{i=1}^n \\varphi_i(X) dx_i \\in \\Omega$, with $\\varphi_i(X)\\in R[[X]]$, we say that $\\omega$  is \\emph{(translation) invariant} if\n\\begin{equation*}\n\\omega(F(X,T)) \\colonequals \\sum_{i=1}^n \\varphi_i(F(X,T)) dF_i(X,T)  = \\omega;\n\\end{equation*}\nhere in $dF_i(X,T)$ we consider $T$ fixed and take partial derivatives with respect to $X$.\n\nThe definition of the formal group law implies that the matrix $\\left(\\frac{\\partial F_i}{\\partial x_j}(0,X)\\right) $ is invertible over $R[[X]]$ and we have: \n\\begin{lemma}\\label{lemma:inv_diff}\nThe differential $\\omega = \\sum_{i=1}^n \\varphi_i(X)dx_i\\in\\Omega$ is invariant if and only if \n\\begin{equation*}\n\\omega = (a_1,\\dots,a_n)\\left(\\frac{\\partial F_i}{\\partial x_j}(0,X)\\right)^{-1}\\left(\\begin{matrix}\n           dx_{1} \\\\\n           \\vdots \\\\\n           dx_{n}\n         \\end{matrix}\\right)\n\\end{equation*}\nfor some $(a_1,\\dots, a_n)\\in R^n$. In particular, the invariant differentials of $F(X,Y)$ form an $R$-module of rank $n$. \n\\end{lemma}\n\n\\begin{proof}\nSee for example \\cite[p.\\,243]{Freije}.\n\\end{proof}\n\n\\begin{lemma}[{\\hspace{1sp}\\cite[Proposition 1.3, Lemma 1.4]{honda}}]\n\\label{lemma:honda}\nIf $R$ is a $\\mathbb{Q}$-algebra and $F$ is a commutative formal group law over $R$, then every invariant differential is exact. \n\\end{lemma}\n\nFrom these two lemmas, one deduces\n\\begin{thm}\\label{thm:log_iso_Ga}\nLet $F$ be a commutative formal group law over a $\\mathbb{Q}$-algebra $R$ and let $\\omega_1,\\dots,\\omega_n$ be an $R$-basis for the invariant differentials of $F$. Let $\\mathcal{L}_i(X)\\in R[[X]]$ be the unique power series such that\n\\begin{equation*}\nd\\mathcal{L}_i = \\omega_i,\\qquad \\mathcal{L}_i(0) = 0.\n\\end{equation*}\nThen $\\mathcal{L} = (\\mathcal{L}_1,\\dots, \\mathcal{L}_n)\\colon F\\to \\mathbb{G}_a^n$ is a formal group isomorphism. Conversely, any isomorphism $F\\to \\mathbb{G}_a^{n}$ is of this form.\n\\end{thm}\n\\begin{proof}\nFor the first statement see for example \\cite[Theorem 1]{Freije}. For the second statement, if $\\alpha = (\\alpha_1,\\dots,\\alpha_n)\\colon F\\to \\mathbb{G}_a^n$ is a homomorphism, then $\\alpha_i(F(X,Y)) = \\alpha_i(X) + \\alpha_i(Y)$, thus $d\\alpha_i$ is an invariant differential.  By \\cite[Lemma 1.4]{zink}, $\\alpha$ is an isomorphism if and only if the Jacobian matrix $ \\big(\\frac{\\partial \\alpha_i}{\\partial x_j}(0)\\big)$ is invertible over $R$. By Lemma \\ref{lemma:inv_diff} this happens if and only if $d\\alpha_1,\\dots,d\\alpha_n$ is a basis for the $R$-module of invariant differentials.\n\\end{proof}\n\n\\begin{mydef}\\label{def:strict_log}\nThe isomorphism of Theorem \\ref{thm:log_iso_Ga} is called a \\emph{logarithm} of $F$. The \\emph{strict logarithm} is the logarithm corresponding to the basis $\\omega_1,\\dots,\\omega_n$ where, for $k\\in\\{1,\\dots,n\\}$,\n\\begin{equation*}\n\\omega_k = (\\delta_{ik})\\left(\\frac{\\partial F_i}{\\partial x_j}(0,X)\\right)^{-1}\\left(\\begin{matrix}\n           dx_{1} \\\\\n           \\vdots \\\\\n           dx_{n}\n         \\end{matrix}\\right).\n\\end{equation*}\nHere $\\delta_{ik}$ is the Kronecker delta. \nThe inverse of a formal group logarithm is called a \\emph{formal group exponential}; the inverse of the strict logarithm is the \\emph{strict exponential}.\n\\end{mydef}\nBy Theorem \\ref{thm:log_iso_Ga}, if $R$ is an integral domain of characteristic $0$ and $F$ is a commutative formal group law over $R$, we can consider an isomorphism to the additive group of a suitable dimension upon base-changing to the fraction field of $R$. For our intended applications, we would like to understand what denominators can occur in the series expansions of the strict formal logarithm and exponential.\n\n\n\\begin{prop}\\label{prop:exp_exp_log}\nLet $R$ be a GCD domain of characteristic $0$ and let $K$ be its field of fractions. Let $F$ be a commutative formal group law over $R$ and let $F_K$ be its base-change to $K$. Then the strict logarithm $\\mathcal{L} = (\\mathcal{L}_1,\\dots, \\mathcal{L}_n)$ and strict exponential $\\mathcal{E} = (\\mathcal{E}_1,\\dots,\\mathcal{E}_n)$ of $F_K$ are of the form: \n\\begin{alignat*}{2}\n\\mathcal{L}_i &= x_i + \\sum_{\\substack{j_1,\\dots, j_n\\in \\mathbb{N}\\\\\nj_1+\\cdots + j_n \\geq 2}} \\frac{a_{j_1,\\dots, j_n}}{\\gcd(j_1,\\dots, j_n)} x_1^{j_1}\\cdots x_n^{j_n}, \\qquad &&\\text{where } a_{j_1,\\dots, j_n}\\in R;\\\\\n\\mathcal{E}_i &=x_i + \\sum_{\\substack{j_1,\\dots, j_n\\in \\mathbb{N}\\\\\nj_1+\\cdots + j_n \\geq 2}} \\frac{b_{j_1,\\dots, j_n}}{j_1!\\cdots j_n!} x_1^{j_1}\\cdots x_n^{j_n}, \\qquad &&\\text{where } b_{j_1,\\dots, j_n}\\in R.\n\\end{alignat*}\n\\end{prop}\n\n\\begin{proof}\nWrite $\\mathcal{L}_i = \\sum c_{j_1,\\dots, j_n} x_1^{j_1}\\cdots x_n^{j_n}$ for $c_{j_1,\\dots, j_n}\\in K$. For every $k\\in \\{1,\\dots, n\\}$, we have\n$\\frac{\\partial \\mathcal{L}_i}{\\partial x_k}\\in R[[X]]$, so\n\\begin{equation*}\nj_k c_{j_1,\\dots, j_n} \\in R.\n\\end{equation*} \nIf $c_{j_1,\\dots,j_n}$ is non-zero, writing $c_{j_1,\\dots, j_n} = \\frac{\\alpha}{\\beta}$ for coprime $\\alpha,\\beta\\in R$, we deduce that $\\beta\\mid j_k$ for every $k$ (by Euclid's lemma for GCD domains) and so $\\beta\\mid \\gcd(j_1,\\dots,j_n)$.  The claim on the leading term follows from the fact that $\\mathcal{L}$ is the strict logarithm. \nFor the coefficients of $\\mathcal{E}_i$, see \\cite[p.\\,68]{cassels_flynn}.\n\\end{proof}\nLet $F$ be a commutative formal group law over a $\\mathbb{Q}$-algebra $R$. Theorem \\ref{thm:log_iso_Ga} characterises any formal group isomorphism $\\mathcal{L}\\colon F\\to \\mathbb{G}_a^n$ by the choice of a basis for the $R$-module of invariant differentials of $F$.  We will now see that the inverse of $\\mathcal{L}$ admits an explicit description in terms of a suitable choice of basis of a related $R$-module: that of invariant derivations of $F$. \nA \\emph{derivation} is an $R$-linear map $D\\colon R[[X]]\\to R[[X]]$ such that $D(fg) = gDf + fDg$, and a derivation $D$ is defined to be \\emph{invariant} for $F$ if\n\\begin{equation*}\nD(f(F(X,T))) = (Df)(F(X,T))\\qquad \\text{for all } f \\in R[[X]].\n\\end{equation*}\n It follows from the definition that any derivation $D$ satisfies\n\\begin{equation}\\label{eq:formula_D}\nD = \\sum_{i=1}^n Dx_i \\cdot \\frac{\\partial}{\\partial x_i};\n\\end{equation}\nthus $D$ is invariant if and only if, for every $j\\in \\{1,\\dots,n\\}$, we have\n\\begin{equation*}\n\\sum_{i=1}^n Dx_i\\cdot \\frac{\\partial F_j(X,T)}{\\partial x_i} = (Dx_j)(F(X,T)).\n\\end{equation*}\nBy \\cite[Theorem 1.15]{zink}, the invariant derivations form an $R$-module of rank $n$. Moreover, denoting by $\\Der R[[X]]^{\\textup{inv}}$ the $R$-module of invariant derivations, and by $\\Omega^{\\textup{inv}}$ the $R$-module of invariant differentials, by \\cite[Theorem 1.19]{zink} there is a perfect pairing\n\\begin{equation}\\label{eq:dual_inv_diff_der}\n(\\cdot, \\cdot)\\colon \\Der R[[X]]^{\\textup{inv}}\\times \\Omega^{\\textup{inv}}\\to R, \\qquad (D,gdf) = gDf. \n\\end{equation}\nIn particular, if $\\mathcal{L}$ is the strict logarithm, the dual to $d\\mathcal{L}_i$ is\n\\begin{equation}\\label{eq:Di}\nD_i = \\sum_{j=1}^n \\frac{\\partial F_j}{\\partial x_i}(0,X)\\frac{\\partial}{\\partial x_j}.\n\\end{equation}\n\\begin{lemma}\n\\label{lemma:log_and_inv_derivations}\nLet $R$ be a $\\mathbb{Q}$-algebra, $F$ a commutative formal group law over $R$. Let $\\mathcal{L}=(\\mathcal{L}_1,\\dots, \\mathcal{L}_n)$ be a formal logarithm $F\\to \\mathbb{G}_a^n$, with inverse $\\mathcal{E}\\colon \\mathbb{G}_a^n\\to F$. Suppose $\\mathcal{L}$ corresponds to the choice of basis for $\\Omega^{\\textup{inv}}$ dual to the basis $(D_1,\\dots,D_n)$ for $\\Der R[[X]]^{\\textup{inv}}$. Then\n\\begin{equation*}\n\\frac{\\partial f(\\mathcal{E}(T))}{\\partial t_i} = (D_i f)(\\mathcal{E}(T))\\qquad \\text{for all } f\\in R[[X]].\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nLet $\\gamma^{(i)}(t_i) = \\mathcal{E}(T)|_{t_j = 0, j\\neq i}$. Then $\\gamma^{(i)}$ defines a formal group homomorphism $\\mathbb{G}_a\\to F$. Therefore, by \\cite[Theorem 1.26]{zink}, the following formula defines an invariant derivation\n\\begin{equation}\\label{eq:Ditilde_formula}\n(\\tilde{D}_i x_j)(X)= \\frac{\\partial F_j(X,\\gamma^{(i)}(t_i))}{\\partial t_i}\\bigg\\vert_{t_i =0}= \\sum_{\\ell=1}^n\\frac{\\partial F_j}{\\partial y_{\\ell}}(X,0) \\frac{\\partial \\mathcal{E}_{\\ell}}{\\partial t_i}(T)\\bigg\\vert_{T = 0} = \\frac{\\partial \\mathcal{E}_j}{\\partial x_i}(X)\\bigg\\vert_{X = \\mathcal{L}(X)}\n\\end{equation}\n(the last equality follows from differentiating $\\mathcal{E}(X+Y) = F(\\mathcal{E}(X),\\mathcal{E}(Y))$ with respect to $Y$ and evaluating at $Y=0$ and $X=\\mathcal{L}(X)$). Therefore, for every $k\\in \\{1,\\dots,n\\}$,\n\\begin{equation*}\n(\\tilde{D}_i, \\omega_k) = \\sum_{j=1}^n \\frac{\\partial \\mathcal{L}_k}{\\partial x_j}\\tilde{D}_i x_j=\\frac{\\partial \\mathcal{L}_k(\\mathcal{E}(X))}{\\partial x_i}\\bigg\\vert_{ X = \\mathcal{L}(X)} = \\delta_{ik},\n\\end{equation*}\nsince $\\mathcal{L}(\\mathcal{E}(X)) = \\mathcal{E}(\\mathcal{L}(X))= X$. This shows that $\\tilde{D}_i = D_i$. The lemma follows from \\eqref{eq:Ditilde_formula} and \\eqref{eq:formula_D}. \n\\end{proof}\n\\begin{rmk}\nWith the notation of Lemma \\ref{lemma:log_and_inv_derivations} and its proof, we have, in particular, that $\\gamma^{(i)}(t_i)$ is the integral curve of $D_i$ (cf. \\cite[Theorems 1.23, 1.26]{zink}).\n\\end{rmk}\n\n\\subsection{Jacobians of genus 2 curves: algebraically, analytically and projectively}\\label{subsec:jac}\nLet $C$ be a hyperelliptic curve of genus $2$ over a field $K$ of characteristic different from $2$, described explicitly by the closure in the weighted projective space $\\mathbb{P}_{1,3,1}$ of a smooth affine curve\n\\begin{equation}\n\\label{eq:Grant}\ny^2 = x^5 + b_1x^4 + b_2x^3 + b_3 x^2 + b_4 x + b_5, \\qquad b_i\\in K.\n\\end{equation}\nIf $K$ is either $\\mathbb{C}$ or an algebraically closed field, any smooth, projective, geometrically irreducible curve of genus $2$ is birationally equivalent over $K$ to one in this form; for other fields $K$, \nthis is not the case; see for example \\cite[Chapter 1]{cassels_flynn}.\n\nThere is a unique point on $C$ that does not belong to the provided affine patch; we will denote this point by $\\infty$. Given a point $P$ we denote by $P^{-}$ the image of $P$ under the hyperelliptic involution, which maps $(x,y)$ to $(x,-y)$ and $\\infty$ to itself. \nThe (algebraic) Jacobian $J$ of $C$ is an abelian variety of dimension $2$ over $K$ whose $K$-rational points are $K$-rational classes of degree zero divisors on $C$ modulo linear equivalence. Every such class can be represented by a $K$-rational divisor of the form\n\\begin{equation*}\nP_1 + P_2 - 2\\infty, \\qquad \\text{for some} \\quad P_1,P_2\\in C.\n\\end{equation*}\nIn fact, such a divisor is unique for every point in $J(K)\\setminus \\{0\\}$, whereas the zero class is represented by \n\\begin{equation*}\nP + P^{-}  - 2\\infty, \\qquad \\text{for every} \\quad P\\in C.\n\\end{equation*}\nTherefore, there is a surjection from the symmetric square of $C$, denoted $C^{(2)}$, to $J$, and we can identify $J$ with $C^{(2)}$ blown down at the origin (see for instance \\cite[IIIa, \\S 2]{MumfordTata2} for the case $K=\\mathbb{C}$). \n We also consider the $\\Theta$ divisor of $J$, whose support is given by\n\\begin{equation*}\n\\Supp(\\Theta) = \\{[P-\\infty]:P\\in C\\}.\n\\end{equation*} \n\nWhile we are ultimately interested in working with Jacobians of curves over finite extensions of $\\mathbb{Q}_p$, for a prime $p$, it is instructive and useful for multiple reasons to first review some of the complex analytic theory. So assume now that $K=\\mathbb{C}$. Our main references are \\cite{Grant1990, Uchida, Baker_multiply_period, MumfordTata1, MumfordTata2}.\nA basis for the space $H^0(C\/\\mathbb{C},\\Omega^1)$ of holomorphic $1$-forms on $C$ is given by\n\\begin{equation}\\label{eq:omega_12}\n\\omega_1 = \\frac{dx}{2y},\\qquad \\omega_2 = \\frac{xdx}{2y};\n\\end{equation}\nusing the Hodge filtration, we identify this with a $2$-dimensional subspace of the first algebraic de Rham cohomology of $C\/\\mathbb{C}$, which we denote by $H^1_{\\dR}(C\/\\mathbb{C})$. Under this identification, we have\n\\begin{equation*}\nH^1_{\\dR}(C\/\\mathbb{C}) \\cong H^0(C\/\\mathbb{C},\\Omega^1) \\oplus W_0,\n\\end{equation*}\nwhere $W_0$ is the space spanned by the classes of the differentials of the second kind\n\\begin{equation}\\label{eq:compl_sub_complex}\n\\eta_1 = (-3x^3 - 2b_1x^2 - b_2 x)\\frac{dx}{2y}, \\qquad \\eta_2 =- \\frac{x^2 dx}{2y}.\n\\end{equation}\nThe space $W_0$ is isotropic with respect to the algebraic cup product pairing; moreover, $[\\eta_1]$ and $[\\eta_2]$ is a dual basis to $[\\omega_1],[\\omega_2]$, in the sense that $[\\eta_i]\\cup [\\omega_j] = \\delta_{ij}$, where $\\delta_{ij}$ is the Kronecker delta, so $[\\omega_1], [\\omega_2], [-\\eta_1],[-\\eta_2]$ is a symplectic basis for $H^1_{\\dR}(C\/\\mathbb{C})$ with respect to the cup product pairing. \n\nAnalogously, we also pick a symplectic basis $A_1,A_2,B_1,B_2$ for $H_1(C,\\mathbb{Z})$ with respect to the intersection product of cycles. It is explained in \\cite[IIIa, \\S 5]{MumfordTata2} how to do this using the fact that $C$ is a double cover of $\\mathbb{P}^1$.\n\nWe then define the period matrices \n\\begin{equation*}\n\\Omega =(\\Omega_{ij}) =  \\biggr(\\int_{A_j} \\omega_i\\biggl), \\qquad \\Omega^{\\prime} =(\\Omega^{\\prime}_{ij}) =  \\biggr(\\int_{B_j} \\omega_i\\biggl).\n\\end{equation*} \nThe matrix $\\tau\\colonequals \\Omega^{-1}\\Omega^{\\prime}$ is well-defined, symmetric and its imaginary part is positive definite; i.e.\\ $\\tau$ belongs to the Siegel upper half space of dimension $2$. Let $\\Lambda = \\Omega \\mathbb{Z}^2  + \\Omega^{\\prime} \\mathbb{Z}^2$. The analytic Jacobian of $C$ is $\\mathbb{C}^2\/\\Lambda$.  The isomorphism between the algebraic and analytic Jacobian is induced by the integration map\n\\begin{equation}\\label{eq:Phi}\n\\Phi\\colon C^{(2)} \\to \\mathbb{C}^2\/\\Lambda, \\qquad (P_1,P_2)\\mapsto \\int_{\\infty}^{P_1}(\\omega_1,\\omega_2) + \\int_{\\infty}^{P_2}(\\omega_1,\\omega_2)  \\bmod{\\Lambda}.\n\\end{equation}\nTo $\\tau$ as above and any $a,b\\in \\mathbb{Q}^2$, we can attach a so-called \\emph{theta function with characteristic}:\n\\begin{equation*}\n\\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(z) = \\sum_{n\\in\\mathbb{Z}^2} \\exp(\\pi i (n+a)^{T}\\tau(n+a) + 2\\pi i (n+a)^T(z+b)), \\qquad z\\in \\mathbb{C}^2.\n\\end{equation*}\nBy \\cite[II, Proposition 1.1 and p.\\,123]{MumfordTata1}, this is a holomorphic function on $\\mathbb{C}^2$. Moreover, it is quasi-periodic with respect to $\\Omega^{-1}\\Lambda$ in the sense of \\cite[p.\\,123]{MumfordTata1}: for every $m\\in \\mathbb{Z}^2$, \n\\begin{align}\\label{al:quasi_period1}\n\\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(z+m) &= \\exp(2\\pi i a^{T} m)\\cdot \\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(z)\\\\\n\\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(z+\\tau m) &= \\exp(-2\\pi i b^{T} m -\\pi i m^T \\tau m - 2\\pi i m^T z)\\cdot \\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(z). \\label{al:quasi_period2}\n\\end{align}\nFrom the quasi-periodicity \\eqref{al:quasi_period1}-\\eqref{al:quasi_period2} it is immediate that, writing $z=(z_1,z_2)$, for every $i,j\\in\\{1, 2\\}$, the following is a meromorphic function on $\\mathbb{C}^2\/\\Lambda$:\n\\begin{equation}\n\\frac{\\partial^2}{\\partial z_i \\partial z_j}\\log\\left(\\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(\\Omega^{-1}z)\\right).\n\\end{equation}\nOther meromorphic functions on $\\mathbb{C}^2\/\\Lambda$ arise in the same way if we replace $\\theta\\begin{bmatrix}\na\\\\\nb\n\\end{bmatrix}(z)$ with itself multiplied by the exponential of a polynomial in $z_1,z_2$ of degree at most $2$. Amongst such choices of polynomials and vectors $a,b\\in \\mathbb{Q}^2$ there are some that are particularly useful. \n\nFirst, when\n\\begin{equation*}\na = \\delta \\colonequals (1\/2,1\/2)\\qquad \\text{and} \\qquad b =\\delta^{\\prime}\\colonequals (1,1\/2), \n\\end{equation*}\nthe theta function is odd \\cite[II, Proposition 3.14]{MumfordTata1} and vanishes to order $1$ precisely on those $z\\in \\mathbb{C}^2$ such that $\\Omega z \\bmod{\\Lambda} = \\Phi(P,\\infty)$ for some $P\\in C$, that is, on the pullback of the (analytic) theta divisor under $\\mathbb{C}^2 \\to \\mathbb{C}^2\/\\Omega^{-1}\\Lambda \\xrightarrow{\\sim}\\mathbb{C}^2\/\\Lambda$. See \\cite[3.80-3.85, 3.89]{MumfordTata2}.\n\nSecondly, if we set\n\\begin{equation*}\n\\tilde{\\sigma}(z) = \\exp\\left(\\frac{1}{2}z^T H\\Omega^{-1}z\\right)\\theta\\begin{bmatrix}\n\\delta\\\\\n\\delta^{\\prime}\n\\end{bmatrix}(\\Omega^{-1} z),\\qquad \\text{where}\\quad  H = (H_{ij}) = \\biggl(\\int_{A_j} \\eta_i\\biggr),\n\\end{equation*}\nthen the second logarithmic derivatives of $\\tilde{\\sigma}$, viewed as functions on $C^{(2)}$, can be described as quotients of elements in $\\mathbb{Z}[b_1,\\dots, b_5][x_1,x_2,y_1,y_2]$; see \\cite[p.\\,38]{Baker_multiply_period}.\n\nFinally, we define the \\emph{hyperelliptic sigma function} of $C$ to be\n\\begin{equation}\\label{eq:sigma}\n\\sigma(z) = c\\tilde{\\sigma}(z),\n\\end{equation}\nwhere $c\\in \\mathbb{C}^{\\times}$ is such that the Taylor expansion of $\\sigma(z)$ around $0$ is of the form\n\\begin{equation*}\n\\sigma(z) = z_1 + O(z_1,z_2)^3.\n\\end{equation*}\nWith such a normalisation, the Taylor expansion has coefficients in $\\mathbb{Q}[b_i]$ (see e.g.\\ \\cite[Proposition 2.1]{Uchida}).\n\n For $i,j,\\dots,k\\in \\{1,2\\}$, let \n\\begin{equation}\n\\label{eq:diff_equation_cx_sigma}\n\\wp_{ij\\cdots k}(z) = -\\frac{\\partial}{\\partial z_i}\\frac{\\partial}{\\partial z_j}\\cdots\\frac{\\partial}{\\partial z_k}\\log(\\sigma(z)) \\qquad \\text{and}\\qquad \\wp = \\wp_{11}\\wp_{22} - \\wp_{12}^2.\n\\end{equation}\nIt follows from above that $\\wp_{ij\\cdots k}$ and $\\wp$ are meromorphic functions on $\\mathbb{C}^2\/\\Lambda$. On the image of $\\Phi$ these are expressible as rational functions dependent on the coefficients of $C$, and these definitions apply more generally when $\\mathbb{C}$ is replaced by any field $K$ of characteristic different from $2$.  For example, if $z = \\Phi((x_1,y_1),(x_2,y_2))$, then\n\\begin{align}\n\\label{eq:p12_p22}\n\\wp_{12}(z) = -x_1x_2,\\qquad \\wp_{22}(z) = x_1+x_2.\n\\end{align}\nSee \\cite[(1.4)]{Grant1990} for the other $\\wp_{ij}$ and for $\\wp_{ijk}$. \n\nFollowing Kanayama \\cite{Kanayama, Kanayama_corrections} and Uchida \\cite{Uchida}, we also introduce the following functions on the Jacobian: for $m\\geq 1$, the \\emph{$m$-th division polynomial} is\n\\begin{equation}\n\\label{eq:def_div_poly}\n\\phi_m(z) = \\frac{\\sigma(mz)}{\\sigma(z)^{m^2}}.\n\\end{equation}\nBy \\cite[Theorem 5.8, Example 5.9]{Uchida}, $\\phi_m(z)$ is a polynomial in $\\wp_{ij}$ and $\\wp_{ijk}$ ($1\\leq i,j,k\\leq 2$) with coefficients in $\\mathbb{Z}\\left[\\frac{1}{2},b_1,\\dots, b_5\\right]$ and, conjecturally, in $\\mathbb{Z}[b_1,\\dots, b_5]$ \\cite[Conjecture 4.14]{Uchida}. In particular, also division polynomials make sense over arbitrary fields of characteristic different from $2$, and given a curve over such a field, we will view $\\phi_m$ as a function on its Jacobian $J$, satisfying\n\\begin{equation}\n\\label{eq:divisor_div_poly}\n\\div(\\phi_m) = [m]^{*}\\Theta - m^2\\Theta.\n\\end{equation}\n\nOther properties of division polynomials that we need are (cf.\\ \\cite[Proposition 4.9, Example 2.16]{Uchida}):\n\\begin{align}\n\\phi_{mn}(z) &= \\phi_m(nz)\\phi_n(z)^{m^2} \\label{eq:div_quad}\\\\\n\\frac{\\phi_m(u+v)\\phi_m(u-v)}{\\phi_m(u)^2\\phi_m(v)^2} &= \\frac{-\\wp_{11}(mu) + \\wp_{11}(mv) - \\wp_{12}(mu)\\wp_{22}(mu) + \\wp_{22}(mu)\\wp_{12}(mv)}{(-\\wp_{11}(u) + \\wp_{11}(v) - \\wp_{12}(u)\\wp_{22}(v) + \\wp_{22}(u)\\wp_{12}(v))^{m^2}}. \\label{eq:div_par}\n\\end{align}\n\n\nThe analytic theory that we have outlined is related to our goal of defining $p$-adic heights in at least two ways. First, suppose that $K$ is a number field and let $v$ be an archimedean place of $K$. Then the (unique up to an additive constant) \\emph{real-valued} N\\'eron function at the place $v$, associated with $2\\Theta$, admits an explicit formula in terms of the hyperelliptic sigma function $\\sigma$: see \\cite[Corollary 2.5]{Yoshitomi}. Moreover, such a N\\'eron function satisfies transformation properties under scalar multiplication and addition on the Jacobian that can be described using the division polynomials and the functions $\\wp_{ij}$, respectively \\cite[Theorem 7.5]{Uchida}. In Section \\ref{sec:sigma_functions}, extending work of Blakestad \\cite{blakestadsthesis},\nwe construct $p$-adic analogues of the genus $2$ complex sigma function, and in Section \\ref{sec:padic_hts} we use these to define $p$-adic N\\'eron functions.\n\n\\begin{rmk}\nHyperelliptic sigma functions, division polynomials and their applications to real-valued N\\'eron functions all admit generalisations to hyperelliptic curves of arbitrary genus: see \\cite{Uchida}. For elliptic curves, see \\cite[Chapter VI]{silvermanadvancedtopics}.\n\\end{rmk}\n\nThe second application of the analytic theory that we are interested in is an explicit set of  equations for the Jacobian as a variety in $\\mathbb{P}^8$. This is worked out in \\cite{Grant1990}: for any field $K$ of characteristic different from $2$, \\cite[Corollary 2.15]{Grant1990} provides\na set of $13$ homogeneous equations\\footnote{\\label{note:typoGrant}There is a typo in $f_{10}$: the first term should be $X_{112}^2$, rather than $X_{122}^2$.} in $K[X_0,X_{11}, X_{12}, X_{22}, X_{111}, X_{112}, X_{122}, X_{222}, X]$, defining in $\\mathbb{P}^8$ the Jacobian of a curve described by an equation of the form \\eqref{eq:Grant} with coefficients in the field $K$. Under this embedding, the identity of $J$ is mapped to\n\\begin{equation*}\nO = (0:0:0:0:1:0:0:0:0).\n\\end{equation*}\nIn \\cite[Theorem 3.3, \\S 4]{Grant1990} Grant also gives explicit formulae for the group law on $J$ with respect to this embedding in $\\mathbb{P}^8$. \n\nNote that the projective coordinates $X_0, X_{ij}, X_{ijk}, X$ are related to the functions $\\wp_{ij},\\wp_{ijk},\\wp$ on $C^{(2)}$ by\n\\begin{align}\n\\label{eq:X_and_p}\n\\frac{X_{ij}}{X_0} = \\wp_{ij},\\qquad \\frac{X_{ijk}}{X_0} = \\frac{1}{2}\\wp_{ijk}, \\qquad \\frac{X}{X_0} = \\frac{1}{2}(\\wp +b_2\\wp_{12} - b_4);\n\\end{align}\nfor notational convenience from now on we dehomogenise with respect to $X_0$, i.e.\\ we set\n\\begin{equation*}\nX_0 \\colonequals 1. \n\\end{equation*}\n\n\n\nWe end this subsection with a remark about invariant differentials and derivations on $J$. The complex analytic isomorphism $J\\xrightarrow{\\sim}\\mathbb{C}^2\/\\Lambda$ is defined using the basis \\eqref{eq:omega_12} for the space of holomorphic differentials on $C$ and the embedding $\\iota\\colon C\\xhookrightarrow{} J$ mapping $\\infty$ to the identity of $J$. \n\nThe differentials $\\omega_1,\\omega_2$ of \\eqref{eq:omega_12} are a basis for $H^0(C\/K,\\Omega^1)$ (that is, not only when $K = \\mathbb{C}$), and the embedding $\\iota$ induces an isomorphism $\\iota^{*}\\colon H^0(J\/K,\\Omega^1)\\to H^0(C\/K,\\Omega^1)$ between the space of holomorphic differentials on $J$ and on $C$. Holomorphic differentials on $J$ are translation-invariant, and they correspond, by duality, to translation-invariant derivations. \n\n\n \n \\begin{mydef}\\label{def:basis_inv_diff_inv_der}\n For each $i\\in \\{1,2\\}$, let $\\Omega_i$ be the invariant differential on $J$ satisfying \n \\begin{equation*}\n \\iota^{*} \\Omega_i = \\omega_i,\n \\end{equation*}\nand let $\\partial_i$ be the invariant derivation dual to $\\Omega_i$.\n \\end{mydef}\n\n\\pagebreak\n \\begin{lemma}\\label{lemma:inv_difder}\\leavevmode\n\\begin{enumerate}\n\\item \\label{lemma_part:inv_difder_1} The basis $\\{\\Omega_1,\\Omega_2\\}$ for $H^0(J\/K,\\Omega^1)$ of Definition \\ref{def:basis_inv_diff_inv_der} is given explicitly by\n \\begin{align*}\n \\Omega_1 = \\frac{1}{2(X_{111}X_{122} - X_{112}^2)} (X_{122}\\cdot dX_{11} - X_{112} \\cdot dX_{12});\\\\\n \\Omega_2 = \\frac{1}{2(X_{111}X_{122} - X_{112}^2)} (X_{111}\\cdot dX_{12} - X_{112}\\cdot dX_{11}).\n \\end{align*}\n\\item \\label{lemma_part:inv_difder_2} The dual invariant derivations $\\partial_1$ and $\\partial_2$ satisfy\n \\begin{equation*}\n \\partial_1(X_{ij}) = 2X_{1ij},\\qquad \\partial_2(X_{ij}) = 2X_{ij2}.\n \\end{equation*}\n \\end{enumerate}\n \\end{lemma}\n \\begin{proof}\nPart \\eqref{lemma_part:inv_difder_2} is proved in \\cite[p.\\,64]{blakestadsthesis}, Part \\eqref{lemma_part:inv_difder_1} then follows from this and an explicit computation. \n \\end{proof}\n\n\\subsection{Grant's formal group law} \\label{subsec:formal}\nWe now apply the general theory of formal group laws presented in \\S \\ref{subsec:formalgps_gen} to the setting of \\S \\ref{subsec:jac}. In particular, we recall Grant's description of a pair of local parameters at the origin of a Jacobian as in \\S \\ref{subsec:jac} (over a suitable field) and the induced formal group law. Moreover, we relate the corresponding strict logarithm to the basis of invariant differentials of Definition \\ref{def:basis_inv_diff_inv_der}. Finally, we consider a group associated to the formal group law.\n\n\nLet $R$ be either the ring of integers of a number field, or a ring of characteristic $0$, complete with respect to a non-archimedean valuation. In both cases, let $K$ be the fraction field of $R$. As in \\S \\ref{subsec:jac}, we consider a hyperelliptic curve $C$ over $K$ defined by an equation of the form \\eqref{eq:Grant}, but we further assume that $b_1,\\dots, b_5\\in R$. We denote by $J$ the projective variety in $\\mathbb{P}^8$ defined in \\S \\ref{subsec:jac}, and we consider the following functions \n\\begin{equation}\\label{eq:local_par}\nT_1 = -\\frac{X_{11}}{X_{111}}, \\qquad T_2 = -\\frac{X}{X_{111}}.\n\\end{equation}\n\\begin{thm}[{\\hspace{1sp}\\cite[Theorem 4.2]{Grant1990}}]\\label{thm:exp_Xij_Xijk}\nThe functions $\\frac{1}{X_{111}}=\\frac{X_0}{X_{111}}, \\frac{X_{ij}}{X_{111}}, \\frac{X_{ijk}}{X_{111}}$ can be expanded as formal power series in $T_1,T_2$ with coefficients in $\\mathbb{Z}[b_1,\\dots, b_5]$. \n\\end{thm}\nIn particular, we have\n\\begin{equation} \\label{eq:Xij_exp}\n\\begin{aligned}\n\\frac{1}{X_{111}} &= \\sum_{i,j}\\alpha_{ij}T_1^iT_2^j = T_1^3\\biggl(-1+\\sum_{\\substack{i\\geq 3\\\\\ni+j>3}}\\alpha_{ij}T_1^{i-3}T_2^j\\biggr)\\\\\n\\frac{X_{22}}{X_{111}} &=\\sum_{i,j}\\beta_{ij}T_1^iT_2^j =  T_1\\biggl(-2T_1T_2 + \\sum_{\\substack{i\\geq 1\\\\\ni+j>3}} \\beta_{ij}T_1^{i-1}T_2^j\\biggr)\\\\\n\\frac{X_{12}}{X_{111}} &=\\sum_{i,j}\\gamma_{ij}T_1^iT_2^j = T_1\\biggl(T_2^2 +\\sum_{\\substack{i\\geq 1\\\\\ni+j>3}}\\gamma_{ij} T_1^{i-1}T_2^j\\biggr), \n\\end{aligned}\n\\end{equation}\nwhere $\\alpha_{ij},\\beta_{ij},\\gamma_{ij}\\in \\mathbb{Z}[b_1,\\dots,b_5]$ can be found recursively as follows. Suppose we have computed $\\alpha_{rs},\\beta_{rs},\\gamma_{rs}$ for all $r,s$ such that $r+s<m$. Then we can compute $\\alpha_{ij}, \\beta_{ij}, \\gamma_{ij}$ for all $i,j$ such that $i+j = m$ using the defining equations of the projective variety $J$. A similar recursion allows one to recover the following expansions:\n\\begin{equation}\\label{eq:Xijk_exp}\n\\begin{aligned}\n\\frac{X_{112}}{X_{111}} &=\\sum_{i,j}\\delta_{ij}T_1^i  T_2^j =  -T_2^2 +\\sum_{i+j\\geq 4}\\delta_{ij}T_1^iT_2^j\\\\\n\\frac{X_{122}}{X_{111}} &=\\sum_{i,j}\\epsilon_{ij}T_1^i  T_2^j = T_1T_2+\\sum_{i+j\\geq 4}\\epsilon_{ij}T_1^i T_2^j\\\\\n\\frac{X_{222}}{X_{111}} &= \\sum_{i,j}\\zeta_{ij}T_1^i  T_2^j =-T_1^2+\\sum_{i+j\\geq 4}\\zeta_{ij}T_1^iT_2^j,\n\\end{aligned}\n\\end{equation}\nfor some $\\delta_{ij},\\epsilon_{ij},\\zeta_{ij}\\in \\mathbb{Z}[b_1,\\dots, b_5]$.\n\n\\begin{rmk}\\label{rmk:odd_and_even_expansions}\nThe functions \n\\begin{equation*}\nT_1 = -\\frac{X_{11}}{X_{111}}, \\quad\\frac{1}{X_{111}}, \\quad  \\frac{X_{22}}{X_{111}},\\quad \\frac{X_{12}}{X_{111}}, \\quad T_2 = -\\frac{X}{X_{111}}\n\\end{equation*}\nare all odd functions on $J$, while\n\\begin{equation*}\n\\frac{X_{112}}{X_{111}}, \\quad \\frac{X_{122}}{X_{111}}, \\quad \\frac{X_{222}}{X_{111}}\n\\end{equation*}\nare even. It follows that\n\\begin{align*}\n&\\alpha_{ij} = \\beta_{ij} = \\gamma_{ij} = 0\\quad &&\\text{for all } i+j\\in 2\\mathbb{Z};\\\\\n&\\delta_{ij} = \\epsilon_{ij} = \\zeta_{ij} = 0 \\quad &&\\text{for all } i+j\\in 1+2\\mathbb{Z}.\n\\end{align*}\n\\end{rmk}\n\nThe group law on $J$ induces a two-dimensional formal group law in $T_1,T_2$ over $R$, as follows. Given points $u,v\\in J$, \\cite[Theorem 3.3]{Grant1990} expresses each of $X_{ij}(u+v), X_{ijk}(u+v), X(u+v)$ as a quotient of polynomials in $\\mathbb{Z}[b_1,\\dots,b_5][X_{ij}(u), X_{ij}(v), X_{ijk}(u), X_{ijk}(v)]$. \n\nLet $T = (T_1,T_2)$ and $S = (S_1,S_2)$ be the local parameters \\eqref{eq:local_par} for $u$ and $v$, respectively. Then one can express each of\n\\begin{equation*}\nF_1(u,v) = -\\frac{X_{11}(u+v)}{X_{111}(u+v)}, \\qquad F_2(u,v) = -\\frac{X(u+v)}{X_{111}(u+v)}.\n\\end{equation*}\nas elements in $\\Frac(\\mathbb{Z}[b_1,\\dots, b_5][T_1,T_2,S_1,S_2])$. To see this, you first have to multiply each of the numerator and denominators of $F_i(u,v)$ by the same suitable rational function.\n\n\n\\begin{thm}[{\\hspace{1sp}\\cite[Theorem 4.6]{Grant1990}}]\\label{thm:formal_gp_law}\nFor each $i\\in \\{1,2\\}$, we have\n\\begin{equation*}\nF_i(T,S) = T_i + S_i + O(T,S)^2\\in R[[T,S]].\n\\end{equation*}\nTherefore $F(T,S) = (F_1,F_2)(T,S)$ is a commutative formal group law over $R$.\n\\end{thm}\nSince $T_1$ and $T_2$ are odd functions on the Jacobian, the formal group inverse is \n\\begin{equation*}\ni(T) = -T;\n\\end{equation*}\ntherefore, for every $i\\in \\{1,2\\}$, we have that $F_i(-T,-S) = - F_i(T,S)$, and hence the expansion of $F_i$ only contains terms of total odd degree. \n\\begin{rmk}\nTechnically speaking the proof in \\cite[Theorem 4.6]{Grant1990} assumes that $R$ is a ring (of characteristic different from $2$), complete under a non-archimedean valuation. This assumption ensures that the Jacobian over $K = \\Frac(R)$ is an analytic group, and hence, locally, a formal group over $K$. Then a version of Gauss's lemma is used to prove that the formal group law is actually defined over $R$. The proof still gives us the desired result when $R$ is the ring of integers of a number field, since if the quotient of two power series in $R[[T,S]]$ is in $K_v[[T,S]]$ for \\emph{a} non-archimedean place $v$, then it is in $K[[T,S]]$, and if it is also in $R_v[[T,S]]$ for \\emph{every} non-archimedean place $v$ of $K$, then it is in $R[[T,S]]$.\n\nIt should also be possible to show that the group law is in fact defined over $\\mathbb{Z}[b_1,\\dots, b_5]$. However, it is not immediately clear how to do this. For example, we could try to re-write $F_i(T,S)$ as $\\frac{G}{H}$ where $G$ and $H$ are power series in $(T,S)$ with coefficients in $\\mathbb{Z}[b_1,\\dots, b_5]$, and $H = 1+ O(T,S)$.\n\nIt is worth mentioning, in this respect, the analogous situation for genus $2$ curves described by an equation $y^2 = f_6 x^6 + f_5 x^5 + f_4 x^4+ f_3 x^3 + f_2 x^2 + f_1 x + f_0$, considered by Flynn in \\cite{Flynn2, Flynn5}. In this case, in \\cite{Flynn2} Flynn describes an explicit embedding of the Jacobian in $\\mathbb{P}^{15}$ (in place of $\\mathbb{P}^8$), as well as a formal group law over a ring that is complete with respect to a non-archimedean valuation. Like Grant, Flynn appeals to the analytic group structure of the Jacobian over the fraction field for the formal group law and to Gauss's lemma. On the other hand, in the subsequent article \\cite{Flynn5}, he gives a more direct proof that the group law is in fact defined over $\\mathbb{Z}[f_1,\\dots, f_6]$ as a corollary of an explicit description of the group law on the Jacobian by biquadratic forms. \n\\end{rmk}\n\nLet $F$ be the formal group law over $R$ of Theorem \\ref{thm:formal_gp_law}. By the general theory of \\S \\ref{subsec:formalgps_gen}, its base-change to $K$ is isomorphic to the additive formal group of dimension $2$. In particular, let $\\mathcal{L} = (\\mathcal{L}_1,\\mathcal{L}_2)$ be the strict logarithm with respect to $T_1,T_2$, let $\\mathcal{E} = (\\mathcal{E}_1,\\mathcal{E}_2)$ be its inverse, and let $(D_1,D_2)$ be the basis of invariant derivations dual to $(d\\mathcal{L}_1,d\\mathcal{L}_2)$ (cf.\\ Theorem \\ref{thm:log_iso_Ga}, Definition \\ref{def:strict_log} and \\eqref{eq:dual_inv_diff_der}).\n\nThe expansions $\\Omega_1(T)$ and $\\Omega_2(T)$ in $T_1,T_2$ of the invariant differentials $\\Omega_1$ and $\\Omega_2$ of Definition \\ref{def:basis_inv_diff_inv_der} are also invariant differentials for $F$. \n\n\\begin{lemma}\\label{lemma:dLi_eq_Omegai}\nFor each $i\\in \\{1,2\\}$, we have\n\\begin{equation*}\nd\\mathcal{L}_i(T) = \\Omega_i(T).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nIt suffices to show that the invariant derivation dual to $\\Omega_i(T)$ is $D_i$. Upon extending $D_i$ to $\\Frac{K[[T_1,T_2]]}$ in such a way that it keeps satisfying the Leibniz rule, it is then sufficient, in view of Lemma \\ref{lemma:inv_difder}\\thinspace{}\\eqref{lemma_part:inv_difder_2}, to show that \\begin{equation*}\n\\alpha D_1(X_{12}) + \\beta D_2(X_{12})\n\\end{equation*}\nis different from $2X_{112}$ for all $(\\alpha,\\beta)\\neq (1,0)$ and that it is different from $2X_{122}$ for all $(\\alpha,\\beta)\\neq (0,1)$. \nBy \\eqref{eq:Xij_exp} and \\eqref{eq:Xijk_exp}, we have\n\\begin{align*}\nX_{12} = -T_1^{-2}T_2^2 + O(T_1,T_2),\\qquad \nX_{112} = T_1^{-3}T_2^{2} + O(1),\\qquad\nX_{122} =-T_1^{-2}T_2 + O(1);\n\\end{align*}\nfor an integer $i$, $O(T_1,T_2)^i$ means terms of total degree $\\geq i$, where we are allowing negative valuation: for example, $T_1^{-1}T_2^{3}$ is $O(T_1,T_2)^{2}$.\nMoreover, by \\eqref{eq:Di},\n\\begin{equation*}\n\\left(\\begin{matrix}D_1\\\\\nD_2\\end{matrix}\\right) = \\left(\\begin{matrix}\n1 + O(T_1,T_2)^2 & O(T_1,T_2)^2\\\\\nO(T_1,T_2)^2 &  1 + O(T_1,T_2)^2 \n\\end{matrix}\\right)\\left(\\begin{matrix}\n\\frac{\\partial}{\\partial T_1}\\\\\n\\frac{\\partial}{\\partial T_2}\n\\end{matrix}\\right).\n\\end{equation*}\nTherefore,\n\\begin{equation*}\nD_1 (X_{12}) = 2T_1^{-3}T_2^2 + O(1), \\qquad D_2(X_{12}) = -2T_1^{-2}T_2 + O(1),\n\\end{equation*}\nwhich proves the lemma. \n\\end{proof}\n\nSo far we have only been concerned with formal group laws. If $R$ is now a characteristic $0$ ring, complete under a non-archimedean absolute value $|\\cdot |_v$, with maximal ideal $\\mathfrak{m}$, we can consider a group $F(\\mathfrak{m})$ associated to the formal group law of Theorem \\ref{thm:formal_gp_law}:\n\\begin{equation*}\nF(\\mathfrak{m}) = \\mathfrak{m}\\times \\mathfrak{m},\\qquad t + s \\colonequals F(t,s),\\text{ for all } t = (t_1,t_2),s = (s_1,s_2)\\in F(\\mathfrak{m}).\n\\end{equation*}\nLet $\\tilde{J}$ be the variety in $\\mathbb{P}^{8}$ obtained by reducing modulo $\\mathfrak{m}$ the coefficients of the equations defining $J$ in $\\mathbb{P}^8$; this may or may not be an abelian variety over the residue field $R\/\\mathfrak{m}$. Nevertheless, we have a reduction map $\\tilde{\\ }\\colon J(K)\\to \\tilde{J}(R\/\\mathfrak{m})$. It follows from Theorem \\ref{thm:exp_Xij_Xijk} and the definition of the group law on $F(\\mathfrak{m})$ that we have the following isomorphism of groups\n\\begin{equation}\\label{eq:kernel_red}\nF(\\mathfrak{m}) \\cong J_1(K)\\colonequals \\{u\\in J(K)\\,:\\, \\tilde{u} = \\tilde{O}\\}.\n\\end{equation}\n(see \\cite[Corollary 4.5, Theorem 4.6]{Grant1990}). Note that this definition is model-dependent. \n\n\\section{$v$-Adic sigma functions}\\label{sec:sigma_functions}\n\\subsection{The naive sigma function}\\label{subsec:naive_sigma}\nIn this subsection, we assume that our genus $2$ curve $C$ is defined over a number field $K$, with ring of integers $\\mathcal{O}$, and that all the coefficients $b_1,\\dots,b_5$ of the defining equation \\eqref{eq:Grant} for $C$ lie in $\\mathcal{O}$.\n\nLet $\\rho$ be an embedding of $K$ in $\\mathbb{C}$ \n and let $\\sigma(z)$ be the hyperelliptic sigma function on $\\mathbb{C}^2$ (in the sense of \\S \\ref{subsec:jac}, \\eqref{eq:sigma}) associated to the curve over $\\mathbb{C}$ obtained from $C$ via $\\rho$.\nIn particular, $\\sigma(z)$ is an odd function and it satisfies the differential equations \\eqref{eq:diff_equation_cx_sigma}. Moreover, it has a Taylor expansion in $z=(z_1,z_2)$ around $0$ with coefficients in $\\mathbb{Q}[\\rho(b_i)]$.\n\nThe algebraicity of the coefficients of the Taylor expansion around $0$ of the \\emph{complex} sigma function is what we would like to use to define a \\emph{$v$-adic} sigma function for a non-archimedean place $v$, thereby extending to genus $2$ Bernardi's definition of a $v$-adic sigma function on elliptic curves \\cite{bernardi}. We will call such a $v$-adic sigma function \\emph{naive}, as it is constructed ad hoc from the complex one, and to distinguish it from other $v$-adic sigma functions that we will consider in \\S \\ref{subsec:infty_sigma}.\n\nBy \\S \\ref{subsec:formal}, we have an explicit description of formal group parameters $T_1,T_2$ and a resulting formal group law $F$ over $\\mathcal{O}$. Consider the base-change to $K$, the strict logarithm $\\mathcal{L} = (\\mathcal{L}_1,\\mathcal{L}_2)$ with respect to $T_1,T_2$ and its inverse $\\mathcal{E} = (\\mathcal{E}_1,\\mathcal{E}_2)$, and let $(D_1,D_2)$ be the basis of invariant derivations dual to $(d\\mathcal{L}_1,d\\mathcal{L}_2)$ (cf.\\ Theorem \\ref{thm:log_iso_Ga}, Definition \\ref{def:strict_log} and \\eqref{eq:dual_inv_diff_der}). We extend $D_i$ to $\\Frac{K[[T_1,T_2]]}$ in such a way that it keeps satisfying the Leibniz rule, and for $f(T)\\in K[[T]]$, we define $D_i(\\log(f(T)))\\in\\Frac{K[[T_1,T_2]]} $ so that Lemma \\ref{lemma:log_and_inv_derivations} holds for $\\log(f(T))$. \n\n We also consider the base-change of $F$ to $\\mathbb{C}$ via $\\rho$.  By \\cite[\\S 10.9.3]{Bost}\\footnote{Technically speaking this reference is for elliptic curves, but as stated by the author the results ``actually admit straightforward generalizations concerning commutative algebraic groups over (local) fields of characteristic zero''.}, there is a formal group logarithm $\\tilde{\\mathcal{L}} \\colon F_{\\mathbb{C}} \\to \\mathbb{G}_a^2$ whose inverse $\\tilde{\\mathcal{E}} \\colon \\mathbb{G}_a^2\\to F_{\\mathbb{C}}$ extends to the surjective homomorphism $\\mathbb{C}^2\\to J$ inducing the inverse of $\\Phi\\colon J\\to \\mathbb{C}^2\/\\Lambda$ (see \\eqref{eq:Phi}).  \n \n\\begin{lemma} \\label{lemma:pijF}\nThe logarithm $\\tilde{\\mathcal{L}}$ and exponential $\\tilde{\\mathcal{E}}$ are the base-change to $\\mathbb{C}$ of $\\mathcal{L}$ and $\\mathcal{E}$. Therefore,\n\\begin{equation*}\nX_{ij}(\\mathcal{E}(z)) = \\wp_{ij}(z), \\qquad X_{ijk}(\\mathcal{E}(z)) = \\frac{\\wp_{ijk}}{2}(z),\n\\end{equation*}\nwhere $\\wp_{ij}(z)$ (resp.\\ $\\wp_{ijk}(z)$) is the quotient of series in $z_1,z_2$ obtained from the Taylor expansion of $\\sigma$ and \\eqref{eq:diff_equation_cx_sigma}.\n\\end{lemma}\n\\begin{proof}\nThis follows from the definition of $\\Phi$ \\eqref{eq:Phi} and Lemma \\ref{lemma:dLi_eq_Omegai}.\n\\end{proof}\n\nLet now $v$ be a prime of $K$ above a fixed rational prime $p$, let $K_v$ be the completion of $K$ with respect to $v$, $\\mathcal{O}_v$ its ring of integers with unique maximal ideal $\\mathfrak{m}_v$. Let $\\mathop{\\mathrm{ord}}\\nolimits_v$ be the valuation on $K_v$ corresponding to $v$. Recall from the end of \\S \\ref{subsec:formal} that there is a group isomorphism between $F(\\mathfrak{m}_v)$ (which, as a set, is $\\mathfrak{m}_v\\times \\mathfrak{m}_v)$ and the group $J_1(K_v)$ of points reducing to $O$ modulo $\\mathfrak{m}_v$. We would like to use this to obtain a $v$-adic sigma function on $J_1(K_v)$, by substituting $\\mathcal{L}(T)$ in the Taylor series expansion of $\\sigma(z)$.\n\nSimilarly to the elliptic curve case \\cite{bernardi}, it will not always be possible to obtain a function on the whole of $J_1(K_v)$: \nwe need to understand how $\\sigma(\\mathcal{L}(T))$ converges $v$-adically. We start with an auxiliary lemma.\n\n\\begin{lemma}\\label{lemma:subs_exponential}\nLet $f(T_1,T_2)\\in \\mathcal{O}_v[[T_1,T_2]]$. Then $f(\\mathcal{E}(z))$ is of the form\n\\begin{equation*}\nf(\\mathcal{E}(z)) = \\sum_{i_1, i_2\\in \\mathbb{N}} \\frac{a_{i_1,i_2}}{i_1! i_2!} z_1^{i_1}z_2^{i_2}, \\qquad \\text{where } a_{i_1,i_2}\\in \\mathcal{O}_v.\n\\end{equation*}\nIn particular, it converges for all $z$ with $\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}>\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$. Moreover,  for such $z$ we have\n\\begin{equation*}\n\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(\\mathcal{E}_i(z))\\} = \\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}.\n\\end{equation*}\nSimilarly, if $\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(T_i)\\}> \\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$, then\n\\begin{equation*}\n\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(\\mathcal{L}_i(T))\\} = \\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(T_i)\\}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nThe first part follows from Proposition \\ref{prop:exp_exp_log} and the fact that if $j_1,j_2\\in \\mathbb{N}$ are such that $j_1+j_2 = n$ then $\\mathop{\\mathrm{ord}}\\nolimits_v(j_1!j_2!) \\leq \\mathop{\\mathrm{ord}}\\nolimits_v(n!)$ by Legendre's formula. For the same reasons, for the second part it suffices to show that if $\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}>\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$ and $ n\\geq 2$, then \n\\begin{equation}\\label{eq:vals}\nn\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}-\\mathop{\\mathrm{ord}}\\nolimits_v(n!) > \\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}.\n\\end{equation}\nLet $s_p(n)$ be the sum of the base $p$ digits of $n$. Then \n\\begin{equation*}\n\\mathop{\\mathrm{ord}}\\nolimits_v(n!) = \\frac{n-s_p(n)}{p-1}\\mathop{\\mathrm{ord}}\\nolimits_v(p) \\leq  \\frac{n-1}{p-1}\\mathop{\\mathrm{ord}}\\nolimits_v(p),\n\\end{equation*}\nso \\eqref{eq:vals} follows. The proof for $\\mathcal{L}_i$ is identical.\n\\end{proof}\nLet $[n]T $ be defined inductively by $[1]T = T$ and for $n\\geq 2$, $[n]T= F([n-1]T, T)$ and let $\\phi_n(T)$ be the expansion in $T$ of the $n$-th division polynomial (see \\eqref{eq:def_div_poly} and surrounding paragraph).\n\\begin{thm}\\label{thm:sigma_naive}\nLet $\\sigma(z) = z_1 + O(z_1,z_2)^3\\in K[[z_1,z_2]]$ be the Taylor expansion around $0$ of the complex sigma function, and let\n\\begin{equation*}\n\\sigma_v^{(0)}(T)\\colonequals \\sigma(\\mathcal{L}(T))\\in K[[T]].\n\\end{equation*}\n\\begin{enumerate}[label=(\\roman*)]\n\\item\\label{thm_sigma:part1} $\\sigma_v^{(0)}(T)$ satisfies the system of differential equations\n\\begin{equation*}\nD_iD_j \\log(\\sigma_v^{(0)}(T)) =-X_{ij}(T), \\qquad \\text{for all } i,j\\in \\{1,2\\} \\qquad (\\text{where } X_{21}\\colonequals X_{12}).\n\\end{equation*}\n\\item\\label{thm_sigma:part2} $\\sigma_v^{(0)}(T)$ converges for all $T=(T_1,T_2)\\in K_v^2$ satisfying $\\min_{i}\\{\\mathop{\\mathrm{ord}}\\nolimits_v(T_i)\\} >\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$.\n\\item\\label{thm_sigma:part3} When all terms are defined, we have\n\\begin{align*}\n\\frac{\\sigma_v^{(0)}(F(T,S))\\sigma_v^{(0)}(F(T,-S))}{\\sigma_v^{(0)}(T)^2\\sigma_v^{(0)}(S)^2}& = -X_{11}(T) + X_{11}(S) - X_{12}(T)X_{22}(S) + X_{22}(T) X_{12}(S);\\\\\n\\frac{\\sigma_v^{(0)}([n]T)}{\\sigma_v^{(0)}(T)^{n^2}} &= \\phi_n(T).\n\\end{align*}\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nBy Lemma \\ref{lemma:log_and_inv_derivations}, we have\n\\begin{equation*}\n(D_iD_j \\log(\\sigma_v^{(0)}(T)))(\\mathcal{E}(z)) = \\frac{\\partial^2}{\\partial z_i\\partial z_j}\\log(\\sigma_v^{(0)}(\\mathcal{E}(z))) = \\frac{\\partial^2 }{\\partial z_i \\partial z_j}\\log(\\sigma(z)) =- \\wp_{ij}(z),\n\\end{equation*}\nwhich gives \\ref{thm_sigma:part1} by Lemma \\ref{lemma:pijF}.\n\nSince the divisor of $\\sigma$ is the preimage of $\\Theta$ under $\\mathbb{C}^2\\to J$, since $T_1$ represents $\\Theta$ in a neighbourhood of the origin \\cite[Lemma 32]{blakestadsthesis} and since $\\sigma(z) = z_1 + O(z_1,z_2)^3$, we have \n\\begin{equation*}\\label{eq:sigma_div_E1}\n\\sigma(z) = \\mathcal{E}_1(z)u(z), \\qquad\\text{for some}\\quad u(z) =1 + O(z_1,z_2)\\in K[[z_1,z_2]].\n\\end{equation*}\nCombining with \\ref{thm_sigma:part1} and the fact that $\\sigma$ is an odd function, we get that\n\\begin{equation*}\n\\sigma_v^{(0)}(T) = T_1 u_v(T),\n\\end{equation*}\nwhere $u_v(T) = 1 + O(T_1,T_2)^2\\in K[[T_1,T_2]]$ satisfies\n\\begin{equation*}\nD_i D_j \\log(u_v(T)) =-X_{ij}(T) - D_iD_j \\log(T_1). \n\\end{equation*}\nSince the left hand side of this equation is a power series, so is the right hand side. Moreover, the latter is easily seen to have coefficients in $\\mathcal{O}$. Evaluating at $\\mathcal{E}(z)$ and using Lemma \\ref{lemma:subs_exponential}, we see that \n\\begin{equation*}\n\\frac{\\partial^2}{\\partial z_i \\partial z_j}\\log(u(z)) = \\sum_{i_1 + i_2\\geq 0} \\frac{a_{i_1,i_2}}{i_1!i_2!} z_1^{i_1}z_2^{i_2}, \\qquad \\text{for some } a_{i_1,i_2}\\in  \\mathcal{O}_v.\n\\end{equation*}\n\nBy properties of integration, the series $\\log(u(z))$ has the same form and since the leading term of $u_v(T)$ is $1$, it has vanishing constant term. Thus, if $z_i =\\mathcal{L}_i(T)$ with $\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(T_i)\\}>\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$, then $\\mathop{\\mathrm{ord}}\\nolimits_v(\\log(u(z))) \\geq \\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}>\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$, so $\\log(u(z))$ is in the domain of the $p$-adic exponential (cf.\\ Lemma \\ref{lemma:subs_exponential}). Applying \\cite[\\S 2 Substitution Theorem]{mattuck}, we find that $u(z)$ converges whenever $\\min_i\\{\\mathop{\\mathrm{ord}}\\nolimits_v(z_i)\\}>\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(p)}{p-1}$. Hence $\\sigma(z)$ converges there too, which is statement \\ref{thm_sigma:part2}. \n\nThe identities stated in part \\ref{thm_sigma:part3} follow from the identities satisfied by the complex sigma function (\\eqref{eq:def_div_poly} and \\cite[Example 2.16]{Uchida}).\n\\end{proof}\n\n\\subsection{Infinitely many sigma functions and Blakestad's canonical one}\\label{subsec:infty_sigma}\nRetain the notation of \\S \\ref{subsec:naive_sigma}. For $1\\leq i,j\\leq 2$, let $c_{ij}\\in K_v$ with $c_{12} = c_{21}$ and let $c = (c_{ij})_{i,j}$. Then the system \n\\begin{equation*}\nD_iD_j(\\log(\\sigma_v^{(c)}(T))) =-X_{ij}(T) + c_{ij}, \\qquad \\text{for all } 1\\leq i,j\\leq 2, \n\\end{equation*}\nhas a unique odd solution $\\sigma_v^{(c)}(T)\\in K_v[[T]]$ of the form $T_1 (1+ O(T_1,T_2))$: if $c = 0$, the zero matrix, then $\\sigma_v^{(0)}  = \\sigma_v^{(c)}$ is the naive sigma function that we studied in \\S \\ref{subsec:naive_sigma}; in general, we have\n\\begin{equation}\\label{eq:formulasigmac}\n\\sigma_v^{(c)}(T) = \\sigma_v^{(0)}(T) \\exp\\biggl(\\frac{1}{2}\\sum_{1\\leq i,j\\leq 2} c_{ij}\\mathcal{L}_i(T)\\mathcal{L}_j(T)\\biggr).\n\\end{equation}\nSee Appendix \\ref{app:sigma_expansion} for the terms up to total degree at most $8$ of the formal power series $ \\sigma_v^{(c)}(T)$.\n\n\\begin{prop} \\label{prop:finite_index_subgp}\\leavevmode\n\\begin{enumerate}[label=(\\roman*)]\n\\item\\label{prop:finite_index_subgp:1} The formal power series $\\sigma_v^{(c)}(T)\\in K_v[[T_1,T_2]]$ induces a function on a finite index subgroup $H_v$ of $J(K_v)$.\n\\item\\label{prop:finite_index_subgp:2} For $P\\in H_v$, $\\sigma_v^{(c)}(T(P))$ vanishes if and only if $T_1(P) = 0$, i.e.\\ it vanishes only on the theta divisor and there it vanishes to order $1$.\n\\item\\label{prop:finite_index_subgp:3} When all terms are defined, we have \n\\begin{align*}\n\\frac{\\sigma_v^{(c)}(F(T,S))\\sigma_v^{(c)}(F(T,-S))}{\\sigma_v^{(c)}(T)^2\\sigma_v^{(c)}(S)^2}& = -X_{11}(T) + X_{11}(S) - X_{12}(T)X_{22}(S) + X_{22}(T) X_{12}(S);\\\\\n\\frac{\\sigma_v^{(c)}([n]T)}{\\sigma_v^{(c)}(T)^{n^2}} &= \\phi_n(T).\n\\end{align*}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\\hfill\n\\begin{enumerate}[label=(\\roman*)]\n\\item The same argument as in \\cite[Chapter 7, \\S 5]{cassels_flynn} (see also \\cite[III, \\S 6]{mattuck}) shows that $J_1(K_v)$ has finite index in $J(K_v)$ (without any assumption on the reduction), and that for every positive integer $n$, $F(\\mathfrak{m}_v^n)$ has finite index in $F(\\mathfrak{m}_v)\\cong J_1(K_v)$.\n  Therefore, it suffices to show that there exists $n\\in\\mathbb{Z}_{>0}$ such that $\\sigma_v^{(c)}(T)$ converges for all $T\\in \\mathfrak{m}_v^{n}\\times \\mathfrak{m}_v^{n}$. By Theorem \\ref{thm:sigma_naive}, there exists $n_1\\in\\mathbb{Z}_{>0}$ such that $\\sigma_v^{(0)}$ converges on $F(\\mathfrak{m}_v^{n_1})$. By Lemma \\ref{lemma:subs_exponential}, there exists $n_2\\in \\mathbb{Z}_{>0}$ such that $ \\exp\\biggl(\\frac{1}{2}\\sum_{1\\leq i,j\\leq 2} c_{ij}\\mathcal{L}_i(T)\\mathcal{L}_j(T)\\biggr)$ converges on $F(\\mathfrak{m}_v^{n_2})$. The conclusion follows for $n=\\max\\{n_1,n_2\\}$ by \\cite[\\S 2 Multiplication Theorem]{mattuck}.\n    \\item We have $\\sigma_v^{(c)}(T) = T_1\\cdot \\exp(f^{(c)}(T))$ for some formal power series $f^{(c)}(T)$. Moreover, we defined $H_v$ in such a way that $f^{(c)}(T)$ and $\\exp(f^{(c)}(T))$ converge on $H_v$. Therefore, the only zeros are at $T_1 = 0$. \n\\item This follows from Theorem \\ref{thm:sigma_naive}\\thinspace{}\\ref{thm_sigma:part3}, Equation \\eqref{eq:formulasigmac}, and the fact that $\\mathcal{L}_i\\colon F\\to \\mathbb{G}_a$ is a homomorphism.\n\n\\end{enumerate}\n\\end{proof}\n\n\nIn other words, for any $2\\times 2$ symmetric matrix $c$ over $K_v$ we obtain a $v$-adically valued function on some finite index subgroup of $J(K_v)$ satisfying properties analogous to the complex sigma function. Given such a $c$, we also define the subspace $W^{(c)}$ of $H^1_{\\dR}(C\/K_v)$ to be the space generated by the classes of the following differentials (compare with \\eqref{eq:compl_sub_complex}):\n\\begin{equation}\\label{eq:eta_12_c}\n\\begin{aligned}\n\\eta_1^{(c)} = (-3x^3 - 2b_1x^2 -b_2x + c_{12} x + c_{11})\\frac{dx}{2y},\\\\\n \\eta_2^{(c)}= (-x^2 + c_{22} x + c_{12})\\frac{dx}{2y}.\n \\end{aligned}\n\\end{equation}\n\n\\begin{prop}\\label{prop:bijection_sigma_is}\nThe map\n\\begin{equation*}\nc = (c_{ij}) \\mapsto W^{(c)}\n\\end{equation*}\ngives a bijection between the set of $2\\times 2$ symmetric matrices over $K_v$ and the set $\\Is(C\/K_v)$ of subspaces of $H^{1}_{\\dR}(C\/K_v)$ that are complementary to $H^{0}(C\/K_v,\\Omega^1)$ and isotropic with respect to the cup product. Moreover, $[\\omega_1],[\\omega_2],[-\\eta_1^{(c)}],[-\\eta_2^{(c)}]$ is a symplectic basis for $H^1_{\\dR}(C\/K_v)$.\n\\end{prop}\n\n\\begin{proof}\nIt is well-known that $\\left\\{[\\omega_1], [\\omega_2], \\left[\\frac{x^i dx}{2y}\\right]\\colon i =2, 3\\right\\}$ is a basis for $H_{\\dR}^1(C\/K_v)$ and that $H^{0}(C\/K_v,\\Omega^1)\\subset H_{\\dR}^1(C\/K_v)$ is generated  by $[\\omega_1]$ and $[\\omega_2]$. Therefore, the space generated by $[\\eta_1^{(c)}]$ and $[\\eta_2^{(c)}]$ is complementary to the space of holomorphic forms and a simple computation of residues at infinity shows that it is isotropic with respect to the cup product, and that $[\\omega_i]\\cup[-\\eta_j^{(c)}] = \\delta_{ij}$. On the other hand, if $W\\in \\Is(C\/K_v)$ and $[\\xi_{1}],[\\xi_{2}]$ is a basis for $W$ such that $[\\omega_i]\\cup[-\\xi_j] = \\delta_{ij}$, then we must have\n\\begin{equation*}\n[\\xi_{i}] = [\\eta_i^{(0)}]  + \\sum_{j=1}^2 b_{ij}[\\omega_j]\n\\end{equation*}\nfor some symmetric matrix  $(b_{ij})$.\n\\end{proof}\n\nBlakestad \\cite{blakestadsthesis} does not consider the infinitely many $v$-adic sigma functions defined by \\eqref{eq:formulasigmac}; he considers just one, which is, however, arguably more interesting than the others. Translated to our setting, his results may be phrased as follows. \n\\begin{thm}[{\\hspace{1sp}\\cite[Propositions 34, 27, Corollary 37\\protect\\footnotemark]{blakestadsthesis}}]\\footnotetext{Corollary 37 of \\emph{loc.\\ cit.}\\ contains some typos. The correct formulae for $c_{11}$ and $c_{12}$ are: $c_{11} = 2b_1b_2 - b_1\\alpha + b_1^2\\beta + 3\\delta - 3b_1\\gamma + 3b_3$,  $c_{12} = b_2 + \\alpha - b_1\\beta$. }\\label{thm:Blakestad_main}\nLet $p\\geq 5$ and suppose that $C$ has good reduction at $v\\mid p$ and that $J$ has good ordinary reduction at $v$. Then\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{thm:Blake_1} There exists a unique $v$-adic sigma function $\\sigma_v^{(c)}(T)$ with $v$-adically integral coefficients. Let $b$ be the corresponding symmetric matrix. \n\\item\\label{thm:Blake_2} The differentials $\\eta_i^{(b)}$ are uniquely determined by the property that the expansion $\\int \\eta_i^{(b)}$ in the local parameter $t = -\\frac{x^2}{y}$ at $\\infty$ has coefficients in $\\mathcal{O}_v$.\n\\item \\label{thm:Blake_3} The matrix $b$ has coefficients in $\\mathcal{O}_v$ and can be computed modulo $p^n$ from explicit expansions in $t = -\\frac{x^2}{y}$ of suitable functions in the Riemann--Roch spaces of $p^n\\infty$ and $3p^n\\infty$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rmk}\\label{rmk:inver_H1}\nSome remarks on our phrasing of Blakestad's results are in order. First, Theorem \\ref{thm:Blakestad_main}\\thinspace{}\\ref{thm:Blake_2} and \\ref{thm:Blake_3} assume the invertibility over the residue field of a $2\\times 2$ matrix $H_1$, defined on \\cite[p.\\,54]{blakestadsthesis}. Blakestad mentions at the beginning of \\S 4.2 of \\emph{loc.\\ cit.}\\ that this condition is related to the ordinarity of $J$. More precisely, we prove in \\cite{BKM22} that if the equation for $C$ has semistable reduction, then $J$ is semistable ordinary if and only if $H_1$ is invertible. \n\nSecondly, Blakestad allows for $p=3$. However, in this case, the construction in \\S 3.2.1 needs to be slightly modified, for example by defining $\\phi_n$ and $\\psi_n$ on p.\\,52 for $n\\geq 2$, instead of $n\\geq 1$. Then the condition on invertibility of $H_1$ could be replaced by invertibility of $H_2$. For simplicity, we stated Theorem \\ref{thm:Blakestad_main} for $p\\geq 5$.\n\\end{rmk}\n\nWe next want to  show that the subspace $W^{(b)}\\subset H^1_{\\dR}(C\/K_v)$ corresponding to Blakestad's $v$-adic sigma function is the unit root subspace of Frobenius (defined below). We achieve this by comparing the property of Theorem \\ref{thm:Blakestad_main}\\thinspace{}\\ref{thm:Blake_2} with the characterisation of the unit root subspace provided by \\cite[Corollary 5.9.6\\thinspace{}(1)]{Katz_crystalline}. We have found the article \\cite{Bogaart} a very useful reference for both theoretical and explicit results on various cohomological theories and we refer the reader to it for details. \n\nRetain the assumptions of Theorem \\ref{thm:Blakestad_main} and assume that $K_v$ is an unramified extension of $\\mathbb{Q}_p$. \nThe equation defining $C$ also defines the affine part of a hyperelliptic curve over $\\mathcal{O}_v$, which we will denote by $\\mathcal{C}\/\\mathcal{O}_v$; denote by $H^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)$ the first cohomology of $\\mathcal{C}\/\\mathcal{O}_v$. By  \\cite{Berthelot74}, there is a canonical isomorphism $H^{1}_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)\\xrightarrow{\\sim} H^1_{\\cris}(\\tilde{\\mathcal{C}}\/\\mathcal{O}_v)$, where $\\tilde{\\mathcal{C}}$ denotes the special fibre of $\\mathcal{C}\/\\mathcal{O}_v$ and $H^1_{\\cris}(\\tilde{\\mathcal{C}}\/\\mathcal{O}_v)$ is its first crystalline cohomology. This isomorphism equips $H^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)$ with an $\\mathcal{O}_v$-linear endomorphism of Frobenius. From the isomorphism \\cite[Proposition 2.2]{Bogaart}\n\\begin{equation*}\nH^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)\\otimes_{\\mathcal{O}_v} K_v\\xrightarrow{\\sim} H^1_{\\dR}(C\/K_v),\n\\end{equation*}\n we also obtain a $K_v$-linear Frobenius endomorphism on the $K_v$-vector space $H^1_{\\dR}(C\/K_v)$. The \\emph{unit root eigenspace of Frobenius} is the slope $0$ subspace of $H^1_{\\dR}(C\/K_v)$ for the Frobenius action. Since we are assuming that $J$ has ordinary reduction at $v$, this space has dimension equal to the dimension of $J$, that is to $2$ \\cite[Theorem 3.1]{yui1978jacobian}.\n \nLet $P$ be an arbitrary point in $\\mathcal{C}(\\mathcal{O}_v)$. Let $\\hat{\\mathcal{C}}_P$ be the formal completion of $\\mathcal{C}\/\\mathcal{O}_v$ along $P$. Then $(\\hat{\\mathcal{C}}_{P}, P)$ is a pointed Lie variety of dimension $1$ over $\\mathcal{O}_v$ \\cite[\\S\\S  5.1, 5.9]{Katz_crystalline}. Let $t$ be a coordinate for it. The first de Rham cohomology $H^1_{\\dR}(\\hat{\\mathcal{C}}_{P}\/\\mathcal{O}_v)$ is the $\\mathcal{O}_v$-module of (closed) differential one-forms in $\\mathcal{O}_v[[t]]dt$ modulo the exact one-forms \\cite[\\S 5.1]{Katz_crystalline}. Since any closed differential form becomes exact upon base-changing to $K_v$, applying $d$ gives an isomorphism of $\\mathcal{O}_v$-modules:\n\\begin{equation}\\label{eq:iso_HdR}\nM_{P}\\colonequals \\frac{\\{f\\in  K_v[[t]]: f(0) = 0 \\text{ and } df\\in \\mathcal{O}_v[[t]]dt\\}}{\\{f\\in \\mathcal{O}_v[[t]]: f(0) = 0\\}}\\xrightarrow{\\sim} H^1_{\\dR}(\\hat{\\mathcal{C}}_{P}\/\\mathcal{O}_v).\n\\end{equation}\n(this is a special case of \\cite[Lemma 5.1.2]{Katz_crystalline}).  \n\n By \\cite[Corollary 5.9.6]{Katz_crystalline}, the unit root eigenspace of Frobenius is the subspace of $H^1_{\\dR}(C\/K_v)$ obtained by tensoring with $K_v$ the kernel of the formal-expansion-at-$P$ map\n     \\begin{equation}\\label{eq:formal_expansion}\n\\beta_P\\colon H^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)\\to H^1_{\\dR}(\\hat{\\mathcal{C}}_{P}\/\\mathcal{O}_v).\n   \\end{equation}   \n   We use the superscript $^{-}$ to indicate $(-1)$-eigenspaces for the action of the hyperelliptic involution. Recall that we can regard $\\eta_1^{(b)}$ and $\\eta_2^{(b)}$ as elements of $H^1_{\\dR}(C\/K_v)$ by using the following isomorphism of $K_v$-vector spaces\n\\begin{equation}\\label{eq:H1dR_sioH0}\nH^1_{\\dR}(C\/K_v)\\cong H^0(C\/K_v,\\Omega^1_{C\/K_v}(4\\infty))^{-};\n\\end{equation}\nsee, for instance, \\cite[\\S 5]{Bogaart}.\n\\begin{prop}\\label{prop:Blakestad_space_unit_root}\nSuppose that the assumptions of Theorem \\ref{thm:Blakestad_main} are satisfied, and that $K_v$ is an unramified extension of $\\mathbb{Q}_p$. Then the differentials $\\eta_1^{(b)}$ and $\\eta_2^{(b)}$ span the unit root eigenspace of Frobenius in $H^1_{\\dR}(C\/K_v)$.\n\\end{prop}\n\n\\begin{proof}\nBy abuse of notation, we write $\\infty$ also for the section at infinity of $\\mathcal{C}\/\\mathcal{O}_v$. \nBy \\cite[Proposition 3.2]{Bogaart}, the $\\mathcal{O}_v$-module $H^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)$ is a lattice in $H^1_{\\dR}(C\/K_v)$, which, in view of the assumption that $p\\geq 5$ and the genus is $2$, we can identify by \\cite[(3.9), Theorem 4.2\\thinspace{}(ii)]{Bogaart}, with $H^0(\\mathcal{C},\\Omega^1_{\\mathcal{C}\/\\mathcal{O}_v}(4\\infty))^{-}\\xhookrightarrow{}H^0(C\/K_v,\\Omega^1_{C\/K_v}(4\\infty))^{-}$, under \\eqref{eq:H1dR_sioH0}. By \\cite[Proposition 5.2]{Bogaart}, the $\\mathcal{O}_v$-module $H^0(\\mathcal{C},\\Omega^1_{\\mathcal{C}\/\\mathcal{O}_v}(4\\infty))^{-}$ is free and spanned by $x^{i}\\frac{dx}{2y}$ for $0\\leq i\\leq 3$. \n\nThe parameter $t=-\\frac{x^2}{y}$ is a coordinate for $(\\hat{\\mathcal{C}}_{\\infty},\\infty)$. Let $M_{\\infty}$ be as in \\eqref{eq:iso_HdR} with respect to $t$.\nThe assumption that $p\\geq 5$ also implies that\nthere is an isomorphism of $\\mathcal{O}_v$-modules\n  \\begin{equation}\n   \\begin{aligned}\n   M_{\\infty,(4\\infty)}\\colonequals  \\frac{\\{f = \\sum_{n=-3}^{\\infty} a_n t^n \\in  t^{-3}K_v[[t]]:a_0 = 0 \\text{ and } df\\in \\mathcal{O}_v((t))dt\\}}{\\{f = \\sum_{n=-3}^{\\infty} a_n t^n \\in  t^{-3}O_v[[t]]:a_0 = 0\\}} \\xrightarrow{\\sim} M_{\\infty}\\\\\n   \\sum_{n=-3}^{\\infty} a_n t^n\\mapsto \\sum_{n=0}^{\\infty} a_n t^n.\n   \\end{aligned}\n   \\end{equation} \nWe obtain a map\n   \\begin{equation}\\label{eq:comp_alpha}\n  H^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)\\xrightarrow{\\sim}H^0(\\mathcal{C},\\Omega^1_{\\mathcal{C}\/\\mathcal{O}_v}(4\\infty))^{-}\\xrightarrow{\\alpha_{\\infty}} M_{\\infty,(4\\infty)}\\xrightarrow{\\sim} M_{\\infty}\\xrightarrow{\\sim} H^1_{\\dR}(\\hat{\\mathcal{C}}_{\\infty}\/\\mathcal{O}_v),\n   \\end{equation}\n  where $\\alpha_{\\infty}$ is the formal-expansion-in-$t$ map followed by formal integration. The differentials $\\eta_1^{(b)}$ and $\\eta_2^{(b)}$ belong to the kernel of $\\alpha_{\\infty}$. Let $\\mathcal{C}^{\\prime}$ be the affine curve over $\\mathcal{O}_v$ obtained from $\\mathcal{C}$ by dehomogenising with respect to $x$. Then the composition  \\eqref{eq:comp_alpha} agrees with composing the restriction map $H^1_{\\dR}(\\mathcal{C}\/\\mathcal{O}_v)\\to H^1_{\\dR}(\\mathcal{C}^{\\prime}\/\\mathcal{O}_v)$ with the expansion in $t$-map $H^1_{\\dR}(\\mathcal{C}^{\\prime}\/\\mathcal{O}_v)\\to H^1_{\\dR}(\\hat{\\mathcal{C}}_{\\infty}\/\\mathcal{O}_v)$, and hence with $\\beta_{\\infty}$. Comparing dimensions, we obtain the proposition. \n\\end{proof}\n\n\\begin{cor}\nUnder the assumptions of Proposition \\ref{prop:Blakestad_space_unit_root}, let  $P\\in C(\\mathcal{O}_v)$ and $t$ be a coordinate for $(\\hat{C}_{P},P)$. Then the expansions of the formal integrals $\\int \\eta_1^{(b)}(t)$  and $\\int \\eta_2^{(b)}(t)$ have coefficients in $\\mathcal{O}_v$.\n\\end{cor}\n\n\\begin{rmk}\nWe will refer to the $v$-adic sigma function of Theorem \\ref{thm:Blakestad_main} as the \\emph{canonical $v$-adic sigma function}. There are two reasons why we care about Proposition \\ref{prop:Blakestad_space_unit_root}. The first one is that it makes the computation of the canonical $v$-adic sigma function a lot more efficient (at least when $K_v \\cong \\mathbb{Q}_p$), since we can compute the unit root subspace using Kedlaya's algorithm: see \\S \\ref{subsec:implementation_canonical_sigma} for details.  Secondly, in the following section, we will use $v$-adic sigma functions to define a $p$-adic height on $J(\\overline{\\mathbb{Q}})$. Now, there are other $p$-adic height constructions in the literature that depend on a choice of space $W_v\\in \\Is(C\/K_v)$, at every $v\\mid p$. For example, the $p$-adic height of Coleman--Gross \\cite{ColemanGross}. In this case, the unit root eigenspace of Frobenius is the ``canonical'' choice of subspace and the corresponding height appears in a $p$-adic analogue of the Birch and Swinnerton-Dyer conjecture \\cite{BaMuSt12}. In Section \\ref{sec:Colmez}, we compare our $p$-adic height defined in terms of sigma functions with the $p$-adic height of Coleman--Gross. In light of Proposition \\ref{prop:Blakestad_space_unit_root}, the comparison restricts to a comparison between the canonical heights. See also Remark \\ref{rmk:comparison_canonical}.\n\nOn the other hand, there are situations where it is preferable to consider the naive $v$-adic sigma function. Indeed, unlike the canonical $v$-adic sigma function, the naive one does not require any assumption on the reduction of $C$ and $J$ at $v$. Furthermore, it is easier to compute. Finally, its tight link with the complex genus 2 sigma function makes it useful in proofs, as a lot of its properties follow formally from the properties of the complex sigma function.\n\\end{rmk}\n\n\\begin{rmk}\\label{rmk:CM_case}\nIn analogy with the elliptic curve situation \\cite{bernardi}, we could ask if the matrix $b$ has algebraic entries, independent of $p$ and $v$, in the special case where $J$ has complex multiplication. See \\cite[Propositions 2.1, 2.14]{BannaiKobayashi} for results in this direction. \nIndeed, under the assumptions of \\emph{loc.\\ cit.}, let $q(z)$ be the unique quadratic form such that $\\theta(z) = \\sigma(z)e^{2\\pi i q(z)}$ is a normalised theta function in the sense of \\cite[p.\\,87]{Lan82}. \nThen, by \\cite[Proposition 2.1]{BannaiKobayashi}, the coefficients of the Taylor expansion of $\\theta(z)$ around $0$ are algebraic. But since $\\sigma(z)$ also has algebraic coefficients, it follows that $2\\pi i q(z) = \\frac{1}{2}\\sum_{1\\leq i,j\\leq 2} c_{ij} z_iz_j$ with $c_{12}= c_{21}$ and algebraic $c_{ij}$. \n Finally by \\cite[Proposition 2.14]{BannaiKobayashi}, the coefficients of $\\theta(\\mathcal{L}(T))$ are integral. See \\S \\ref{subsec:eg_large_p} for an explicit example. \n\\end{rmk}\n\n\n\\section{$p$-adic heights}\\label{sec:padic_hts}\nWe keep the notation of Section \\ref{sec:sigma_functions}: $K$ is a number field and the coefficients of the defining equation for $C$ lie in its ring of integers. Fix a prime number $p$ and a continuous idele class character\n\\begin{equation*}\n\\chi = \\sum_v\\chi_v\\colon \\mathbb{A}_K^{\\times}\/K^{\\times} \\to \\mathbb{Q}_p.\n\\end{equation*}\nBy continuity, $\\chi_v$ is identically zero if $v$ is an archimedean place, and $\\chi_v$ is identically zero on $\\mathcal{O}_v^{\\times}$ if $v$ is a non-archimedean place not dividing $p$. See \\cite[\\S 2.1]{QCnfs} for more properties and for a discussion on how to construct the finite-dimensional $\\mathbb{Q}_p$-vector space of all such $\\chi$, for a given $K$. If $L$ is a finite extension  of $K$, we denote by $\\chi_L$ the continuous idele class character for $L$ obtained by composing $\\chi$ with the idele norm $N_{L\/K}\\colon \\mathbb{A}_L^{\\times}\\to \\mathbb{A}_K^{\\times}$. Note that, if $w$ is a place of $L$ above the place $v$ of $K$ and $x\\in K_v^{\\times}$, then $\\chi_{L,w}(x) = [L_w:K_v]\\chi_v(x)$. \n\nLet $J_{\\Theta}\\colonequals J \\setminus \\Supp(\\Theta)$. The goal of this section is to define a quadratic form \n\\begin{equation*}\nh_p\\colon J(\\overline{\\mathbb{Q}}) \\to \\mathbb{Q}_p,\n\\end{equation*}\ndependent on $\\chi$. We will do so in the following way. Suppose that $P$ is a point in $J_{\\Theta}(L)$, for some finite extension $L$ of $K$. Given a non-archimedean place $w$ of $L$, denote by $n_w$ the degree of the field extension $L_w\/\\mathbb{Q}_{\\ell}$, where $\\ell$ is the rational prime below $w$. Then \n\\begin{equation*}\nh_p(P) = \\frac{1}{[L:\\mathbb{Q}]}\\sum_{w}n_w\\lambda_w(P),\n\\end{equation*}\nwhere the sum runs over all the non-archimedean places $w$ of $L$ and $\\lambda_w\\colon J_{\\Theta}(L_w) \\to \\mathbb{Q}_p$ is a ``quasi--quadratic'' function, dependent on $\\chi_{L,w}$, which we call a $p$-adic N\\'eron function at $w$ with respect to $2\\Theta$. The reason for this terminology will be explained in \\cite{BKM22}; see also Remark \\ref{rmk:neron_away_p} below.\n\nBefore embarking on defining $p$-adic N\\'eron function, we state a useful lemma. See also Appendix \\ref{app:stevan} for more general, related, results. \n\\begin{lemma}\\label{lemma:exists_m}\nLet $P\\in J(L_w)$ be a non-torsion point and let $H$ be a finite index subgroup of $J(L_w)$. Then there exists a positive integer $m$ such that $mP\\in H\\setminus \\Supp(\\Theta)$.\n\\end{lemma}\n\\begin{proof}\nSince $H$ has finite index in $J(L_w)$ and $P$ is non-torsion, there exists $m^{\\prime}$ such that $Q\\colonequals m^{\\prime}P\\in H\\setminus\\{0\\}$. If $Q \\in\\Supp(\\Theta)$, then $Q = [Q_1-\\infty]$ for a non-Weierstrass point $Q_1\\in C(L_{w})$. Thus, $2Q = [2Q_1-2\\infty]\\in H\\setminus\\Supp(\\Theta)$.\n\\end{proof}\nIf $v$ is a place of $K$ lying above $p$, and $w\\mid v$, the $p$-adic N\\'eron function $\\lambda_w$ will be defined in terms of a $v$-adic sigma function from \\S \\ref{subsec:infty_sigma}. Recall from Proposition \\ref{prop:bijection_sigma_is} that the set of $v$-adic sigma functions is in bijection with the set $\\Is(C\/K_v)$. We thus fix, once and for all, a choice of subspace $W_v\\in \\Is(C\/K_v)$, at every $v\\mid p$. \n\\subsection{$p$-adic N\\'eron functions}\n\\label{subsec:local_Neron}\nLet $L$ be a finite extension of $K$, let $v$ be a place of $K$ and let $w$ be a place of $L$ above $v$.\n\nAssume first that $v\\mid p$. \nOur choice of subspace $W_v$ induces a choice of $v$-adic sigma function $\\sigma_{v}$. By Proposition \\ref{prop:finite_index_subgp}, $\\sigma_v$ converges on a subgroup $H_v$, dependent on $W_v$, of finite index in $J(K_v)$. Clearly it also converges on a finite index subgroup $H_{w}$ of $J(L_w)$.\n\n\n\n\\begin{mydef}\\label{def:Neron_fct_above_p}\nThe \\emph{(local) $p$-adic N\\'eron function at $w\\mid v\\mid p$}, with respect to the divisor $2\\Theta$, the character $\\chi$ and the subspace $W_v$, is the function\n\\begin{equation*}\n\\lambda_w\\colon J_{\\Theta}(L_w) \\to \\mathbb{Q}_p,\n\\end{equation*}\ndefined as follows. If $P\\in J_{\\Theta}(L_w)$ is a non-torsion point and $m$ is a positive integer such that $mP\\in H_w\\setminus \\Supp(\\Theta)$, then \n\\begin{equation*}\n\\lambda_w(P) = -\\frac{2}{n_w m^2}\\cdot \\chi_{L,w}\\left(\\frac{\\sigma_v(T(mP))}{\\phi_m(P)}\\right);\n\\end{equation*}\nwe may extend the definition of $\\lambda_w$ to torsion-points in $J_{\\Theta}(L_w)$ by continuity. \n\\end{mydef}\n\nBy Lemma \\ref{lemma:exists_m}, an integer $m$ satisfying the conditions of Definition \\ref{def:Neron_fct_above_p} always exists, and by Proposition \\ref{prop:finite_index_subgp}\\thinspace{}\\ref{prop:finite_index_subgp:3} and Equation \\eqref{eq:div_quad}, the definition of $\\lambda_w$ is independent of the choice of $m$. Moreover, by Proposition \\ref{prop:finite_index_subgp}\\thinspace{}\\ref{prop:finite_index_subgp:2} and by Equation \\eqref{eq:divisor_div_poly}, $\\sigma_v\\circ T\\circ m$ and $\\phi_m$ have the same zeros on $J_{\\Theta}$. This is what allows to extend to torsion-points.  \n\nAssume now that $v\\nmid p$. As the name suggests, we are looking for a $\\mathbb{Q}_p$-valued analogue of a classical real-valued N\\'eron function of divisor $2\\Theta$ on $J_{\\Theta}(L_w)$ (in the sense of \\cite[Chapter 11]{Lang_diophantine_geometry},\\cite[\\S B.9]{Hindry_silverman}, \\cite[\\S 7]{Uchida}, and in particular \\cite{Uchidacanloc} for the genus $2$ case). In this case (i.e.\\ when $w\\nmid p$), it is possible to construct such a $p$-adic function starting from a suitable real-valued one, just by replacing a real-valued continuous homomorphism $L_w^{\\times}\\to\\mathbb{R}$ with $\\chi_{L,w}$. We refer to \\cite{BKM22} for details. \n\nWe take a slightly different approach here. Denote by $J_{1,w}(L_w)$ the model-dependent kernel of reduction for $J$ over $L_w$ (as in \\eqref{eq:kernel_red}).\n\n\\begin{lemma}\\label{lemma:equals_naive}\nFor $P\\in J_{1,w}(L_w)\\setminus \\Supp(\\Theta)$, let $\\lambda_w(P) = -\\frac{2}{n_w}\\chi_{L,w}(T_1(P))$. Then \n\\begin{equation}\\label{eq:equals_naive}\n\\lambda_{w}(P) = -\\frac{1}{n_w}\\chi_{L,w}(\\max_{i,j}\\{|X_{ij}(P)|_w, 1\\}).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor such $P$,\n\\begin{equation*}\n\\chi_{L,w}(T_1(P)) = \\chi_{L,w}(|T_1(P)|_w^{-1}) = \\frac{1}{2}\\chi_{L,w}(\\max_{i,j}\\{|X_{ij}(P)|_w, 1\\}),\n\\end{equation*}\n (cf.\\ the expansions \\eqref{eq:Xij_exp}). \n\\end{proof}\n\n\\begin{cor} \\label{cor:quasi_quad_in_formal}\nLet $\\lambda_w\\colon J_{1,w}(L_w)\\setminus \\Supp(\\Theta)\\to \\mathbb{Q}_p$ as in Lemma \\ref{lemma:equals_naive}. Then\n\\begin{enumerate}[label=(\\roman*)]\n\\item for all integers $m$ such that $mP\\not\\in \\Supp(\\Theta)$,\n\\begin{equation*}\n\\lambda_w(mP) = m^2\\lambda_w(P) - \\frac{2}{n_w}\\chi_{L,w}(\\phi_{|m|}(P)).\n\\end{equation*}\n\\item for all $P,Q\\in J_{1,w}(L_w)\\setminus \\Supp(\\Theta)$ such that $P+Q,P-Q\\not\\in \\Supp(\\Theta)$, \n\\begin{align*}\n\\lambda_w(P+Q) + \\lambda_w(P-Q) = 2\\lambda_w(P) + 2\\lambda_w(Q)\\\\\n - \\frac{2}{n_w}\\chi_{L,w}(-X_{11}(P)+X_{11}(Q)-X_{12}(P)X_{22}(Q) + X_{22}(P)X_{12}(Q)).\n\\end{align*}\n\\end{enumerate}\n\\end{cor}\n\\begin{proof}\nThe equality of Lemma \\ref{lemma:equals_naive} implies this, in view of \\cite[Theorem 4.1]{Stoll} (see also \\cite[Corollary 7.2]{Uchidacanloc}), \\cite[Proposition 6.2]{muller_de_jong} and \\cite[Theorem 7.5]{Uchida}.\n\\end{proof}\n\n\\begin{mydef}\\label{def:Neron_fct_away_p}\nThe \\emph{(local) $p$-adic N\\'eron function at $w\\mid v\\nmid p$}, with respect to the divisor $2\\Theta$ and the character $\\chi$, is the function\n\\begin{equation*}\n\\lambda_w\\colon J_{\\Theta}(L_w) \\to \\mathbb{Q}_p,\n\\end{equation*}\ndefined as follows. If $P\\in J_{\\Theta}(L_w)$ is a non-torsion point and $m$ is a positive integer such that $mP\\in J_{1,w}(L_w)\\setminus \\Supp(\\Theta)$, then \n\\begin{equation}\\label{eq:lambdaw_away_p}\n\\lambda_w(P) = -\\frac{2}{n_w m^2}\\cdot \\chi_{L,w}\\left(\\frac{T_1(mP)}{\\phi_m(P)}\\right);\n\\end{equation}\nwe may extend the definition of $\\lambda_w$ to torsion-points in $J_{\\Theta}(L_w)$ by continuity. \n\\end{mydef}\nSimilarly to Definition \\ref{def:Neron_fct_above_p}, also in this case Lemma \\ref{lemma:exists_m} guarantees the existence of $m$ and the extension to torsion points is meaningful. By Corollary \\ref{cor:quasi_quad_in_formal} and Equation \\eqref{eq:div_quad}, the definition is independent of $m$.  \n\n\\begin{rmk}\\label{rmk:neron_away_p}\n\\leavevmode\n\\begin{enumerate}[label=(\\roman*)]\n\\item  For all $P\\in J_{\\Theta}(L_w) $, we have $\\frac{n_w\\lambda_w(P)}{\\chi_{L,w}(\\ell)}=: \\lambda_w^{\\prime}(P)\\in \\mathbb{Q}$, where $\\ell$ is the rational prime below $w$. In fact, the real-valued N\\'eron function of divisor $2\\Theta$ of \\cite{Uchidacanloc} at $P$ is $\\lambda_w^{\\prime}(P)\\log|\\ell|_w$, where $\\log$ is the real logarithm.\n\\item\\label{rmk:neron_away_p_good_reduction} If $C$ has good reduction at $w$, the equality of Lemma \\ref{lemma:equals_naive} holds for all $P\\in J_{\\Theta}(L_{w})$, since in this case the right hand side of \\eqref{eq:equals_naive} defines a function satisfying Corollary \\ref{cor:quasi_quad_in_formal} on the whole of $J_{\\Theta}(L_w)$ (cf.\\ \\cite[Lemma 1]{FlynnSmart} for the case $K=L=\\mathbb{Q}$ and \\cite[Proposition 5.2]{Stoll} for the general case; see also Section \\ref{sec:application_bihyper} and, in particular, Theorem \\ref{thm:properties_mu}). \n\\item\\label{rmk:neron_unramified} Equivalently, we could have used Definition \\ref{def:Neron_fct_above_p} to define the local $p$-adic N\\'eron function also in the case $v\\nmid p$, since $\\chi_{L,w}$ vanishes on $\\mathcal{O}_{w}^{\\times}$ and $\\sigma_v(T) = T_1 (1+ O(T_1,T_2))$.  We preferred stating the simplified formula, which does not involve the choice of a subspace $W_v$. Conversely, if $v\\mid p$ and $\\chi$ is unramified at $v$, Definition \\ref{def:Neron_fct_above_p} simplifies to Definition \\ref{def:Neron_fct_away_p}.\n\\end{enumerate}\n\\end{rmk}\n\nWe now state the properties satisfied by our local $p$-adic N\\'eron function at $w$, regardless of whether $w$ divides or does not divide $p$. These are completely analogous to the properties of the real-valued N\\'eron function: see \\cite[Theorem 5.3, Theorem 5.6]{Uchidacanloc} for the genus $2$ case, and \\cite[Theorem 7.5]{Uchida} for arbitrary genus (in the latter, the N\\'eron function is with respect to $\\Theta$, rather than $2\\Theta$).\n\\begin{prop}\\label{prop:properties_neron_fcts}\nFor any place $w$ of $L$, the local $p$-adic N\\'eron function at $w$ has the following properties:\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{it:extension} If $F$ is a finite extension of $L$ and $w^{\\prime}$ a place of $F$ above $w$, then $\\lambda_{w^{\\prime}}$ restricts to $\\lambda_w$ on $J_{\\Theta}(L_w)$.\n\\item\\label{it:quasi_quadraticity} For all $P\\in J_{\\Theta}(L_w)$ and for all integers $m$ such that $mP\\in J_{\\Theta}(L_w)$,\n\\begin{equation*}\n\\lambda_w(mP) = m^2\\lambda_w(P) - \\frac{2}{n_w}\\chi_{L,w}(\\phi_{|m|}(P)).\n\\end{equation*}\n\\item\\label{it:quasi_parallelogram}  For all $P,Q\\in J_{\\Theta}(L_w)$ such that $P+Q,P-Q\\in J_{\\Theta}(L_w)$,\n\\begin{align*}\n\\lambda_w(P+Q) + \\lambda_w(P-Q) = 2\\lambda_w(P) + 2\\lambda_w(Q)\\\\\n - \\frac{2}{n_w}\\chi_{L,w}(-X_{11}(P)+X_{11}(Q)-X_{12}(P)X_{22}(Q) + X_{22}(P)X_{12}(Q)).\n\\end{align*}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n\\ref{it:extension} is clear. \nBy Proposition \\ref{prop:finite_index_subgp}\\thinspace{}\\ref{prop:finite_index_subgp:3} and Corollary \\ref{cor:quasi_quad_in_formal}, properties \\ref{it:quasi_quadraticity} and \\ref{it:quasi_parallelogram} hold in a subgroup of finite index $H_w$. Since we used \\ref{it:quasi_quadraticity} to extend $\\lambda_w$ outside of this subgroup, part \\ref{it:quasi_quadraticity} holds everywhere in view of \\eqref{eq:div_quad}. As regards \\ref{it:quasi_parallelogram}, assume $P,Q,P+Q,P-Q$ are non-torsion points. Since $H_w$ has finite index in $J(L_w)$, we can find a positive integer $m$ such that $mR\\in H_w\\setminus\\{0\\}$ for each  $R\\in \\{P,Q,P+Q,P-Q\\}$. By Lemma \\ref{lemma:stevan}, we can also ensure, possibly after replacing $m$ with a multiple, that each $mR$ does not lie on $\\Theta$. Then \\ref{it:quasi_parallelogram} holds when $P$ and $Q$ are replaced by $mP$ and $mQ$. \nFinally, by \\eqref{eq:div_par}, the equality \\ref{it:quasi_parallelogram} holds for $P,Q$. The torsion case follows by continuity.\n\\end{proof}\n\n\\subsection{Global height}\n\\begin{mydef}\\label{def:global_ht}\nThe \\emph{global $p$-adic height} for $J$ with respect to the character $\\chi$ and the subspaces $W_v\\in \\Is(C\/K_v)$ (at $v\\mid p$) is the function\n\\begin{equation*}\nh_p\\colon J(\\overline{\\mathbb{Q}}) \\to \\mathbb{Q}_p,\n\\end{equation*}\ndefined as follows. If $P$ belongs to $J_{\\Theta}(L)$, for some finite extension $L$ of $K$, then\n\\begin{equation}\\label{eq:hp_sum}\nh_p(P) = \\frac{1}{[L:\\mathbb{Q}]}\\sum_{w}n_w\\lambda_w(P),\n\\end{equation}\nwhere the sum runs over all the non-archimedean places $w$ of $L$ and $\\lambda_w$ is the local $p$-adic N\\'eron function of Definition \\ref{def:Neron_fct_above_p} if $w\\mid p$ and of Definition \\ref{def:Neron_fct_away_p} if $w\\nmid p$. Since by Proposition \\ref{prop:properties_neron_fcts}\\thinspace{}\\ref{it:quasi_quadraticity}, $h_p(mP) = m^2h_p(P)$ for $P,mP\\in J_{\\Theta}(L)$, we extend $h_p$ to $J(L)$ using Lemma \\ref{lemma:exists_m} and by requiring that\n\\begin{equation*}\nh_p(mP) = m^2h_p(P)\\qquad  \\text{continues to hold and}\\qquad h_p(0) = 0.\n\\end{equation*}\n\\end{mydef}\nBy Remark \\ref{rmk:neron_away_p}\\thinspace{}\\ref{rmk:neron_away_p_good_reduction}, the sum in \\eqref{eq:hp_sum} is finite and by Proposition \\ref{prop:properties_neron_fcts}\\thinspace{}\\ref{it:extension}, it is independent of the choice of $L$. \nOur extension of the $p$-adic N\\'eron functions to torsion points together with the vanishing of $\\chi_L$ on $L^{\\times}$ also guarantees that $h_p(P) = 0$ for a torsion point $P\\in J_{\\Theta}(L)$. Therefore the extension to $J(L)$ is well-defined. \n\n By Proposition \\ref{prop:properties_neron_fcts}\\thinspace{}\\ref{it:quasi_parallelogram}, the global $p$-adic height also satisfies the parallelogram law:\n \\begin{equation*}\n h_p(P+Q) + h_p(P-Q) = 2h_p(P) + 2h_p(Q).\n \\end{equation*}\n Therefore it induces a symmetric bilinear pairing $J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})\\to\\mathbb{Q}_p$.\n\n\\begin{rmk}\nBy \\cite{BKM22}, if $p$ and $C$ satisfy the assumptions of Theorem \\ref{thm:Blakestad_main} and if we pick, for every $v$ at which $\\chi$ is ramified, the canonical $v$-adic sigma function of Blakestad, the $p$-adic height $h_p$ coincides with the canonical global Mazur--Tate height \\cite{mazur-tate}.\n\\end{rmk}\n\n\n\\begin{rmk} Assume $K=\\mathbb{Q}$. Extending work of Perrin--Riou for elliptic curves \\cite{Perrin-Riou-comptes, Perrin-Riou}, Trip has recently defined a quadratic form $h_p^{\\naive}\\colon J(\\mathbb{Q})\\to \\mathbb{Q}_p$ as a limit of a naive height: see \\cite[Theorem 3.3.1, \nDefinition 3.3.17, Theorem 3.3.19]{Trip} (in her thesis, $h_p^{\\naive}$ is denoted $h_p$). Let $\\chi$ be the cyclotomic character for $\\mathbb{Q}$  (cf.\\ \\S \\ref{subsec:implementation_Neron_fcts}), let $W_p\\in \\Is(C\/\\mathbb{Q}_p)$ be the subspace corresponding to the naive sigma function, and let $h_p$ be the global $p$-adic height from above with respect to these choices.  Then Trip shows that $h_p(u) = h_p^{\\naive}(u)$ for all $u\\in J(\\mathbb{Q})$ \\cite[Theorem 3.3.25]{Trip}.\n\\end{rmk}\n\n\\section{Local $p$-adic N\\'eron functions vs Colmez's Green functions}\\label{sec:Colmez}\nIn the first part of this section (\\S\\S \\ref{subsec:Colm_int}--\\ref{subsec:Colm_heights}) we review some of the results of Colmez\\cite{Colm96}. To be more precise, in \\S \\ref{subsec:Colm_int} we sketch Colmez's construction of a $p$-adic integration theory on smooth algebraic varieties over a $p$-adic field, which satisfies ``natural'' properties and does not require any assumption on the reduction. Similar techniques can be used to attach to divisors on abelian varieties certain $p$-adic Green functions, which are, essentially, double $p$-adic integrals. In \\S \\ref{subsec:Colm_curves}, single $p$-adic integrals of differentials of the third kind on curves are related to $p$-adic Green functions of divisor $\\Theta$ on their Jacobians. \\S \\ref{subsec:Colm_int} and \\S \\ref{subsec:Colm_curves} are entirely expository. The results of \\S \\ref{subsec:Colm_curves} are applied in \\S \\ref{subsec:Colm_heights} to define an adelic height on Jacobians of curves over a number field $K$, whose component at $p$ coincides with the $p$-adic height of Coleman--Gross and Besser \\cite{ColemanGross, BesserPairing}. Such adelic heights are defined in \\cite{Colm96}, but we phrase Colmez's results in a slightly more general form: in particular, our adelic heights depend on the choice, at every place $p$ of $\\mathbb{Q}$, of a continuous $\\mathbb{Q}_p$-valued idele class character for $K$.\n\nIn \\S \\ref{subsec:comparison}, we then restrict to the situation of Section \\ref{sec:padic_hts} and compare our local N\\'eron functions to Colmez's Green functions of divisor $\\Theta$: see Theorem \\ref{thm:comparison} below. In view of \\S \\ref{subsec:Colm_heights}, we also obtain, in our odd degree genus $2$ setting, a formula relating the Coleman--Gross local $p$-adic height at $v$ with our $p$-adic N\\'eron function at $v$ (Corollary \\ref{cor:lambda_eq_CG}).\n\nWe use this to show that the global $p$-adic height $h_p$ of Definition \\ref{def:global_ht} is equal to the Coleman--Gross global $p$-adic height (Corollary \\ref{cor:global_CG_same_as_this}).\n   \n\\begin{rmk}\nWe have chosen to devote considerable space to summarising Colmez's results for a few reasons. First, we crucially use a lot of them (and their proofs). Secondly, in order to state our comparison results to a larger degree of generality, we need Colmez's height construction to be extended to  arbitrary idele characters. Finally, we believe it would be difficult for the reader to contextualise the results of \\S \\ref{subsec:comparison} without having some familiarity with \\cite{Colm96}.\n   \\end{rmk}\n   \n\n\n\\subsection{Colmez's $p$-adic integration theory and Green functions}\\label{subsec:Colm_int}\nLet $K$ be a closed subfield of $\\mathbb{C}_p$ (for instance, a finite extension of $\\mathbb{Q}_p$). There is a unique locally analytic function $\\mathop{\\mathrm{Log}}\\nolimits\\colon \\mathcal{O}_K^{\\times}\\to K$ satisfying:\n\\begin{equation}\\label{eq:Log_colmez}\nd\\mathop{\\mathrm{Log}}\\nolimits{t} = \\frac{dt}{t}, \\qquad \\text{and}\\qquad \\mathop{\\mathrm{Log}}\\nolimits(xy) = \\mathop{\\mathrm{Log}}\\nolimits(x) + \\mathop{\\mathrm{Log}}\\nolimits(y), \\quad \\text{for all } x,y\\in \\mathcal{O}_K^{\\times}.\n\\end{equation}\nBy viewing $\\mathop{\\mathrm{Log}}\\nolimits{p}$ as a variable, we can extend $\\mathop{\\mathrm{Log}}\\nolimits$ to a homomorphism $\\mathop{\\mathrm{Log}}\\nolimits\\colon K^{\\times}\\to K_{\\st} \\colonequals K[\\mathop{\\mathrm{Log}}\\nolimits{p}]$. \nOne of the main results of \\cite{Colm96} is the development of a $K_{\\st}$-valued theory of $p$-adic integration of closed differential $1$-forms on smooth geometrically connected algebraic varieties over $K$, without any assumption on the reduction.  Given such a variety $X$, recall that a rational closed differential 1-form is \n\\begin{enumerate}[label = (\\roman*)]\n\\item of the \\emph{first kind} if it is holomorphic;\n\\item of the \\emph{second kind} if its residue divisor is zero;\n\\item of the \\emph{third kind} if it has at most simple poles and the residue at each pole is an integer. \n\\end{enumerate}\nSee \\cite[Lemme I.1.12, Remarque\nI.1.18\\thinspace{}(ii)]{Colm96} for a precise definition of residue divisor. Any closed differential $1$-form is a linear combination of differentials of the second and third kind  \\cite[Corollaire I.1.16]{Colm96}.\n\nLet $K(X)$ be the field of rational functions on $X$ and $K(X)_{\\st}$ the ring generated by $K(X)$ and $\\mathop{\\mathrm{Log}}\\nolimits(f)$, for $f\\in K(X)^{\\times}$. \nA \\emph{locally log-meromorphic} function is a function which locally (with respect to the $p$-adic topology) can be written as a polynomial in logarithms of analytic functions with meromorphic coefficients. \n\n\\begin{thm}[{\\hspace{1sp}\\cite[Th\\'eor\\`eme 1]{Colm96}}]\\label{thm:uniquetheory}\nThere exists a unique theory of integration on smooth algebraic varieties over $K$ satisfying the following properties, for all such varieties $X$, closed differential $1$-forms $\\omega$, $\\omega_1$ and $\\omega_2$ on $X$, points $a,b,c\\in X(K)$ and scalars $\\mu_1,\\mu_2\\in K$:\n\\begin{enumerate}[label = (\\roman*)]\n\\item \\label{it:loc_an} $F(x) = \\int_{a}^x \\omega$ is a locally log-meromorphic function of $x\\in X(K)$ with values in $K\\oplus K \\mathop{\\mathrm{Log}}\\nolimits{p}$ satisfying $dF = \\omega$; \n\\item $\\int_a^b (\\mu_1\\omega_1 + \\mu_2\\omega_2) = \\mu_1 \\int_a^b\\omega_1 +\\mu_2 \\int_a^b\\omega_2$;\n\\item $\\int_a^c\\omega = \\int_a^b\\omega + \\int_b^c \\omega$;\n\\item\\label{it:exact} If $\\omega = dg$ (with $g\\in K(X)_{\\st}$) is exact, then $\\int_a^b \\omega = g(b) - g(a)$;\n\\item\\label{change_of_vars} If $f\\colon X\\to Y$ is a morphism of smooth varieties over $K$, $\\eta$ is a closed differential $1$-form on $Y$, then $\\int_a^b f^{*}\\eta = \\int_{f(a)}^{f(b)}\\eta$.\n\\end{enumerate}\n\\end{thm}\n\nThe strategy is to first consider the case where $X$ is an abelian variety, and to then use an Albanese morphism combined with functoriality \\ref{change_of_vars} to extend to general varieties. In view of \\cite[Remarque\n2\\thinspace{}(iv)]{Colm96}, it is essentially enough to consider the case $K=\\mathbb{C}_p$.\n\nAssume now that $X$ is an abelian variety over $K$. We give a rough summary of Colmez's construction in this case. \n\nA holomorphic differential $\\omega$ on $X$ is translation-invariant. Therefore, by \\ref{change_of_vars}, the primitive of $\\omega$ vanishing at $0$ must be a group homomorphism $X(K)\\to K$.  We can construct such a primitive as follows. There is an open subgroup $V$ of $X(K)$ such that the unique formal primitive $\\mathcal{L}_{\\omega}$ of $\\omega$ that satisfies $\\mathcal{L}_{\\omega}(0) = 0$ converges on $V$ and is a group homomorphism $V\\to K$. \n Since $X(K)\/V$ is torsion \\cite[{Th\\'eor\\`eme II.1.9}]{Colm96}, we can extend $\\mathcal{L}_{\\omega}$ to a locally analytic group homomorphism $\\mathcal{L}_{\\omega}\\colon X(K)\\to K$ by insisting that $\\mathcal{L}_{\\omega}(na) = n\\mathcal{L}_{\\omega}(a)$ for all integers $n$ and points $a$. \nConversely, given a locally analytic group homomorphism $\\mathcal{G}\\colon X(K)\\to K$, its differential is a translation-invariant, and hence holomorphic, differential. That is, the map $\\mathcal{G}\\mapsto d\\mathcal{G}$ induces an isomorphism between the $K$-vector space of locally analytic group homomorphisms $X(K)\\to K$ (which we call \\emph{one-logarithms} of $X$) and $H^0(X,\\Omega^1)$ \\cite[Lemme II.1.13]{Colm96}.\n\n\\begin{example}\nWhen $X$ is the Jacobian of the curve $C$ of \\S \\ref{subsec:formal}, and $\\omega = \\Omega_i$ for $i\\in\\{1,2\\}$, the primitive $\\mathcal{L}_{\\Omega_i}$ is an extension to $J(K)$ of the $i$-th component $\\mathcal{L}_i$ of the strict logarithm $\\mathcal{L}$. See Proposition \\ref{prop:exp_exp_log} (and Lemma \\ref{lemma:subs_exponential}) for convergence properties of $\\mathcal{L}_i$. \nThe $K$-vector space of one-logarithms of $X$ is $2$-dimensional and spanned by $\\mathcal{L}_{\\Omega_1}$ and $\\mathcal{L}_{\\Omega_2}$.\n\\end{example}\n\nThe case of differentials of the second and third kind is more involved. Given a positive integer $n$ and a subset $I$ of $\\{1,\\dots, n\\}$, let \n\\begin{equation*}\nm_I\\colon X^{n+1}\\to X, \\qquad (x,h_1,\\dots,h_n)\\mapsto x + \\sum_{i\\in I} h_i\n\\end{equation*}\nand given $\\alpha$, a differential form, a function, or a divisor on $X$, set\n\\begin{equation*}\n\\Delta^{[n]}\\alpha = \\sum_{I\\subseteq \\{1,\\dots, n\\}} (-1)^{n-|I|} m_I^{*} \\alpha.\n\\end{equation*}\nBy the theorem of the square \\cite[Propositions I.3.14, I.3.15]{Colm96}, if $\\omega$ is a differential form of the second or third kind on $X$, then $\\Delta^{[2]}\\omega$ is exact on $X^{3}$. \nLet $F_{\\Delta^{[2]}\\omega}$ be the element of $K(X^3)_{\\st}$ satisfying $dF_{\\Delta^{[2]}\\omega} = \\Delta^{[2]}\\omega$ and $F_{\\Delta^{[2]}\\omega}(x,0,0) = 0$. By \\ref{change_of_vars} and the choice of $F_{\\Delta^{[2]}\\omega}$, we must have\n\\begin{equation*}\n\\Delta^{[2]}\\biggl(\\int \\omega\\biggr) = F_{\\Delta^{[2]}\\omega}.\n\\end{equation*}\nColmez proves a sufficient criterion for a locally log-meromorphic function $g$ on $X(\\mathbb{C}_p)^{n+1}$ to be of the form $\\Delta^{[n]}f$, for a locally log-meromorphic function $f$ on $X(\\mathbb{C}_p)$; when this applies, $f$ is unique up to addition by a polynomial of degree at most $n-1$ in the one-logarithms of $X$: see \\cite[Th\\'eor\\`eme II.1.16]{Colm96} for a precise statement. Applying this technical result to $g=F_{\\Delta^{[2]}\\omega}$, one solves for a primitive $f = \\int \\omega$ (see \\cite[Proposition II.1.17]{Colm96}). See \\S \\ref{subsec:diff_first_2nd_third} for explicit formulae when $X$ is the Jacobian of the curve $C$. \n\nTheorem \\ref{thm:uniquetheory} concerns single integration only. However, applying a similar strategy to the one we have just outlined for integrals of differentials of the second and third kind, Colmez also constructs certain $p$-adic Green functions on the abelian variety $X$, which are double integrals in disguise (see Proposition \\ref{prop:dpartial_second_kind} below).\n\nThe theorem of the cube states the following:\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition I.3.18]{Colm96}}]\\label{prop:Delta3_principal}\nIf $X$ is an abelian variety over an algebraically closed field of characteristic $0$ and $D$ is a divisor of $X$, then $\\Delta^{[3]}D$ is principal.\n\\end{prop}\nLet $D$ be a divisor of $X$ and let $f_D^{(4)}$ be the rational function on $X^4$ whose divisor is $\\Delta^{[3]}D$ and whose restriction to $(X^3\\times\\{0\\})\\cup (X^2\\times \\{0\\}\\times X) \\cup (X\\times\\{0\\}\\times X^2)$ is equal to the constant function $1$.  \nThe aforementioned criterion of \\cite[Th\\'eor\\`eme II.1.16]{Colm96} applied to $\\mathop{\\mathrm{Log}}\\nolimits f_D^{(4)}$ allows one to go from $X^4$ to $X$:\n\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition 4, Proposition II.1.19]{Colm96}}]\\label{prop:padicGreen}\nThere exists a locally log-meromorphic function $G_D$ on $X$, with values in $K\\oplus \\mathbb{Q}\\mathop{\\mathrm{Log}}\\nolimits{p}$, such that $\\Delta^{[3]}G_D = \\mathop{\\mathrm{Log}}\\nolimits{f_D^{(4)}}$. Moreover, $G_D$ is unique up to addition by a polynomial of degree at most $2$ in the one-logarithms of $X$, it has a logarithmic singularity along $D$ and it is locally analytic away from the support of $D$.\n\\end{prop}\n\n\\begin{mydef}\nA function $G_D$ satisfying Proposition \\ref{prop:padicGreen} is called a \\emph{Green function of divisor $D$}.\n\\end{mydef}\n\nThe following proposition tells us that the Green function $G_D$ is a sort of double $p$-adic integral:\n\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition II.1.20]{Colm96}}]\\label{prop:dpartial_second_kind}\nLet $\\partial$ be an invariant derivation on $X$. If $G_D$ is a Green function of divisor $D$, then $d(\\partial G_D)$ is a differential form of the second kind.\n\\end{prop}\n\n\\subsection{Colmez's theory on curves and Jacobians}\\label{subsec:Colm_curves}\nAs in \\S \\ref{subsec:Colm_int}, let $K$ be a closed subfield of $\\mathbb{C}_p$. Let $X$ be a smooth, projective, geometrically irreducible curve of positive genus $g$ over $K$ such that $X(K)$ is non-empty, let $J$ be its Jacobian, and given a fixed $P_0\\in X(K)$, consider the embedding $\\iota\\colon X\\xhookrightarrow{} J$ with base point $P_0$. This induces an isomorphism $\\iota^{*}\\colon H_{\\textup{dR}}^1(J)\\to H_{\\textup{dR}}^1(X)$. \n\n\\begin{rmk}\\label{rmk:from_X_to_C}\nWhen $X$ is the genus $2$ curve given explicitly by an equation of the form \\eqref{eq:Grant}, a lot of the notation that we are going to introduce now has already been introduced or fixed (e.g.\\ $\\iota$ is the embedding with respect to $P_0 = \\infty$). Since there is no clear advantage in specialising to that setting here, we keep the exposition of Colmez's results general. The rule to keep in mind when we return to the case $X = C$ in \\S \\ref{subsec:comparison} is that all the notation of this section should be specialised to that previously fixed for $C$.\n\\end{rmk}\n\nFix a basis $(\\Omega_1,\\dots, \\Omega_g)$ for $H^0(J,\\Omega^1)$ and denote by $(\\partial_1,\\dots,\\partial_g)$ its dual basis of invariant derivations. Consider the function $\\varpi\\colon X^g\\to J$, defined by $\\varpi(P_1,\\dots,P_g) = \\iota(P_1)+ \\cdots + \\iota(P_g)$. Then $\\varpi$ is surjective and induces a birational equivalence of the $g$-th symmetric power $X^{(g)}$ of $X$ with $J$. We call a point $u\\in J$ \\emph{generic} if there is a unique element of $X^{(g)}$ mapping to $u$ under the map induced by $\\varpi$ and the latter has distinct $P_i$. \nThe \\emph{theta divisor}, denoted $\\Theta$, is the divisor of $J$ which is the image of\n\\begin{equation*}\nX^{g-1}\\to J, \\qquad  (P_1,\\dots,P_{g-1} )\\mapsto - \\iota(P_1)\\cdots - \\iota(P_{g-1}).\n\\end{equation*}\nFor generic $u = \\iota(P_{1,u})+ \\cdots +\\iota(P_{g,u})$, we consider the following divisor on $X$ (cf.\\ \\cite[Lemma 6.7]{milneAV}):\n\\begin{align*}\nX_u &=\\sum_{P\\in \\iota^{-1}(u+ \\Theta)} P\n = P_{1,u}+\\dots+ P_{g,u}.\n\\end{align*}\nThe theta divisor is \\emph{symmetric}, i.e.\\ there exists a unique $w\\in J$ such that $x$ belongs to $\\Theta$ if and only if $w- x$ belongs to $\\Theta$ (by \\cite[Corollaire I.4.7]{Colm96} and Lefschetz principle). \nIf $G$ is a Green function of divisor $\\Theta$, so is $G_w(x) \\colonequals G(w-x)$ and we say that $G$ is \\emph{symmetric} if $G(x) = G_w(x)$. \n\nRecall that, by Proposition \\ref{prop:dpartial_second_kind}, $d(\\partial_i G)$ is a differential of the second kind on $J$.\n\n\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition II.2.4]{Colm96}}]\\label{prop:bijection}\nThe function that maps a Green function $G$ to the subspace of $H^1_{\\textup{dR}}(X)$ generated by $\\iota^*d(\\partial_1 G), \\dots, \\iota^* d(\\partial_g G)$ induces a bijection between the set of symmetric Green functions of divisor $\\Theta$ (up to addition by a constant) and the set $\\Is(X)$ of subspaces of $H^1_{\\textup{dR}}(X)$ which are complementary to $H^0(X,\\Omega^1)$ and isotropic for the cup product. \n\\end{prop}\n\nUsing this proposition and properties of Green functions, Colmez expresses integrals of differentials of the third kind on $X$ (as in Theorem \\ref{thm:uniquetheory}) in terms of pullbacks of symmetric Green functions for the theta divisor of $J$. We now explain how.\n\n\n\nGiven a differential $\\omega$ of the third kind on $X$, let $\\Res_P(\\omega)$ be the residue of $\\omega$ at $P$ and let\n\\begin{equation*}\n\\operatorname{Div}(\\omega) = \\sum_{P\\in X}\\Res_P(\\omega) P \\in \\operatorname{Div}^0(X).\n\\end{equation*}\nIn particular, \n\\begin{equation*}\n\\operatorname{Div}\\left(\\frac{df}{f}\\right) = \\div(f),\n\\end{equation*}\nwhere the latter is the divisor of a rational function, in the usual sense. \n \nFix an element $W\\in \\Is(X)$ and let $G_{\\Theta,W}$ be a corresponding symmetric Green function (by Proposition \\ref{prop:bijection}, this is unique up to addition by a constant); set $F_{i,W} = \\partial_i G_{\\Theta,W}$.  \nGiven generic $u,v\\in J$, consider the differential\n\\begin{equation}\n\\omega_u - \\omega_v = \\iota^*\\left(\\sum_{i=1}^g (F_{i,W}(x- u) - F_{i,W}(x-v)) \\Omega_i\\right).\n\\end{equation}\n\\begin{lemma}\n$\\omega_u - \\omega_v$ is of the third kind and\n\\begin{equation*}\n\\operatorname{Div}(\\omega_u - \\omega_v) = X_u - X_v.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSee \\cite[p.\\,92]{Colm96}.\n\\end{proof}\n\nWe consider the $\\mathbb{Z}$-module generated by the differentials of the form $\\omega_u -\\omega_v$ for $u,v\\in J$ generic points; we call a differential in this module a \\emph{differential of type $(1,1)$}. This definition depends on the choice of $W\\in \\Is(X)$.\n\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition II.2.9]{Colm96}}]\\label{prop:dec_third_kind}\nLet $\\omega$ be a differential of the third kind on $X$. Then there exist a unique holomorphic differential $\\omega^{1,0}$ and a unique differential $\\omega^{1,1}$ of type $(1,1)$ such that $\\omega = \\omega^{1,0} + \\omega^{1,1}$. Moreover,\n\\begin{enumerate}[label = (\\roman*)]\n\\item If $\\omega = \\frac{df}{f}$, then $\\omega^{1,0} = 0$.\n\\item\\label{it:dec_4} If $\\omega^{1,1} = \\sum_i n_i \\omega_{u_i}$, then the image of $\\operatorname{Div}(\\omega)$ in $J$ is $\\sum_i n_iu_i$ and \n\\begin{equation*}\n\\int^x \\omega = \\int^x \\omega^{1,0} + \\sum_i n_{i} G_{\\Theta,W}(\\iota(x)-u_i).\n\\end{equation*}\n\\end{enumerate}\n\\end{prop}\n\\begin{example}\nIf $\\operatorname{Div}(\\omega) = P_1 - P_2$, pick $Q_1, \\dots, Q_{g-1}$ such that $u = \\iota(P_1) + \\iota(Q_1) +  \\cdots + \\iota(Q_{g-1})$ and $v = \\iota(P_2) + \\iota(Q_1) + \\cdots + \\iota(Q_{g-1})$ are both generic. Then\n\\begin{equation*}\n\\operatorname{Div}(\\omega_u -\\omega_v) = X_u - X_v = P_1 - P_2;\n\\end{equation*}\nhence \n\\begin{equation*}\n\\omega - (\\omega_u - \\omega_v) = \\omega^{1,0}\\in H^0(X, \\Omega^1)\n\\end{equation*}\nand\n\\begin{equation*}\n\\int^x \\omega = \\int^x \\omega^{1,0} +G_{\\Theta,W}(\\iota(x)-u)- G_{\\Theta,W}(\\iota(x)-v).\n\\end{equation*}\nThis example should clarify that, although the differential $\\omega^{1,1}$ is unique, its decomposition as a sum $\\sum_i n_i\\omega_{u_i}$ is not. The choices of the generic points $u_i\\in J$ affect the logarithmic singularities of the individual terms $G_{\\Theta,W}(\\iota(x)-u_i)$; however, overall, $\\sum_i n_{i} G_{\\Theta,W}(\\iota(x)-u_i)$ is locally analytic outside $\\operatorname{Div}(\\omega)$.\n\\end{example}\nFor every $i\\in\\{1,\\dots,g\\}$, denote by $\\mathcal{L}_i$ the one-logarithm of $J$ whose differential is the basis element $\\Omega_i$ for $H^0(J,\\Omega^1)$. \n\n\\begin{mydef}\\label{def:LogJtilde}\nFor a differential $\\omega$ of the third kind on $X$, we define $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}(\\omega)\\in H^1_{\\textup{dR}}(X)$ as the class of the differential\n\\begin{equation*}\n \\omega^{1,0} + \\sum_{i=1}^g \\mathcal{L}_i(\\operatorname{Div}(\\omega))\\eta_i,\n\\end{equation*}\nwhere $\\omega^{1,0}$ is as in Proposition \\ref{prop:dec_third_kind} and, for every $i\\in \\{1,\\dots,g\\}$, $\\eta_i = \\iota^*d(\\partial_i G_{\\Theta,W})$. \n\\end{mydef}\nAs a corollary to Proposition \\ref{prop:dec_third_kind} and since $\\eta_i\\in W$ by construction, we obtain \n\\begin{lemma}\\label{lemma:prop_LogJtilde}\\leavevmode\n\\begin{enumerate}[label = (\\roman*)]\n\\item $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}(\\omega) = [\\omega]$ if $\\omega$ is holomorphic.\n\\item If $\\omega$ is of type $(1,1)$ with respect to $W$, then $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}(\\omega)$ belongs to $W$.\n\\item \\label{it:prop_LogJtilde3}$\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}\\left(\\frac{df}{f}\\right) = 0$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{rmk}\\label{rmk:logJtilde}\nThe function $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}$ is defined more conceptually in \\cite[Th\\'eor\\`eme II.2.11]{Colm96} and it is then shown to satisfy Definition \\ref{def:LogJtilde} and Lemma \\ref{lemma:prop_LogJtilde}. Denoting by $T(K)$ the group of differentials of the third kind on $X$, and by $T_{\\ell}(K)$ the subgroup of logarithmic differentials (i.e.\\ those of the form $\\frac{df}{f}$ for some $f\\in K(X)^{\\times}$), we see that $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}$ induces a group homomorphism\n\\begin{equation*}\n\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}\\colon T(K)\/T_{\\ell}(K)\\to H^1_{\\dR}(X).\n\\end{equation*}\nIn fact, this agrees with the canonical homomorphism $\\Psi$ of \\cite[Proposition 2.5]{ColemanGross} (see \\cite[Remarque II.2.12]{Colm96}).\n\\end{rmk}\n\n\\subsection{Colmez's adelic heights}\\label{subsec:Colm_heights}\n\\subsubsection{Notation and preliminary choices}\n\\label{subsubsec:notation_colm_heights}\nLet $K$ be a number field. For every place $v$ of $K$, we define a $\\mathbb{Q}$-vector space $\\mathscr{L}(K_v^{\\times})$ and a logarithm $\\mathop{\\mathrm{Log}}\\nolimits_v\\colon K_v^{\\times}\\to \\mathscr{L}(K_v^{\\times})$, as follows.\n\nIf $v$ is a non-archimedean place, let $p$ be the rational prime below $v$ and let $\\mathop{\\mathrm{Log}}\\nolimits_v$ be the logarithm $\\mathop{\\mathrm{Log}}\\nolimits$ for the field $K_v$, as defined at the beginning of \\S \\ref{subsec:Colm_int}. Then $\\mathop{\\mathrm{Log}}\\nolimits_v$ takes values in $\\mathscr{L}(K^{\\times}_v) \\colonequals K_v\\oplus \\mathbb{Q} \\mathop{\\mathrm{Log}}\\nolimits_v{p}$. An element $x\\in \\mathscr{L}(K_v^{\\times})$ can be written uniquely as $x = x^{(0)} + x^{(1)}\\mathop{\\mathrm{Log}}\\nolimits_v{p}$ with $x^{(0)}\\in K_v$ and $x^{(1)}\\in \\mathbb{Q}$. We will use the superscripts $(0)$ and $(1)$ with this meaning. \n\nIf $v$ is an archimedean place, we let $\\mathscr{L}(K_v^{\\times}) = \\mathbb{R}$ and $\\mathop{\\mathrm{Log}}\\nolimits_v\\colon K_v^{\\times}\\to \\mathscr{L}(K_v^{\\times})$ the map $\\mathop{\\mathrm{Log}}\\nolimits_v{x} = \\log{|x|_{\\infty}}$, where $|\\cdot |_{\\infty}$ is the standard absolute value on $\\mathbb{R}$ or $\\mathbb{C}$. \n\nWe then define the $\\mathbb{Q}$-vector space\n\\begin{equation*}\n\\mathscr{L}(\\mathbb{A}_{K}^{\\times}) = {\\prod}^{\\prime} \\mathscr{L}(K_v^{\\times})= \\{(x_v)_v\\in \\prod \\mathscr{L}(K_v^{\\times}) : x_v\\in K_v \\ \\text{for almost all } v\\}.\n\\end{equation*}\nSince $\\mathop{\\mathrm{Log}}\\nolimits_v(\\mathcal{O}_v^{\\times})\\subset K_v$, applying $\\mathop{\\mathrm{Log}}\\nolimits_v$ component-wise gives a well-defined map\n$\\mathop{\\mathrm{Log}}\\nolimits\\colon \\mathbb{A}_K^{\\times}\\to \\mathscr{L}(\\mathbb{A}_{K}^{\\times})$.\nDenoting by $\\mathscr{L}(K^{\\times})$ the $\\mathbb{Q}$-vector subspace of $\\mathscr{L}(\\mathbb{A}_{K}^{\\times})$ generated by $\\mathop{\\mathrm{Log}}\\nolimits(K^{\\times})$, we set\n\\begin{equation*}\n\\mathscr{L}(\\mathbb{A}_{K}^{\\times}\/K^{\\times})\\colonequals \\mathscr{L}(\\mathbb{A}_{K}^{\\times})\/\\mathscr{L}(K^{\\times}).\n\\end{equation*}\n\nNext, we define a $\\mathbb{Q}$-linear map $T\\colon \\mathscr{L}(\\mathbb{A}_{K}^{\\times}\/K^{\\times})\\to \\prod_{p\\leq \\infty} \\mathbb{Q}_p$ (where $\\mathbb{Q}_{\\infty}\\colonequals \\mathbb{R}$). The map $T$ will depend on the choice, at every $p$, of a continuous idele class character $\\chi^{(p)}\\colon \\mathbb{A}_{K}^{\\times}\/K^{\\times} \\to \\mathbb{Q}_p$ (cf.\\ Section \\ref{sec:padic_hts} for the case $p<\\infty$). We denote by $\\mathscr{I}(K)$ the set of all possible choices of $\\chi = (\\chi^{(p)})_{p\\leq \\infty}$.\n\n\\begin{rmk}\nIn Colmez's set-up there is no dependence on an element of $\\mathscr{I}(K)$. On the other hand, implicit in the construction is the choice of (a scalar multiple of) the cyclotomic character at $p<\\infty$ (see \\cite[Example 2.7]{QCnfs}) and a choice of a continuous real-valued idele class character. The latter choice amounts to the choice of a multiplicative constant.\n\\end{rmk}\n\\begin{lemma}\\label{lemma:Tvp}\nLet $\\chi^{(p)}\\colon \\mathbb{A}_{K}^{\\times}\/K^{\\times} \\to \\mathbb{Q}_p$ be a continuous idele class character. For every place $v$ of $K$, there exists a unique $\\mathbb{Q}$-linear map $T_v^{(p)}\\colon \\mathscr{L}(K_v^{\\times})\\to \\mathbb{Q}_p$ making the following diagram commutative\n\\begin{equation*}\n   \\xymatrix{\n      {K_{v}^\\times}  \\ar[rr]^{\\chi_{v}^{(p)}} \\ar[dr]_{\\mathop{\\mathrm{Log}}\\nolimits_v} & &   \\mathbb{Q}_p.\\\\\n      & \\mathscr{L}(K_v^{\\times})\\ar[ur]_{T_v^{(p)}}\n  }\n  \\end{equation*}\n  Moreover, for all but finitely many $v$, $T_v^{(p)}$ vanishes identically on $K_v$. \n\\end{lemma}\n\\begin{proof}\n$\\mathbb{Q}$-linearity gives uniqueness. \nFor existence,\n\\begin{itemize}\n\\item if $v$ and $p$ are infinite, then $\\chi_v^{(\\infty)} = \\alpha\\mathop{\\mathrm{Log}}\\nolimits_v$ for some $\\alpha\\in \\mathbb{R}$, so we let $T_v^{(\\infty)}(x) = \\alpha x$ for all $x\\in \\mathscr{L}(K_v^{\\times}) = \\mathbb{R}$;\n\\item if $p$ is finite and $v\\mid p$, there exists a unique $\\mathbb{Q}_p$-linear map $t_v^{(p)}\\colon K_v\\to \\mathbb{Q}_p$ such that $\\chi_v^{(p)} = t_v^{(p)}\\circ \\mathop{\\mathrm{Log}}\\nolimits_v$ on $\\mathcal{O}_v^{\\times}$ (see, for example, \\cite[\\S 2.1]{QCnfs}). We then let\n\\begin{equation*} \nT_v^{(p)}(x) = t_v^{(p)}(x^{(0)})  + x^{(1)}\\chi_v^{(p)}(p),\\qquad  \\text{for all } x\\in \\mathscr{L}(K_v^{\\times});\n\\end{equation*}\n\\item if $p$ is finite and $v$ is infinite, $\\chi_v^{(p)}$ is identically zero, so $T_v^{(p)}(x) = 0$;\n\\item in all the remaining cases, $\\chi_v^{(p)}$ is identically zero on $\\mathcal{O}_v^{\\times}$, so\n\\begin{equation*}\nT_v^{(p)}(x) = x^{(1)} \\chi_v^{(p)}(\\ell),\n\\end{equation*}\nwhere $\\ell$ is the rational prime below $v$.\\qedhere\n\\end{itemize}\n\\end{proof}\n\\begin{mydef}\\label{def:Tmap}\nFix $\\chi = (\\chi^{(p)})_p\\in \\mathscr{I}(K)$ and, for every pair $(v,p)$ of a place of $K$ and one of $\\mathbb{Q}$, denote by $T_v^{(p)}$ the map of Lemma \\ref{lemma:Tvp} for $\\chi_v^{(p)}$. By the lemma, the following map (whose definition depends on $\\chi$) is well-defined and $\\mathbb{Q}$-linear:\n\\begin{equation}\\label{eq:maps_prod_Qp}\nT \\colon \\mathscr{L}(\\mathbb{A}_{K}^{\\times}\/K^{\\times})\\to \\prod_{p\\leq \\infty} \\mathbb{Q}_p, \\qquad T((x_v))_p = \\sum_v T_v^{(p)}(x_v).\n\\end{equation}\n\\end{mydef}\nSuppose $L$ is a finite extension of $K$. Then for every place $v$ of $K$ and place $w\\mid v$ of $L$, there is a unique $\\mathbb{Q}$-linear map $\\Tr_{L_{w}\/K_v}\\colon \\mathscr{L}(L_w^{\\times})\\to \\mathscr{L}(K_v^{\\times})$ satisfying\n\\begin{equation*}\n\\Tr_{L_w\/K_v}(\\mathop{\\mathrm{Log}}\\nolimits_w(x)) = \\mathop{\\mathrm{Log}}\\nolimits_v(N_{L_w\/K_v}(x)) \\qquad \\text{for all } x\\in L_w^{\\times}.\n\\end{equation*}\nThese local trace maps induce a $\\mathbb{Q}$-linear map\n\\begin{equation*}\n\\Tr_{L\/K}\\colon \\mathscr{L}(\\mathbb{A}_L^{\\times})\\to \\mathscr{L}(\\mathbb{A}_K^{\\times}), \\qquad \\Tr_{L\/K}(\\dots,x_w,\\dots) = \\biggl(\\dots, \\sum_{w\\mid v} \\Tr_{L_w\/K_v} (x_w),\\dots\\biggr)\n\\end{equation*}\nthat factors through\n\\begin{equation*}\n\\Tr_{L\/K}\\colon \\mathscr{L}(\\mathbb{A}_L^{\\times}\/L^{\\times})\\to \\mathscr{L}(\\mathbb{A}_K^{\\times}\/K^{\\times}).\n\\end{equation*}\n\nTherefore, given $\\chi\\in \\mathscr{I}(K)$ as above, the composition of $T$ with $\\Tr_{L\/K}$ gives a $\\mathbb{Q}$-linear $\\mathscr{L}(\\mathbb{A}_L^{\\times}\/L^{\\times})\\to \\prod_{p\\leq \\infty}\\mathbb{Q}_p$. \n\\begin{rmk}\\label{rmk:trace_ext}\nIn fact,  $T\\circ \\Tr_{L\/K}$ is equal to the $\\mathbb{Q}$-linear map $\\mathscr{L}(\\mathbb{A}_{L}^{\\times}\/L^{\\times})\\to \\prod_{p\\leq \\infty} \\mathbb{Q}_p$ that Definition \\ref{def:Tmap} associates to the character $\\chi_{L}\\in \\mathscr{I}(L)$ given by $\\chi_L^{(p)} = \\chi^{(p)}\\circ N_{L\/K}$ where $N_{L\/K}$ is the norm on the idele class group.\n\\end{rmk}\n\n\\subsubsection{Adelic heights with respect to $\\chi$}\nLet now $X$ be a smooth, projective, geometrically irreducible curve over $K$, of genus greater than $0$, with Jacobian $J$. We assume that we have fixed $\\chi\\in \\mathscr{I}(K)$, which by Definition \\ref{def:Tmap} determines a choice of $\\mathbb{Q}$-linear map $T \\colon \\mathscr{L}(\\mathbb{A}_{K}^{\\times}\/K^{\\times})\\to \\prod_{p\\leq \\infty} \\mathbb{Q}_p$. We warn the reader that this choice will not be reflected in our notation.\n\nThe goal is to construct a symmetric bilinear pairing $J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})\\to  \\prod_{p\\leq \\infty} \\mathbb{Q}_p$, using $T$. Projecting to $\\mathbb{Q}_p$, for a chosen prime $p$, (or to $\\mathbb{R}$) we will also obtain a $\\mathbb{Q}_p$-valued (or real-valued) height function in the usual sense.\n\nBesides the choice of $\\chi\\in \\mathscr{I}(K)$, the adelic height depends on a choice, for each non-archimedean place $v$, of subspace $W_v$ of $H_{\\textup{dR}}^1(X\/K_v)$, complementary to the space of holomorphic forms and isotropic with respect to the cup product; in other words, on a subspace $W_v$ in the set $\\Is(X\/K_v)$. We denote by $\\Is_K(X)$ the set of such choices; an element $W\\in \\Is_K(X)$ has a component $W_v$ at each finite place $v$ of $K$. Fix $W\\in \\Is_K(X)$; we can and will view $W$ as an element in $\\Is_L(X)$ for any finite extension $L$ of $K$. \n\nGiven a divisor $D$ of degree $0$ on $X\/K$ and a non-archimedean place $v$, there is a unique differential $\\omega_{D,W_v}$ of the third kind of type $(1,1)$ with respect to $W_v$ whose residue divisor is $D$ (Proposition \\ref{prop:dec_third_kind}). Equivalently, by Lemma \\ref{lemma:prop_LogJtilde}, $\\omega_{D,W_v}$ is the unique differential of the third kind whose residue divisor is $D$ and whose image under $\\mathop{\\mathrm{Log}}\\nolimits_{\\tilde{J}}$ belongs to $W_v$. We denote by $G_{D,W,v}$ a primitive of $\\omega_{D,W_v}$, which, by Proposition \\ref{prop:dec_third_kind}\\thinspace{}\\ref{it:dec_4}, is, up to a constant, of the form\n\\begin{equation*}\n\\sum_{i} n_{i} G_{\\Theta,W_v}(\\iota(x)-u_i),\n\\end{equation*}\nfor some generic $u_i\\in J$ and $n_i\\in \\mathbb{Z}$.\n\nIf $v$ is archimedean, there exists a unique differential $\\omega_{D,W_v}$ of the third kind whose residue divisor is $D$ and all of whose periods are purely imaginary (the notation is a bit misleading, because $W_{\\infty}$ has technically not been defined). We write $G_{D,W,v}$ for the real part of a primitive of $\\omega_{D,W_v}$. \n\nThe \\emph{adelic Green function} associated to $D$ with respect to $W$ is the function \n\\begin{equation*}\nG_{D,W}\\colon \\prod_v X(K_v) \\to \\prod_v \\mathscr{L}(K_v^{\\times}), \\qquad G_{D,W}((\\dots, x_v,\\dots)) = (\\dots, G_{D,W,v}(x_v),\\dots).\n\\end{equation*}\nThe Green function $G_{D,W}$ is only well-defined on $(x_v)\\in \\prod_v X(K_v) $ such that $x_v$ does not belong to the support of $D$, for any $v$. Moreover, each $G_{D,W,v}$ is unique only up to addition of a constant function. This lack of uniqueness disappears if we view $G_{D,W}$ as a function on degree $0$ divisors, after extending by linearity.\n\n\\begin{prop}[{\\hspace{1sp}\\cite[Lemme II.2.16]{Colm96}}] \\label{prop:prop_global_Green}\nIf $D_1$ and $D_2$ are disjoint degree $0$ divisors over $K$, then\n\\begin{enumerate}[label = (\\roman*)]\n\\item\\label{it:adG_1} $G_{D_1,W}(D_2) = G_{D_2,W}(D_1)$.\n\\item\\label{it:adG_2} $G_{D_1,W}(D_2)\\in \\mathscr{L}(\\mathbb{A}_K^{\\times})$.\n\\item\\label{it:adG_3} If $D_1 = \\div(f)$, then $G_{D_1,W,v}(D_2) = \\mathop{\\mathrm{Log}}\\nolimits_v(f(D_2))$. In particular, if $D_1$ or $D_2$ is principal, then $G_{D_1,W}(D_2)\\in \\mathscr{L}(K^{\\times})$. \n\\item\\label{it:adG_4} If $L\/K$ is a finite extension, then $\\Tr_{L\/K}(G_{D_1\\otimes L, W}(D_2\\otimes L)) = [L:K]G_{D_1,W}(D_2)$. \n\\end{enumerate}\n\\end{prop}\n\\begin{mydef}\\label{def:pairingW}\nThe \\emph{adelic height with respect to $T$} is the symmetric bilinear pairing\n\\begin{equation*}\n\\langle \\cdot, \\cdot\\rangle_{W}\\colon J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})\\to \\prod_{p\\leq \\infty}\\mathbb{Q}_p\n\\end{equation*}\ndefined as follows. Given $x,y\\in J(\\overline{\\mathbb{Q}})$, pick $D_1,D_2\\in \\operatorname{Div}^0(X)$ such that the class of $D_1$ and $D_2$ equals $x$ and $y$, respectively. Assume that $D_1$ and $D_2$ are defined over $L$, for some $L\/K$ of finite degree, and that the supports of $D_1$ and $D_2$ are disjoint. Then \n\\begin{equation*}\n\\langle x,y \\rangle_{W}\\colonequals \\frac{1}{[L:\\mathbb{Q}]} \\cdot T\\circ \\Tr_{L\/K}(G_{D_1,W}(D_2)).\n\\end{equation*}\n\\end{mydef}\n\n\\begin{rmk}\nProposition \\ref{prop:prop_global_Green} guarantees that $\\langle x, y\\rangle_W$ is symmetric and well-defined, in the sense that it is independent of the choices of $D_1$ and $D_2$ and of $L$. By symmetry, by the definition of $G_{D_1,W}$ and by $\\mathbb{Q}$-linearity of $T\\circ \\Tr_{L\/K}$, we get bilinearity of $\\langle \\cdot, \\cdot \\rangle_W$.\n\\end{rmk}\nWe also obtain, for each place $p$, a pairing\n\\begin{equation*}\n\\langle\\cdot, \\cdot \\rangle_{W,p} \\colon J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})\\to \\mathbb{Q}_p\n\\end{equation*}\nby taking the $p$-th component of $\\langle\\cdot, \\cdot \\rangle_W$. \nThe $\\mathbb{Q}_p$-valued pairing $\\langle\\cdot, \\cdot \\rangle_{W,p}$ decomposes as a sum of local height pairings. In particular, retaining the notation of Definition \\ref{def:pairingW}, we have\n\\begin{equation*}\n\\langle x,y \\rangle_{W,p} = \\frac{1}{[L:\\mathbb{Q}]}\\sum_w \\langle D_1,D_2\\rangle_{p,w,W},\n\\end{equation*}\nwhere $\\langle \\cdot ,\\cdot \\rangle_{p,w,W}$ is the $\\mathbb{Q}_p$-valued pairing on relatively prime degree zero divisors with support in $X(L_w)$ defined as follows:\n\\begin{equation}\\label{def:loc_ht_Colm}\n\\langle D_1,D_2\\rangle_{p,w,W} = T_{v}^{(p)}\\circ \\Tr_{L_{w}\/K_v}(G_{D_1,W,w}(D_2));\n\\end{equation}\nhere $v$ is the place of $K$ below $w$.\n\nThe real (resp.\\ $p$-adic) height pairing of N\\'eron--Tate (resp.\\ Coleman--Gross) is also defined in terms of local height pairings on relatively prime degree zero divisors. \n\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition 2.3]{Gross86}, \\cite[Proposition 1.2]{ColemanGross}}]\\label{prop:NT_pairing}\nLet $p$ be a place of $\\mathbb{Q}$ and let $\\chi^{(p)}\\colon \\mathbb{A}_K^{\\times}\/K^{\\times}\\to \\mathbb{Q}_p$ be a continuous idele class character. Let $L$ be a finite extension of $K$ and let $\\chi_L^{(p)} = \\chi^{(p)}\\circ N_{L\/K}\\colon \\mathbb{A}_L^{\\times}\/L^{\\times}\\to \\mathbb{Q}_p$. Let $w$ be a place of $L$, such that, if $p \\neq \\infty$, then $w\\nmid p$. There exists a unique $\\mathbb{Q}_p$-valued pairing \n$\\langle\\cdot, \\cdot \\rangle_{p,w}^{N}$ on relatively prime degree zero divisors with support in $X(L_w)$ such that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\langle D_1,D_2\\rangle_{p,w}^N = \\langle D_2,D_1\\rangle_{p,w}^N$;\n\\item $\\langle D_1, D_2 + D_3\\rangle_{p,w}^N = \\langle D_1,D_2\\rangle_{p,w}^N + \\langle D_1,D_3\\rangle_{p,w}^N$;\n\\item $\\langle\\div(f), D \\rangle_{p,w}^N = \\chi_{L,w}^{(p)}(f(D))$;\n\\item $\\langle(x)-(x_0),D \\rangle_{p,w}^N$ is a continuous function of $x$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{mydef}\nThe \\emph{N\\'eron--Tate real height pairing} with respect to the idele character $\\chi^{(\\infty)}$ is  the pairing\n\\begin{equation*}\n\\langle\\cdot, \\cdot \\rangle^{NT}\\colon J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})\\to \\mathbb{R}\n\\end{equation*}\ndefined as follows. If $x,y, D_1,D_2, L$ are as in Definition \\ref{def:pairingW}, then\n\\begin{equation*}\n\\langle x,y\\rangle^{NT} = \\frac{1}{[L:\\mathbb{Q}]}\\sum_w \\langle D_1,D_2\\rangle_{\\infty,w}^{N}.\n\\end{equation*}\n\\end{mydef}\n\\begin{rmk}\nUp to a multiplicative constant, there is only one choice of continuous idele class character $\\chi^{(\\infty)}$.\n\\end{rmk}\nIn the statement of Proposition \\ref{prop:NT_pairing} we excluded pairs $(p,w)$ where $p$ is finite and $w\\mid p$, because uniqueness fails in this case. Nevertheless, Coleman--Gross \\cite{ColemanGross} defined a local pairing dependent on some choice of auxiliary data.  Let $v$ be the place of $K$ below $w$, let $W_v$ be a subspace of $H_{\\textup{dR}}^1(X\/K_v)$ isotropic with respect to the cup product and complementary to $H^0(X,\\Omega^1)$. Unravelling definitions, we see that the local height of Coleman--Gross at $w$ satisfies  \n\\begin{equation}\\label{eq:equality_colm_ht_CG_above_p}\n\\langle D_1,D_2\\rangle_{p,w,W_v}^{N} = \\langle D_1,D_2\\rangle_{p,w,W},\n\\end{equation}\nprovided that the $v$-adic component of $W$ is $W_v$ and the $p$-adic component of $\\chi$ is $\\chi^{(p)}$. \nTechnically speaking, Coleman--Gross assumed good reduction at each $v\\mid p$ as they defined the left hand side of \\eqref{eq:equality_colm_ht_CG_above_p} using Coleman integration \\cite{Col82, ColemandeShalit}, but Besser \\cite[Section 7]{Besser_pArakelov} \\cite{BesserPairing} extended the definition to arbitrary reduction type, using the $p$-adic integration theory of Vologodsky \\cite{Vologodsky}. Single Vologodsky integration agrees with Colmez's (summarised in \\S \\ref{subsec:Colm_int}). The only difference between (single) Coleman and Colmez integration in good reduction is that the former depends on the branch of the logarithm, while the latter treats the value of the logarithm at $p$ as a variable. This difference is lost in the local height parings of \\eqref{eq:equality_colm_ht_CG_above_p} since we are applying compatible trace maps to $\\mathbb{Q}_p$. The equality \\eqref{eq:equality_colm_ht_CG_above_p} therefore indeed  holds.\n\nIn view of Besser's extension, we may state Colmez's results without restricting to good reduction. \n\n\\begin{mydef}\nFor every $v\\mid p$, let $W_v$ be a subspace of $H_{\\textup{dR}}^1(X\/ K_v)$, complementary to $H^0(X, \\Omega^1)$ and isotropic with respect to the cup product. Then the \\emph{(extended) Coleman--Gross $p$-adic height pairing} with respect to $(W_v)_{v\\mid p}$ and $\\chi^{(p)}$ is the pairing\n\\begin{equation*}\n\\langle\\cdot, \\cdot \\rangle^{CG}_{(W_v)} \\colon J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})\\to \\mathbb{Q}_p\n\\end{equation*}\ndefined as follows. If $x,y,D_1,D_2,L$ are as in Definition \\ref{def:pairingW}, then\n\\begin{equation*}\n\\langle x,y \\rangle^{CG}_{(W_v)} =  \\frac{1}{[L:\\mathbb{Q}]}\\biggl(\\sum_{w\\mid v\\mid p}\\langle D_1,D_2\\rangle_{p,w,W_v}^N + \\sum_{w<\\infty,w\\nmid p}\\langle D_1,D_2 \\rangle_{p,w}^{N} \\biggr).\n\\end{equation*}\n\\end{mydef}\n\n\\begin{thm}[{\\hspace{1sp}\\cite[Th\u00e9or\u00e8me II.2.18]{Colm96} $+$ $\\epsilon$}]\\label{thm:NT_CG_eq_Colmez}\nThe pairing $\\langle \\cdot, \\cdot \\rangle_{W,p}$ is equal to the N\\'eron--Tate real height pairing  with respect to $\\chi^{(\\infty)}$ if $p=\\infty$ and to the extended $p$-adic height pairing of Coleman--Gross with respect to $(W_v)_{v\\mid p}$ and $\\chi^{(p)}$ if $p$ is finite. Moreover, if $p$ is infinite, or $p$ is finite and $w\\nmid p$, then\n\\begin{equation*}\n\\langle \\cdot, \\cdot \\rangle_{p,w,W} = \\langle \\cdot, \\cdot\\rangle_{p,w}^N,\n\\end{equation*}\nand so $\\langle \\cdot, \\cdot \\rangle_{p,w,W}$ is independent of $W$; otherwise,\n\\begin{equation*}\n\\langle \\cdot, \\cdot \\rangle_{p,w,W} = \\langle \\cdot, \\cdot\\rangle_{p,w,W_v}^N.\n\\end{equation*}\n\\end{thm}\n\n\\begin{proof}\nColmez proves the result for a specific choice of $\\chi\\in \\mathscr{I}(K)$; given the definitions and results above, the proof carries over to an arbitrary $\\chi$. In particular, it suffices to show equality of the local height pairings for all the pairs $(p,w)$ satisfying the assumptions of Proposition \\ref{prop:NT_pairing}. For example, if $p, w$ are non-archimedean and $w\\mid \\ell\\neq p$, by \\eqref{def:loc_ht_Colm} and the proof of Lemma \\ref{lemma:Tvp}, we have\n\\begin{equation}\\label{eq:G_instead_of_int}\n\\langle D_1, D_2\\rangle_{p,w,W} =G_{D_1,W,w}(D_2)^{(1)}\\chi_{L,w}^{(p)}(\\ell).\n\\end{equation}\nBy uniqueness, it suffices to show that \\eqref{eq:G_instead_of_int} satisfies all the properties of Proposition \\ref{prop:NT_pairing}, and this is independent of the choice of $\\chi^{(p)}$.\n\\end{proof}\n\n\n\\begin{rmk}\nAs Colmez observes, by the comparison result of Theorem \\ref{thm:NT_CG_eq_Colmez} and Equation \\eqref{eq:G_instead_of_int}, we obtain a formula for the N\\'eron--Tate local height which does not use intersection theory. This is analogous to the difference between Definition \\ref{def:Neron_fct_away_p} and the definition of $p$-adic N\\'eron functions away from $p$ of \\cite{BKM22}. \n\nRecently, Besser, M\\\"uller and Srinivasan \\cite{BMP_adelic_metric} gave a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory \\cite{Besser_pArakelov}, and, in particular, $p$-adic log functions. Their height satisfies properties that generalise what we have just observed concerning Colmez's heights: namely, the local contributions at all places are defined analytically. In the special case of Jacobians of curves, they compare their construction to Colmez's. For this, they use a result of Besser \\cite[Appendix B]{Besser_pArakelov} that relates the theory of log functions on line bundles on abelian varieties to Colmez's Green functions. \n\\end{rmk}\n\n\\begin{rmk}\nIf $p$ is finite, $w\\mid v\\mid p$, and $\\chi^{(p)}$ is unramified at $v$ (i.e.\\ $\\chi_v^{(p)}(\\mathcal{O}_v^{\\times}) = 0$), then $\\langle D_1, D_2\\rangle_{p,w,W}$ only depends on $G_{D_1,W,w}(D_2)^{(1)}$ and not on $G_{D_1,W,w}(D_2)^{(0)}$, since the local trace map $t_v^{(p)}$ of the proof of Lemma \\ref{lemma:Tvp} is trivial. In fact, also in this case the local height is independent of the choice of $W_v$, and is given by formula \\eqref{eq:G_instead_of_int}.\nCompare this with Remark \\ref{rmk:neron_away_p}\\thinspace{}\\ref{rmk:neron_unramified}.\n\\end{rmk}\n\n\n\n\n\n\n\n\\subsection{$p$-Adic N\\'eron functions, Green functions and Coleman--Gross heights}\\label{subsec:comparison}\nIn this subsection, we go back to the situation of Section \\ref{sec:padic_hts}, where $p$ is a finite prime of $\\mathbb{Q}$ and $C$ is a smooth genus $2$ curve over a number field $K$, given by an equation of the form \\eqref{eq:Grant}, with coefficients in the ring of integers of $K$. Let $v$ be a place of $K$, and consider the base-change of $C$ to $K_v$. \n\n\nWe apply Colmez's results from \\S\\S \\ref{subsec:Colm_curves}--\\ref{subsec:Colm_heights} according to Remark \\ref{rmk:from_X_to_C}: in particular,\n\\begin{itemize}\n\\item $\\iota$ is the embedding with respect to the base-point $P_0 = \\infty$.\n\\item $(\\Omega_1,\\Omega_2)$ is the basis for $H^0(J,\\Omega^1)$ of Definition \\ref{def:basis_inv_diff_inv_der}, described explicitly in terms of the $\\mathbb{P}^8$ embedding of $J$ in Lemma \\ref{lemma:inv_difder}\\thinspace{}\\eqref{lemma_part:inv_difder_1}. The dual basis $(\\partial_1,\\partial_2)$ of invariant derivations is also described in Lemma \\ref{lemma:inv_difder}\\thinspace{}\\eqref{lemma_part:inv_difder_2}.\n\\item $x\\in\\Theta$ if and only if $-x\\in \\Theta$, so the element $w\\in J$ defined just before Proposition \\ref{prop:bijection} is just $0$. In particular, a $v$-adic Green function of division $\\Theta$ is symmetric if and only if it is even. \n\\end{itemize}\nAccording to Proposition \\ref{prop:bijection}, the set of symmetric $v$-adic Green functions of divisor $\\Theta$ (up to addition by a constant function) is in bijection with the subset $\\Is(C\/K_v)$ of the set of subspaces of $H^1_{\\dR}(C\/ K_v)$. By Proposition \\ref{prop:bijection_sigma_is}, there is also a bijection between $\\Is(C\/K_v)$ and the set of $v$-adic sigma functions. \n\nSuppose $\\chi\\colon \\mathbb{A}_K^{\\times}\/K^{\\times}\\to \\mathbb{Q}_p$ is a continuous idele class character. Let $T_v\\colon \\mathscr{L}(K_v^{\\times}) \\to \\mathbb{Q}_p$ be the trace map at $v$ induced by $\\chi$ as in Lemma \\ref{lemma:Tvp}. Fix a subspace $W_v\\in \\Is(C\/K_v)$. \nIn this subsection, we prove:\n\\begin{thm}\\label{thm:comparison} \nThere exists a symmetric $v$-adic Green function $G\\colonequals G_{\\Theta,W_v}$ of divisor $\\Theta$ simultaneously satisfying the following properties: \n\\begin{enumerate}[label=(\\roman*)]\n \\item $\\langle\\iota^{*}[d(\\partial_1 G)],\\iota^{*}[d(\\partial_2 G)] \\rangle = W_v$; \n \\item Let $L$ be a finite extension of $K$, let $w\\mid v$ be a place of $L$, let $\\lambda_w\\colonequals \\lambda_{w,W_v}$ be the local $p$-adic N\\'eron function at $w$ with respect to $W_v\\in \\Is(C\/K_v)$ and $\\chi$, and let $T_w\\colonequals T_{v}\\circ \\Tr_{L_{w}\/K_v}$ .  Then\n \\begin{equation}\\label{eq:eq_lambda_Gtheta}\n n_w\\lambda_{w} = -2T_w(G).\n \\end{equation}\n\\end{enumerate}\n\\end{thm}\n\\begin{rmk}\nThe theorem also holds true if $v$ is a prime at which $\\chi$ is unramified; in this case, both the left and right hand side of \\eqref{eq:eq_lambda_Gtheta} are independent of $W_v$ (cf.\\ Definition \\ref{def:Neron_fct_away_p}, Remark \\ref{rmk:neron_away_p}\\thinspace{}\\ref{rmk:neron_unramified} and Theorem \\ref{thm:NT_CG_eq_Colmez}).\n\\end{rmk}\nThe comparison of the $p$-adic component of the adelic-Green-function height and the (extended) Coleman--Gross $p$-adic height then gives, as a corollary, a comparison result between the local N\\'eron function $\\lambda_{w,W_v}$ and the local height pairing $\\langle\\cdot, \\cdot \\rangle_{p,w,W_v}^N$ with respect to $\\chi$. \n\n\\begin{cor}\\label{cor:lambda_eq_CG} Let $L$ be a finite extension of $K$, and $w$ a place of $L$ above the place $v$ of $K$. Let $D_1 = \\sum_{i=1}^{r} k_i P_i$ and $D_2 = \\sum_{j=1}^{s} m_j Q_j$ be disjoint degree $0$ divisors supported on $C(L_w)$. Choose points $u_1,\\dots,u_r\\in J(L_w)$ such that $u_1,\\dots,u_r$ are generic, $\\sum_{i} k_i C_{u_i} = D_1$ and $\\iota(Q_j)-u_i\\not \\in \\Supp(\\Theta)$ for all $i,j$. Then\n\\begin{align*}\n\\langle D_1, D_2 \\rangle_{p,w,W_v}^N\n= -\\frac{n_w}{2}\\sum_{i=1}^r \\sum_{j=1}^{s}k_i m_j\\lambda_{w,W_v}(\\iota(Q_j) -u_i).\n\\end{align*}\n\\end{cor}\n\\begin{proof}\nThis follows from Theorem \\ref{thm:comparison} and Theorem \\ref{thm:NT_CG_eq_Colmez}.\n\\end{proof}\n\\begin{example}\\label{eg:finding_u1_u2}\nWith the notation of Corollary \\ref{cor:lambda_eq_CG}, \nlet $R\\in C(L_w)$ such that $R\\neq P_i,P_i^{-}, Q_j$ for all $i\\in\\{1,\\dots,r\\}$ and all $j\\in\\{1,\\dots,s\\}$. Then we may choose  $u_i  = \\iota(P_i) + \\iota(R)$. \n\\end{example}\n\\begin{rmk}\nThe comparison also holds at the primes at which $\\chi$ is unramified. For these, there is no dependence on $W_v$.\n\\end{rmk}\n\n\\begin{cor}\\label{cor:global_CG_same_as_this}\nThe Coleman--Gross global height pairing on $J(\\overline{\\mathbb{Q}})\\times J(\\overline{\\mathbb{Q}})$ is equal to the symmetric bilinear pairing induced by $h_p$, provided that we make the same choices of $W_v\\in \\Is(C\/K_v)$ at all places $v$ of ramification for $\\chi$. \n\\end{cor}\n\\begin{rmk}\\label{rmk:comparison_canonical}\nSuppose that $p\\geq 5$, that $C$ has good ordinary reduction at every place $v\\mid p$ of ramification for $\\chi$, and that, for each such $v$, $K_v$ is an unramified extension of $\\mathbb{Q}_p$. Then, at every such $v$, we may take $W_v$ to be the unit root eigenspace of Frobenius. By Proposition \\ref{prop:Blakestad_space_unit_root}, this corresponds to choosing the canonical $v$-adic sigma function of Blakestad. Under these assumptions, Corollary \\ref{cor:global_CG_same_as_this} thus relates the Coleman--Gross $p$-adic height with respect to the unit root eigenspaces of Frobenius to the $p$-adic height with respect to the canonical $v$-adic sigma functions. \n\\end{rmk}\n\\begin{proof}[Proof of Corollary \\ref{cor:global_CG_same_as_this}]\nWe omit the choices of subspace $W_v$ from the notation. Let $L$ be an arbitrary finite extension of $K$. Since $C(L)$ is finite by Faltings's theorem \\cite{faltings}, there are finitely many points in $J(L)$ which are either of the form $[2P-2\\infty]$ or $[P+W-2\\infty]$ for a Weierstrass point $W$.  Therefore, any point in $J(L)\\setminus J_{\\tors}(L)$ admits a multiple $v$ of the form\n\\begin{equation*}\nv = [P_1 - P_2],\n\\end{equation*}\nfor some non-Weierstrass points $P_1,P_2$ such that $P_1\\neq P_2^{-}$. Thus, by quadraticity, it suffices to show that $h_p(v) = \\langle v, v\\rangle^{CG}$ for all $v$ in this form. \n \n\n In this case, the divisors $D_1 = P_1 - P_2$ and $D_2 = P_2^{-} - P_1^{-}$ are disjoint and satisfy $[D_1] = [D_2] = v\\not\\in \\Supp(\\Theta)$. By possibly replacing $L$ with a field extension, we may assume that the supports of $D_1$ and $D_2$ are defined over $L$. Let $w$ be a non-archimedean place of $L$, and let $v_1 =[P_2^{-} -P_1] $, $v_2 = [P_2^{-}-P_2]$, $v_3 = [P_1^{-}-P_1]$. By Corollary \\ref{cor:lambda_eq_CG} and Example \\ref{eg:finding_u1_u2}, we have\n\\begin{equation*}\n\\langle D_1, D_2\\rangle_{p,w}^N = - \\frac{n_w}{2} \\left(2\\lambda_w(v_1) - \\lambda_w(v_2) - \\lambda_w(v_3) \\right).\n\\end{equation*}\nThus, by Proposition \\ref{prop:properties_neron_fcts}\\thinspace{}\\ref{it:quasi_parallelogram}, \n\\begin{equation*}\n\\langle D_1, D_2\\rangle_{p,w}^N =  n_w\\lambda_w(v) - \\chi_{L,w}(-X_{11}(v_1) + X_{11}(v) - X_{12}(v_1)X_{22}(v) + X_{22}(v_1)X_{12}(v)).\n\\end{equation*}\nSumming over all places $w$ gives the result.\n\\end{proof}\n\nIt remains to prove Theorem \\ref{thm:comparison}.\n This will be a corollary of some intermediate results. In order to keep the notation more legible, for now, let $\\sigma_v$ be just some $v$-adic sigma function for $C\/K_v$.\n\\begin{lemma}\\label{lemma:log_sigma_loc_Green}\nLet $V$ be a finite index subgroup of $J(K_v)$ on which $\\sigma_v$ converges. Then there exists a finite index subgroup $V^{\\prime}$ of $J(K_v)$ such that $V^{\\prime}\\subseteq V$ and the restriction of $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v)$ to $V^{\\prime}$ satisfies Proposition \\ref{prop:padicGreen} for $D = \\Theta$; that is, \n\\begin{equation*}\n\\Delta^{[3]}(\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v)) = \\mathop{\\mathrm{Log}}\\nolimits g_{\\Theta}^{(4)},\n\\end{equation*}\nwhere $g_{\\Theta}^{(4)}$ is the restriction to $V^{\\prime, 4}$ of $f_{\\Theta}^{(4)}$, $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v)$ is locally analytic outside $\\Theta$ and has a logarithmic singularity along $\\Theta$. \n\\end{lemma}\n\\begin{proof}\nIf $\\sigma_v$ and $\\sigma_v^{\\prime}$ are two different sigma functions, then $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v) - \\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v^{\\prime})$ is a polynomial of degree two in the formal one-logarithms of $J$, so, since the statement of the lemma allows us to move to a smaller $V^{\\prime}$, it suffices to consider the naive $v$-adic sigma function of Theorem \\ref{thm:sigma_naive}. Upon switching variables to the additive group, the latter is the Taylor expansion of the complex sigma function $\\sigma$, so it is enough to show that \n\\begin{enumerate}[label = (\\roman*)]\n\\item \\label{it:periodic_sigma} the function $\\sigma$ is such that\n\\begin{equation*}\nF_{\\Theta}^{(4)} (z_0,z_1,z_2,z_3) = \\frac{\\sigma(z_0 +z_1 + z_2+z_3) \\sigma(z_0 + z_1) \\sigma(z_0 + z_2)\\sigma(z_0 + z_3)}{\\sigma(z_0 + z_1 + z_2)\\sigma(z_0 + z_1 + z_3)\\sigma(z_0 + z_2+z_3)\\sigma(z_0)} \n\\end{equation*}\nis periodic with period $\\Lambda^4$;\n\\item \\label{it:divisor_sigma} the divisor of $F_{\\Theta}^{(4)}$ is $\\Delta^{[3]}\\Theta$. \n\\end{enumerate}\nPart \\ref{it:periodic_sigma} follows from the translation properties of $\\sigma$ \\cite[Proposition 2.2]{Uchida}, while part \\ref{it:divisor_sigma} by \\cite[Proposition 2.7]{Uchida}.\nNote that on $J^3\\times\\{0\\}\\cup (J^2\\times\\{0\\}\\times J)\\cup (J\\times \\{0\\}\\times J^2)$ we have $F_{\\Theta}^{(4)} = 1$. \n\\end{proof}\n\nLemma \\ref{lemma:log_sigma_loc_Green} tells us that, locally, $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v)$ is a $v$-adic Green function of divisor $\\Theta$. Next, we want to show that we can extend $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v)$ to a $v$-adic Green function on $J$, and that the right way to do so is by using a quasi-quadraticity formula involving division polynomials, just like we did when defining local N\\'eron functions. \n\nIn order to achieve this, we observe that the existence of $G_{\\Theta}$ (and uniqueness, up to addition by polynomials of degree at most $2$ in the one-logarithms of $J$), as stated in Proposition \\ref{prop:padicGreen}, is proved by Colmez using \\cite[Th\\'eor\\`eme II.1.16]{Colm96}. The proof of this technical result comprises two steps: first one solves the problem locally, and then extends the solution using \\cite[Proposition II.1.23]{Colm96}.  \n\n\\begin{prop}\\label{prop:sigma_extends}\nThe function $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v)$ extends to a unique Green function $G_{\\Theta}$ for $\\Theta$ on $J$. Moreover, $G_{\\Theta}$ is an even function and \n\\begin{equation}\\label{eq:G_quasi_quad}\nG_{\\Theta}(nx) - n^2 G_{\\Theta}(x) = \\mathop{\\mathrm{Log}}\\nolimits (\\phi_n(x)),\n\\end{equation}\nwhere $\\phi_n$ is the $n$-th division polynomial of \\eqref{eq:def_div_poly}.\n\\end{prop}\n\n\\begin{proof}\nBy \\cite[Proposition II.1.23]{Colm96}, $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma)$ extends to a unique Green function $G_{\\Theta}$ on $J$. \nNow, $\\mathop{\\mathrm{Log}}\\nolimits(\\sigma)$ is an even function, and since $[-1]^{*}\\Theta = \\Theta$ we have that $\\mathop{\\mathrm{Log}}\\nolimits f_{\\Theta}^{(4)}(-z_0,-z_1,-z_2,-z_3)$  and $\\mathop{\\mathrm{Log}}\\nolimits f_{\\Theta}^{(4)}(z_0, z_1,z_2,z_3)$ differ by a constant. Evaluating at $z_3 = 0$, we find that the constant is $0$. Thus the formula of \\cite[Proposition II.1.23]{Colm96} with $f = \\mathop{\\mathrm{Log}}\\nolimits(\\sigma)$ gives an even function, satisfying\n\\begin{align}\\label{eq:def_green}\nG_{\\Theta}(z_0 + z_1 + z_2 + z_3) - G_{\\Theta}(z_0+ z_1+ z_2) - G_{\\Theta}(z_0+ z_2+ z_3) - G_{\\Theta}(z_0+ z_1+ z_3) \\\\\n+ G_{\\Theta}(z_0 + z_1) + G_{\\Theta}(z_0+ z_2) + G_{\\Theta}(z_0+ z_3) - G_{\\Theta}(z_0) = \\Delta^{[3]} G_{\\Theta}(z_0,z_1,z_2,z_3) = \\mathop{\\mathrm{Log}}\\nolimits(f_{\\Theta}^{(4)})\\nonumber\n\\end{align}\nWe now use this to prove \\eqref{eq:G_quasi_quad}. The proof will be similar to the one for elliptic curves given in the proof of \\cite[Proposition II.2.20]{Colm96}.\n\nBoth the right and left hand side of \\eqref{eq:def_green} have a logarithmic singularity at $\\{0\\}\\times J^3$. We have\n\\begin{equation}\\label{eq:Gz1z2z3}\n\\begin{aligned}\nG_{\\Theta}( z_1 + z_2 + z_3) - G_{\\Theta}(z_1+ z_2) - G_{\\Theta}(z_2 + z_3) - G_{\\Theta}(z_1 + z_3) \\\\\n+ G_{\\Theta}(z_1) + G_{\\Theta}( z_2) + G_{\\Theta}( z_3)  = \\lim_{z_0\\to 0}(  \\mathop{\\mathrm{Log}}\\nolimits(f_\\Theta^{(4)})(z_0,z_1,z_2,z_3)+ G_{\\Theta}(z_0))= \\mathop{\\mathrm{Log}}\\nolimits(h(z_1,z_2,z_3)),\n\\end{aligned}\n\\end{equation}\nfor some rational function $h$. Once again, letting $z_1 = z_2$, taking limits $z_3\\to  -z_1$ and using that $G_{\\Theta}$ is even, we get\n\\begin{equation}\\label{eq:G2z1}\n4G_{\\Theta}(z_1)-G_{\\Theta}(2z_1) = \\mathop{\\mathrm{Log}}\\nolimits(f_2^{-1}(z_1))\n\\end{equation}\nfor some rational function $f_2$. Going back to \\eqref{eq:Gz1z2z3} and letting $z_1 = (n-1)x, z_2 = z_3 = x$, we get\n\\begin{equation*}\nG_{\\Theta}((n+1)x) - 2G_{\\Theta}(nx) - G_{\\Theta}(2x) + G_{\\Theta}((n-1)x) + 2G_{\\Theta}(x) = \\mathop{\\mathrm{Log}}\\nolimits(h((n-1)x,x,x)).\n\\end{equation*}\nBy induction, we conclude that for all $n\\geq 1$ there exists a rational function $f_n$ such that\n\\begin{equation*}\nG_{\\Theta}(nx) - n^2G_{\\Theta}(x) = \\mathop{\\mathrm{Log}}\\nolimits(f_n(x)).\n\\end{equation*}\nComparing log-singularities, we see that $f_n$ equals $\\phi_n$, up to multiplication by a constant, which, by comparing around $0$, we see we can take to be equal to $1$.\n\\end{proof}\n\nIt remains to show that if $\\sigma_v$ corresponds to $W_v$ via Proposition \\ref{prop:bijection_sigma_is}, then the Green function $G_{\\Theta}$ of Proposition \\ref{prop:sigma_extends} also corresponds to $W_v$ under the bijection of Proposition \\ref{prop:bijection}.\n\\begin{lemma}\\label{lemma:explicit_subspace_H1dRJ}\nLet $\\sigma_v$ be the sigma function satisfying the differential equations\n\\begin{equation*}\nD_iD_j(\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v))= - X_{ij} + c_{ij}, \\qquad c_{12} = c_{21}.\n\\end{equation*}\nLet $G_{\\Theta}$ be the Green function associated to $\\sigma_v$ by Proposition \\ref{prop:sigma_extends}. \nThen $\\eta_{i,J} =  d(\\partial_i G_{\\Theta})$ is given explicitly by \n\\begin{equation*}\n\\eta_{i,J} = (-X_{1i} + c_{1i}) \\Omega_1 + (-X_{i2} + c_{i2})\\Omega_2.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{lemma:dLi_eq_Omegai}, $D_i$ is the restriction of $\\partial_i$ to the formal group law.  So we have\n\\begin{align*}\n\\partial_i\\partial_j(\\mathop{\\mathrm{Log}}\\nolimits(\\sigma_v(nx))) = n^2(-X_{ij}(nx)  + c_{ij})\\\\\n\\partial_i \\partial_j(\\mathop{\\mathrm{Log}}\\nolimits(\\phi_n(x))) = -n^2X_{ij}(nx) + n^2X_{ij}(x)\n\\end{align*}\n(cf.\\ \\cite[Proposition 4.10]{Uchida}), and thus $\\partial_i\\partial_j(G_{\\Theta}) = -X_{ij} + c_{ij}$, as desired.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:explicit_subspace_H1dRX}\nWith the notation of Lemma \\ref{lemma:explicit_subspace_H1dRJ}, let $[\\eta_1]  =\\iota^*[\\eta_{1,J}], [\\eta_2] = \\iota^{*}[\\eta_{2,J}]$. Then we may choose\n\\begin{equation*}\n\\eta_1 = (-3x^3 - 2b_1x^2 - b_2x + c_{12} x + c_{11})\\frac{dx}{2y},\\qquad \\eta_2= (-x^2 + c_{22} x + c_{12})\\frac{dx}{2y},\n\\end{equation*}\ni.e.\\ $\\eta_1 = \\eta_1^{(c)}$ and $\\eta_2 = \\eta_2^{(c)}$ where $\\eta_1^{(c)}$ and $\\eta_2^{(c)}$ are the differentials of \\eqref{eq:eta_12_c}.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\pi_i\\colon C^2\\to C$ be the projection map, and let $\\varpi\\colon C^2\\to J$ be the map sending $(P_1,P_2)$ to $\\iota(P_1)+ \\iota(P_2)$. According to \\cite[Lemme II.2.1]{Colm96}, given a differential $\\omega$ of the second kind on $C$, there exists a unique differential $\\omega_J$ of the second kind on $J$ such that $\\varpi^{*}\\omega_J = \\pi_1^{*}\\omega + \\pi_2^{*}\\omega$. Moreover the class of $\\iota^{*}\\omega_J$ equals that of $\\omega$ in $H^1_{\\dR}(C)$. \nWe deduce that any differential $\\xi$ of the second kind on $J$ is equivalent to one whose image under $\\varpi^{*}$ is of the form\n\\begin{equation}\\label{eq:pullback}\n\\pi_1^*\\omega + \\pi_2^*\\omega,\n\\end{equation}\nand, moreover, that $[\\iota^* \\xi] = [\\omega]$. Therefore, in order to compute the classes of $[\\eta_i]$, it will suffice to find a representative for $\\varpi^*[\\eta_{i,J}]$ of the form \\eqref{eq:pullback}. By \\cite[p.\\,66]{blakestadsthesis}, we have\n\\begin{equation*}\nd(X_{222}) = (3X_{12}X_{22} - X_{11} + 2b_1X_{12}) \\Omega_1 + (3X_{22}^2 + 2X_{12} + 2b_1X_{22} + b_2) \\Omega_2;\n\\end{equation*}\nmoreover, by definition, $\\varpi^{*}\\Omega_1= \\frac{dx_1}{2y_1} + \\frac{dx_2}{2y_2}$, $\\varpi^{*}\\Omega_2 = x_1\\frac{dx_1}{2y_1} + x_2\\frac{dx_2}{2y_2}$ and $\\varpi^{*}X_{12} = -x_1x_2$, $\\varpi^{*}X_{22} =x_1+x_2$ . A computation then shows that\n\\begin{equation}\\label{eq:eta_iJ_eta_i}\n\\begin{aligned}\n\\varpi^{*}(\\eta_{1,J}  - d(X_{222})) &= \\pi_1^{*}\\eta_1 + \\pi_2^{*}\\eta_1 \\\\\n \\varpi^{*}(\\eta_{2,J}) &=\\pi_1^{*}\\eta_2 + \\pi_2^{*}\\eta_2. \n \\end{aligned} \\qedhere\n\\end{equation}\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:comparison}]\nThis follows from Lemma \\ref{lemma:Tvp}, Remark \\ref{rmk:trace_ext}, Proposition \\ref{prop:sigma_extends} and Lemmas \\ref{lemma:explicit_subspace_H1dRJ}, \\ref{lemma:explicit_subspace_H1dRX}.\n\\end{proof}\n\n\n\n\\section{Implementation and examples}\\label{sec:implementation}\n\\subsection{Formal group and sigma functions}\\label{subsec:implementation_formal_gp_sigma}\nLet $C$, $R$ and $K$ be as in \\S \\ref{subsec:formal}. Recall that the functions $\\frac{1}{X_{111}}$, $\\frac{X_{ij}}{X_{111}}$ and $\\frac{X_{ijk}}{X_{111}}$ can be expanded as power series over $R$ in the formal group parameters $T_1$ and $T_2$. Grant proved this by explaining how to use the equations defining $J$ in $\\mathbb{P}^8$ to obtain recursive formulae for the coefficients in the expansion (cf.\\ \\eqref{eq:Xij_exp}, \\eqref{eq:Xijk_exp} and the surrounding discussion).\n\nWe implemented this recursion (in conjunction with Remark \\ref{rmk:odd_and_even_expansions}). As a result, we can compute the expansions of $\\frac{1}{X_{111}}$, $\\frac{X_{ij}}{X_{111}}$ and $\\frac{X_{ijk}}{X_{111}}$ as power series in $T_1$ and $T_2$, up to any desired precision.  Combining this with the addition formulae on $J$ provided by \\cite[Theorem 3.2]{Grant1990}, we can then compute the formal group law $F(T,S) = (F_1,F_2)(T,S)$ of Theorem \\ref{thm:formal_gp_law} explicitly, up to any desired precision.\n\n\nWith the formal group law at hand, the computation of the strict logarithm $\\mathcal{L} = (\\mathcal{L}_1,\\mathcal{L}_2)$ and of the naive sigma function are an easy exercise in solving systems of differential equations over a power series ring.  Note that \\eqref{eq:Di} expresses the derivation $D_i$, for each $i\\in\\{1,2\\}$, in terms of the derivatives with respect to $T_1$ and $T_2$ and the formal group law. \n\nGiven the naive sigma function and the strict logarithm $\\mathcal{L}$, we can compute the expansion of any other sigma function using the formula \\eqref{eq:formulasigmac}. We included in Appendix \\ref{app:sigma_expansion} the first terms in the expansion of $\\sigma_v^{(c)}(T)$ for an arbitrary curve $C$ and for arbitrary constants $c_{11},c_{12},c_{22}$ (so, in particular, the coefficients belong to $\\mathbb{Q}[b_1,\\dots, b_5][c_{11},c_{12},c_{22}]$). \n\n\n\\subsection{The canonical sigma function}\\label{subsec:implementation_canonical_sigma}\nAssume now that $K$ is the completion of a number field at a non-archimedean place $v$, and that the assumptions of Theorem \\ref{thm:Blakestad_main} are satisfied. We would like to compute the canonical $v$-adic sigma function of Blakestad. By Proposition \\ref{prop:Blakestad_space_unit_root}, if $K$ is an unramified extension of $\\mathbb{Q}_p$, this is the $v$-adic sigma function corresponding to the unit root eigenspace of Frobenius. While it is possible to compute this space using Theorem \\ref{thm:Blakestad_main}\\thinspace{}\\ref{thm:Blake_3}, it is computationally very inefficient. \n\nIf $K$ is isomorphic to $\\mathbb{Q}_p$,  for a prime $p$ greater than $3$, we can alternatively compute the unit root eigenspace of Frobenius using Kedlaya's algorithm \\cite{kedlaya} and \\cite[Proposition 6.1]{Bes-Bal10}. We explain how to do this in practice and how to deduce the symmetric matrix $b$ corresponding to the unit root eigenspace.\n\nSo assume now that $C$ is defined over $\\mathbb{Q}_p$, where $p$ is a prime greater than or equal to $5$, of good reduction for $C$ and ordinary reduction for $J$. Consider the characteristic polynomial $\\chi(t)$ of Frobenius of the Jacobian of the base-change of $C$ to $\\mathbb{F}_p$, which is of the form \n\\begin{equation*}\n\\chi(t) = t^4 + a_1t^{3}+a_2t^2+pa_1t+p^2\\in \\mathbb{Z}[t].\n\\end{equation*}\nThe Jacobian over $\\mathbb{F}_p$ is ordinary if and only if $a_2$ is coprime with $p$ (see for instance \\cite[\\S 3]{yui1978jacobian}). In this case, the factorisation  \n\\begin{equation*}\n\\chi(t) \\equiv t^4 +a_1t^3+a_2t^2 \\bmod{p} = t^2(t^2+a_1t+a_2)\n\\end{equation*}\n lifts to a factorisation over $\\mathbb{Z}_p$ by Hensel's lemma. Let $\\chi_{W^{(b)}}(t) = t^2+\\hat{a}_1t + \\hat{a}_2\\in \\mathbb{Z}_p[t]$ be the lift of $t^2+a_1t+a_2 \\bmod{p}$ in such a factorisation of $\\chi(t)$.\n  \n Let $\\mathcal{C}\/\\mathbb{Z}_p$ be the hyperelliptic curve over $\\mathbb{Z}_p$ whose complement of the section at infinity is the closed subscheme of $\\mathbb{A}_{\\mathbb{Z}_p}^2$ described by our usual affine equation for $C$. Recall that we have\n \\begin{equation*}\nH^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)\\hookrightarrow{} H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)\\otimes \\mathbb{Q}_p \\cong H^1_{\\dR}(C\/\\mathbb{Q}_p),\n \\end{equation*}\n and that this induces a $\\mathbb{Q}_p$-linear Frobenius endomorphism on $H^1_{\\dR}(C\/\\mathbb{Q}_p)$. The characteristic polynomial of this endomorphism is equal to $\\chi(t)$ (see, for instance,  \\cite[Theorem 5.3.2]{Edix_pt_counting} combined with Remark \\ref{rmk:compatibility_frob} below). \nThe unit root eigenspace of Frobenius is the $2$-dimensional Frobenius-invariant $\\mathbb{Q}_p$-vector subspace $W^{(b)}$ of $H^1_{\\dR}(C\/\\mathbb{Q}_p)$ on which Frobenius acts with characteristic polynomial $\\chi_{W^{(b)}}(t)$.\n \\begin{rmk}\\label{rmk:compatibility_frob}\n Technically speaking, Kedlaya's algorithm computes the Frobenius action on the Monsky--Washnitzer cohomology $H_{\\textup{MW}}^1(\\tilde{C}^{\\prime}\/\\mathbb{F}_p, \\mathbb{Q}_p)$ of the curve $\\tilde{C}^{\\prime}$ obtained from the reduction of $C$ modulo $p$ by removing the Weierstrass points. Since the odd part of $H_{\\textup{MW}}^1(\\tilde{C}^{\\prime}\/\\mathbb{F}_p,\\mathbb{Q}_p)$ is isomorphic to $H^1_{\\dR}(C\/\\mathbb{Q}_p)$ and the isomorphism is compatible with the Frobenius action on $H^1_{\\dR}(C\/\\mathbb{Q}_p)$ introduced in \\S \\ref{subsec:infty_sigma}, we will ignore this subtlety. For details, see for instance \\cite[Theorem 2.6]{Bogaart}.\n \\end{rmk}\n In order to compute the differentials $\\eta_1^{(b)}$ and $\\eta_2^{(b)}$ of Theorem \\ref{thm:Blakestad_main}, we work directly with $H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)$ and its $\\mathbb{Z}_p$-linear Frobenius action. As we saw in the proof of Proposition \\ref{prop:Blakestad_space_unit_root}, under our running assumptions, this $\\mathbb{Z}_p$-module is isomorphic to $H^0(\\mathcal{C},\\Omega^1_{\\mathcal{C}\/\\mathbb{Z}_p}(4\\infty))^{-}$ and the latter is freely generated by \n \\begin{equation}\\label{eq:mathcalB}\n\\mathcal{B} = \\left\\{\\frac{x^i dx}{2y}\\colon i =0,\\dots, 3\\right\\}.\n\\end{equation}\nFrom now on, we identify $H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)$ with the $\\mathbb{Z}_p$-span of $\\mathcal{B}$. The action of Frobenius, denoted by $\\phi^{*}$, on $H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)$ with respect to $\\mathcal{B}$ can be computed using Kedlaya's algorithm \\cite{kedlaya}.\n\nThe $\\mathbb{Q}_p$-vector space $W^{(b)}$ is obtained by tensoring with $\\mathbb{Q}_p$ the $\\mathbb{Z}_p$-submodule $W^{(b)}_0\\subset H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)$ defined as the kernel of the expansion at $P$-map $\\beta_P\\colon H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)\\to H^1_{\\dR}(\\hat{\\mathcal{C}}_{P}\/\\mathbb{Z}_p)$, for any $P\\in \\mathcal{C}(\\mathbb{Z}_p)$. The differentials $\\eta_1^{(b)}$, $\\eta_2^{(b)}$ belong to $W^{(b)}_0$ and, in fact, freely generate it as a $\\mathbb{Z}_p$-module. Moreover, $H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)$ is isomorphic to $W^{(b)}_0 \\oplus \\langle\\frac{dx}{2y}, \\frac{xdx}{2y}\\rangle_{\\mathbb{Z}_{p}}$.\n\nNow, \n\\begin{equation}\\label{eq:holo_pH1}\n\\phi^{*}\\left(\\frac{dx}{2y}\\right), \\phi^{*}\\left(\\frac{xdx}{2y}\\right)\\in pH^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p);\n\\end{equation}\nsee, for instance, \\cite[Proof of Lemma 3.4]{harrison}.\nIn addition, since the determinant of the matrix of Frobenius on any basis for $W^{(b)}_0$ is equal to $\\tilde{a}_2\\in \\mathbb{Z}_p^{\\times}$, we have $\\phi^{*}(W^{(b)}_0) = W^{(b)}_0$ by \\cite[XIII, Propositions 3.1 and 4.16]{Lang:algebra}. From these considerations, one deduces the following algorithm to compute a basis for $W^{(b)}_0$ modulo a prescribed precision. Note that essentially the same result is stated in higher generality in \\cite{Bes-Bal10}; however, no distinction seems to be made there between the $\\mathbb{Q}_p$-vector space $W^{(b)}$ and the $\\mathbb{Z}_p$-module $W^{(b)}_0$.\n\\begin{prop}[{\\hspace{1sp}\\cite[Proposition 6.1]{Bes-Bal10}}]\\label{cor:basis_using_Kedlaya}\nLet $M\\in M_4(\\mathbb{Z}_p)$ be the matrix of Frobenius with respect to $\\mathcal{B}$, and let $n$ be a positive integer. There is a basis for $W^{(b)}_0$ whose coefficients with respect to $\\mathcal{B}$ reduce modulo $p^n$ to the third and fourth column of $M^n$. \n\\end{prop}\nUsing Corollary \\ref{cor:basis_using_Kedlaya}, we can compute a basis for $W^{(b)}_0$ modulo $p^n$, from which we can deduce $\\eta_1^{(b)}$, $\\eta_2^{(b)}$ modulo $p^n$.\n\n\\begin{rmk}\nIf $p=3$, the set $\\mathcal{B}$ is not a basis for $H^1_{\\dR}(\\mathcal{C}\/\\mathbb{Z}_p)$ (see \\cite[\\S 5]{Bogaart}). \nHowever, with some extra care, it should be possible to compute the unit root eigenspace in a similar way and to relate this to the canonical sigma function of Blakestad (see Remark \\ref{rmk:inver_H1}).\n\\end{rmk}\n\n\\subsection{Division polynomials} \\label{subsec:division_polynomials}\nIn the previous two subsections, we discussed how to compute the expansion of various sigma functions. Recall that we used division polynomials to extend a sigma function outside of its domain of convergence.\n\nIn fact, if we want to compute the value of a local N\\'eron function using Definition \\ref{def:Neron_fct_above_p} or \\ref{def:Neron_fct_away_p}, we do not need to compute the division polynomial, but just its value at the point of interest. \\texttt{Magma} \\cite{magma} code to compute values of division polynomials is provided with\\cite{muller_de_jong}. We translated this into \\texttt{SageMath}, with some minor modifications (see Remark \\ref{rmk:code_div_poly_below}).\n\nFor the purpose of this problem, we may more generally assume that $C$ is defined over any characteristic $0$ field. Let $P$ be a point on the Jacobian, away from the theta divisor. The strategy is as follows:\n\\begin{enumerate}\n\\item For $1\\leq n\\leq 5$, Uchida computed $\\phi_n$ as a polynomial in the variables $\\wp_{ij}$ and $\\wp_{ijk}$ with coefficients in $\\mathbb{Z}[b_1,\\dots,b_5]$. Evaluating at $\\wp_{ij}(P),\\wp_{ijk}(P)$ gives $\\phi_n(P)$ for $1\\leq n\\leq 5$. \n\\item \\label{it:div_poly_2} Compute $\\phi_n(P)$ for $6\\leq n\\leq 8$ using the recurrence relation \\cite[Theorem 9 (corrected)]{Kanayama_corrections}. \n Since the computation of $\\phi_6(P)$ and $\\phi_8(P)$ requires division by $\\phi_2(P)$, we must assume that $2P\\not\\in \\Supp(\\Theta)$.\n\\item \\label{it:div_poly_3} Compute $\\phi_n(P)$ recursively for $n\\geq 8$ using \\cite[Example 6.6]{Uchida}. This requires division by\n\\begin{equation*}\n\\phi_5(P) - \\phi_4(P)\\phi_2(P)^3+\\phi_3(P)^3,\n\\end{equation*}\nso the formulae can only be applied when this is non-zero. For $n$ even we further need that $\\phi_2$ does not vanish at $P$. \n\\end{enumerate}\nFor certain values of $n$, the code by de Jong--M\\\"uller is not directly applicable if either $\\phi_2$ or $\\phi_5-\\phi_4\\phi_2^3 + \\phi_3^3$ vanishes at our point $P$ of interest. If we are only interested in the value at $P$ of the global height $h_p$, we may remedy the situation by computing $h_p(mP)$ for a non-zero integer $m$, and deducing the value $h_p(P)$ by quadraticity. Otherwise, we may compute $\\phi_n(P)$ by taking limits over points approaching $P$ of the recurrence relations of Steps \\eqref{it:div_poly_2} and \\eqref{it:div_poly_3}.\n\n\\begin{rmk}\\label{rmk:code_div_poly_below}\nThe recursion formulae of Step \\eqref{it:div_poly_3} express $\\phi_{2m}$ and $\\phi_{2m+1}$ in terms of $\\phi_k$ for $k\\in\\{2,3,4,5\\}\\cup \\{m-4,\\dots, m+4\\}$. In particular, in order to compute $\\phi_n$, it is not necessary to compute $\\phi_k$ for all $k\\leq n-1$; rather, it suffices to compute $\\phi_k$ for $k$ in some intervals. We implemented this improvement. The same trick is used in Harvey's implementation \\cite{harvey} on \\texttt{SageMath} of $p$-adic heights on elliptic curves.\n\\end{rmk}\n\n\n\n\\subsection{$p$-adic N\\'eron functions}\\label{subsec:implementation_Neron_fcts}\nIn order to compute the $p$-adic N\\'eron functions of Definitions \\ref{def:Neron_fct_above_p} and \\ref{def:Neron_fct_away_p}, it remains to discuss how to compute a suitable integer $m$ and how to work with an idele class character for a number field $K$ explicitly. \n\nThe latter problem is discussed in detail in \\cite[\\S 2.1]{QCnfs}. For the purpose of this work we restrict the implementation to the case $K=\\mathbb{Q}$, for which the $\\mathbb{Q}_p$-vector space of continuous idele class characters is one-dimensional, and generated by the cyclotomic character $\\chi^{\\textup{cyc}}$ (cf.\\ \\cite[Example 2.7]{QCnfs}). Explicitly, we have\n\\begin{equation*}\n\\chi^{\\textup{cyc}}_{p}(x) = \\log_p(x) \\qquad \\text{for all } x\\in \\mathbb{Q}_p^{\\times}, \n\\end{equation*}\nwhere $\\log_p$ is the branch of the $p$-adic logarithm that vanishes at $p$, and, for $q\\neq p$,\n\\begin{equation*}\n\\chi^{\\textup{cyc}}_q(x) = \\log_p|x|_q, \\qquad \\text{for all } x\\in \\mathbb{Q}_q^{\\times}.\n\\end{equation*}\nThe $p$-adic sigma functions that we are mostly interested in are the naive one and the canonical one (when this exists). If $p$ is odd, they both converge on the whole of $J_1(\\mathbb{Q}_p)$. Indeed, the canonical sigma function has integral coefficients, while the convergence of the naive sigma function is given by Theorem \\ref{thm:sigma_naive}\\thinspace{}\\ref{thm_sigma:part2}.\n\nTherefore, given $P\\in J(\\mathbb{Q}_p)$, we need to compute (a multiple of) its order in $J(\\mathbb{Q}_p)\/J_1(\\mathbb{Q}_p)$. If $p$ is a prime of good reduction, this is the order of $\\tilde{P}$ in the finite group $\\tilde{J}(\\mathbb{F}_p)$. \n\nIf $p$ is an odd prime of bad reduction, denote by $J_0(\\mathbb{Q}_p)$ the subgroup of $J(\\mathbb{Q}_p)$ consisting of the points that reduce to the component of the origin in the smooth part of  $\\tilde{J}$.\nWe can apply \\cite[Remark 5.12, Figure 1]{Bruin-Stoll:MWSieve} to determine the index of $J_1(\\mathbb{Q}_p)$ inside $J_0(\\mathbb{Q}_p)$, and the discussion on p.\\,294--295 of \\emph{loc.\\ cit.}\\ to determine a multiple of $P$ that belongs to $J_0(\\mathbb{Q}_p)$. Since it is easy to check whether a point belongs to $J_1(\\mathbb{Q}_p)$, we can alternatively compute successive multiples of $ [J_0(\\mathbb{Q}_p):J_1(\\mathbb{Q}_p)]P$, until we land in $J_1(\\mathbb{Q}_p)$. A trial and error strategy can also be applied when $p=2$.\n\n\\subsection{Example: A prime greater than $10^{6}$}\\label{subsec:eg_large_p}\nWe now consider the genus $2$ curve over $\\mathbb{Q}$ with LMFDB label \\href{http:\/\/www.lmfdb.org\/Genus2Curve\/Q\/160000\/c\/800000\/1}{160000.c.800000.1} \\cite{lmfdb}\n\\begin{equation*}\nC\\colon y^2 = x^5 - 1.\n\\end{equation*}\nIts Jacobian $J$ is geometrically simple and has complex multiplication by $\\mathbb{Q}(\\zeta_5)$, where $\\zeta_5$ is a primitive fifth root of unity. We shall see that every $p\\equiv 1\\bmod{10}$ is of good ordinary reduction and that the constants $c_{ij}$ corresponding to the unit root eigenspace for $p$ are all equal to zero. In particular, they are algebraic and independent of $p$: compare with Remark \\ref{rmk:CM_case}. This enables us to compute the canonical $p$-adic N\\'eron function at $p$ for large primes without using Proposition \\ref{cor:basis_using_Kedlaya} explicitly. \n\n\\begin{lemma}[{\\hspace{1sp}\\cite{yui1978jacobian}}]\nLet $p$ be a prime congruent to $1$ modulo $10$. Then $C$ has good reduction at $p$, and $J$ has good ordinary reduction at $p$.\n\\end{lemma}\n\\begin{proof}\nThe only primes at which the given equation for $C$ has bad reduction are $2$ and $5$. For ordinarity, it suffices to show that the Cartier--Manin matrix of $C$ modulo $p$ has non-zero determinant: see \\cite[Example 3.3]{yui1978jacobian}.\n\\end{proof}\n\n\\begin{lemma}\nLet $p$ be a prime congruent to $1$ modulo $10$. Then the classes of the differentials \n\\begin{equation*}\n\\eta_1^{(0)} = -3x^3\\frac{dx}{2y}, \\qquad \\eta_2^{(0)} = -x^2\\frac{dx}{2y}\n\\end{equation*}\nspan the unit root eigenspace of Frobenius. In particular, the canonical sigma function is equal to the naive sigma function.\n\\end{lemma}\n\\begin{proof}\nFirst note that $p$ splits completely in $\\mathbb{Q}(\\zeta_5)$. The $\\mathbb{Q}(\\zeta_5)$-automorphism $\\psi\\colon (x,y)\\mapsto (\\zeta_5 x, y)$ of the curve $C$ reduces to an automorphism $\\tilde{\\phi}$ over $\\mathbb{F}_p$, and this commutes with the $p$-th power Frobenius endomorphism $\\pi$. Therefore, the lifts of $\\pi\\circ \\tilde{\\psi}$ and $\\tilde{\\psi}\\circ \\pi$ induce homotopic maps on differentials \\cite[Theorem (2.4.4) (iii)]{vanderPut} and hence the same map in cohomology. In other words, for every differential $\\omega$ of the second kind,\n\\begin{equation}\\label{eq:Frobpsi_psiFrob}\n\\psi^{*}(\\phi^*([\\omega])) = \\phi^*(\\psi^{*}([\\omega])).\n\\end{equation}\nFor each $i\\in\\{0,1,2,3\\}$, the differential $\\omega_i = x^{i}\\frac{dx}{2y}$ is an eigenvector for $\\psi^{*}$ with eigenvalue $\\zeta_5^{i+1}$. Therefore, by \\eqref{eq:Frobpsi_psiFrob}, the matrix of Frobenius with respect to $\\mathcal{B}$ is diagonal. The result then follows by \\eqref{eq:holo_pH1}. \n\\end{proof}\n The lemmas apply to the prime $p = 10^6 + 81$ and we shall use them to compute the canonical $p$-adic N\\'eron function at $p$ for the point\n \\begin{equation*}\n P = [(-1-i,-2-i ) - (-1+i,2-i )]\\in J_{\\Theta}(\\mathbb{Q}).\n \\end{equation*}\n The first terms of the canonical $p$-adic sigma function can be deduced from the expansion of Appendix \\ref{app:sigma_expansion} by specialising to $b_1=b_2 = b_3=b_4 =c_{11} = c_{12} = c_{22}= 0$, $b_5 = -1$. For this reason, we omit here the expansion (although the precision of the computations below requires higher $T$-adic precision than the one given in the appendix). \n \nLet $m$ be a positive integer such that $mP\\in J_1(\\mathbb{Q}_p)\\setminus \\Supp(\\Theta)$. Since $m$ is large, the computation of $mP$ is expensive. However, if we compute the canonical $p$-adic sigma function up to $O(T_1,T_2)^n$, we can only determine $\\sigma_p(T(mP))$ modulo $p^n$, and hence there is no advantage in computing $T_i(mP)$ as a rational number, compared to a $p$-adic number to precision $O(p^n)$. In other words, we may perform multiplication-by-$m$ on the Jacobian over the $p$-adics, to a suitable $p$-adic precision.\n\nThe order of $J(\\mathbb{F}_p)$ is $m^{\\prime} = 1001600512000$ and the computation of $m^{\\prime} P$ shows that we may take $m=m^{\\prime}$.\n\nSimilarly, it suffices to compute the value of the $m$-th division polynomial at $P$ to finite $p$-adic precision; we do this as outlined in \\S \\ref{subsec:division_polynomials}. Combining the above intermediate computations, we conclude that the $p$-adic N\\'eron function for the prime $p$, at the point $P$, with respect to the idele class character $\\chi^{\\textup{cyc}}$ and the unit root eigenspace of Frobenius is:\n \\begin{align*}\n\\lambda_p(P) &=  790065 \\cdot p + 875980 \\cdot p^{2} + 899921 \\cdot p^{3} + 943161 \\cdot p^{4} + 701712 \\cdot p^{5} \\\\\n&+ 507099 \\cdot p^{6} + 399164 \\cdot p^{7} + 725683 \\cdot p^{8} + 423209 \\cdot p^{9} + 174881 \\cdot p^{10} \\\\\n&+ 96387 \\cdot p^{11} + 973189 \\cdot p^{12} + 88349 \\cdot p^{13} + 970515 \\cdot p^{14} + 117600 \\cdot p^{15} \\\\\n&+ 519019 \\cdot p^{16} + 639751 \\cdot p^{17} + 971144 \\cdot p^{18} + 996211 \\cdot p^{19} + O(p^{20}). \n\\end{align*}\nThe computation was run on a single core of a $32$-core $2.3$GHz AMD Opteron 6276 processor with 256GB RAM. \nAll the computations were run in \\texttt{SageMath} \\cite{sage} and took approximately 14.3 seconds in total. Of these, 12.6 seconds were needed for the computation of the expansion of the sigma function up to $O(T_1,T_2)^{21}$. We computed this as a power series with coefficients in $\\mathbb{Q}$, rather than approximating the coefficients $p$-adically.\n\nIn order to compute the global $p$-adic height $h_p(P)$, we also need to compute the values at $P$ of the $p$-adic N\\'eron functions at the primes $q\\neq p$. First, since\n\\begin{equation*}\nX_{11}(P) =5 ,\\qquad X_{12}(P) = -2,\\qquad X_{22}(P) =-2 ,\n\\end{equation*}\nand $C$ has good reduction at all the primes different from $2$ and $5$, by Remark \\ref{rmk:neron_away_p}\\thinspace{}\\ref{rmk:neron_away_p_good_reduction} the only non-trivial contributions can occur at $q\\in\\{2,5\\}$.\n\nThe point $2P$ reduces to the identity modulo $2$, and does not lie on the theta divisor. \nTherefore, by Definition \\ref{def:Neron_fct_away_p},\n\\begin{equation*}\n\\lambda_2(P) = -\\frac{1}{2}\\log_p\\left\\vert\\frac{T_1(2P)}{\\phi_2(P)}\\right\\vert_2 =  -\\frac{1}{2}\\log_p\\left\\vert\\frac{2 + O(2^7)}{-30} \\right\\vert_2= 0.\n\\end{equation*}\nAs far as $q = 5$ is concerned, by \\cite[Remark 5.12, Figure 1]{Bruin-Stoll:MWSieve}, the group $J_0(\\mathbb{Q}_5)\/J_1(\\mathbb{Q}_5)$ has order $25$. An explicit computation shows that $25P$ does not belong to $J_1(\\mathbb{Q}_5)$, but $50P$ does. Hence,\n\\begin{equation*}\n\\lambda_5(P) = - \\frac{2}{50^2}\\log_p\\left\\vert\\frac{T_1(50P)}{\\phi_{50}(P)}\\right\\vert_5 = -\\frac{1}{2\\cdot 625}\\log_p\\left\\vert\\frac{4\\cdot 5^2 + O(5^3)}{4\\cdot 5^{627} + O(5^{628})}\\right\\vert_5 = -\\frac{1}{2}\\log_p(5).\n\\end{equation*}\n\nIn conclusion, the global canonical $p$-adic height of $P$ is\n\\begin{align*}\nh_p(P) &= \\lambda_p(P) + \\lambda_5(P)\\\\\n&= 227482 \\cdot p + 997009 \\cdot p^{2} + 96340 \\cdot p^{3} + 795588 \\cdot p^{4} + 602398 \\cdot p^{5} \\\\\n&+ 562446 \\cdot p^{6} + 378071 \\cdot p^{7} + 977705 \\cdot p^{8} + 744905 \\cdot p^{9} + 778414 \\cdot p^{10} \\\\\n&+ 506461 \\cdot p^{11} + 834642 \\cdot p^{12} + 129041 \\cdot p^{13} + 687989 \\cdot p^{14} + 134678 \\cdot p^{15} \\\\\n&+ 452034 \\cdot p^{16} + 429426 \\cdot p^{17} + 552523 \\cdot p^{18} + 572577 \\cdot p^{19} + O(p^{20}).\n\\end{align*}\n\n\\subsection{Example: Coleman--Gross local heights via N\\'eron functions}\\label{subsec:example_hts_Neron_fcts}\nLet\n\\begin{equation*}\nC\\colon y^2 = x^3(x-1)^2 + 1,\n\\end{equation*}\nlet $P_1 = (1,-1)$, $P_2 = (0,1)\\in X(\\mathbb{Q})$, and let $D_1 = P_1-P_2$, $D_2 = P_2^{-} - P_1^{-} \\in \\operatorname{Div}^0(C)$. \nThe prime $p=11$ is of good reduction for $C$ (and hence for its Jacobian $J$), and of ordinary reduction for $J$. Let $W\\colonequals W^{(b)}\\subset H^1_{\\dR}(C\/\\mathbb{Q}_p) $ be the unit root eigenspace of Frobenius.\n\nIn this subsection, we verify the equalities of Corollaries \\ref{cor:lambda_eq_CG} and \\ref{cor:global_CG_same_as_this} numerically for the pair of divisors $D_1$, $D_2$, as follows. The Coleman--Gross local $p$-adic height pairings $\\langle D_1,D_2 \\rangle_{p,p,W}^N$ and $\\langle D_1,D_2 \\rangle_{p,q}^N$ ($q\\neq p$) were computed directly from their definition in \\cite[\\S 7.2.1]{BBM0} (the notation in \\emph{loc.\\ cit.}\\ differs from ours). We check that we obtain the same answer if we take a linear combination of values of $p$-adic N\\'eron functions satisfying the assumptions of  Corollary \\ref{cor:lambda_eq_CG}. \n\nSecondly, the divisors $D_1$ and $D_2$ are linearly equivalent, so, by Corollary \\ref{cor:global_CG_same_as_this}, the sum of the Coleman--Gross local $p$-adic height pairings on $D_1,D_2$ is equal to the canonical global $p$-adic height $h_p([D_1])$. We verify this numerically by computing $h_p([D_1])$ (almost) directly from its definition. \n\nTo ease notation, we will from now on omit the unit root subspace $W$ from the subscripts. \nConsider the following points in $J(\\mathbb{Q})$:\n\\begin{align*}\nu_1 = \\iota(P_1) = [P_1 - \\infty], \\qquad u_2 = \\iota(P_2) = [P_2-\\infty],\\\\\nv_1 = [P_2^{-} - P_1],\\qquad v_2 = [P_2^{-} - P_2], \\qquad v_3 = [P_1^{-}-P_1].\n\\end{align*}\nBy Example \\ref{eg:finding_u1_u2}, the points $u_1,u_2$ satisfy the assumptions of Corollary \\ref{cor:lambda_eq_CG}, for any rational prime $w=q$. \nWith this choice, the equality of the corollary translates to\n\\begin{equation}\\label{eq:eq_CG_Neron_in_practice}\n\\langle D_1,D_2\\rangle_{p,q}^N = -\\frac{1}{2}(2\\lambda_q(v_1) - \\lambda_q(v_2) - \\lambda_q(v_3)).\n\\end{equation}\nConsider first the case $q = p$. As a preliminary step in the computation of the values of $\\lambda_p$ appearing in \\eqref{eq:eq_CG_Neron_in_practice}, we need to determine the symmetric matrix $b$ corresponding to the choice of unit root eigenspace of Frobenius. In this case, we use Kedlaya's algorithm as explained in \\S \\ref{subsec:implementation_canonical_sigma}. We get\n\\begin{align*}\nb_{11} &= 6 + 6 \\cdot 11 + 3 \\cdot 11^{2} + 6 \\cdot 11^{4} + 2 \\cdot 11^{5} + 10 \\cdot 11^{6} + 11^{7} + 6 \\cdot 11^{8} + 9 \\cdot 11^{9} + O(11^{10}),\\\\\n b_{12}  &= 3 + 10 \\cdot 11 + 10 \\cdot 11^{2} + 11^{4} + 11^{5} + 5 \\cdot 11^{6} + 11^{7} + 3 \\cdot 11^{8} + 4 \\cdot 11^{9} + O(11^{10}) , \\\\\n  b_{22} &=  4 + 3 \\cdot 11 + 6 \\cdot 11^{2} + 6 \\cdot 11^{3} + 9 \\cdot 11^{4} + 10 \\cdot 11^{5} + 4 \\cdot 11^{6} + 5 \\cdot 11^{7} + 2 \\cdot 11^{8} + 2 \\cdot 11^{9} + O(11^{10}).\n\\end{align*}\nThe first terms of the canonical $p$-adic sigma function can be deduced from the expansion of Appendix \\ref{app:sigma_expansion} by specialising to $b_1=-2,b_2 =1, b_3=b_4 =0, b_5 = 1, c_{ij} = b_{ij}$ for all $1\\leq i\\leq j\\leq 2$.\n\nUsing Definition \\ref{def:Neron_fct_above_p} with $m=116$, $29$ and $58$, respectively, and the explicit techniques of \\S \\ref{subsec:implementation_formal_gp_sigma}--\\ref{subsec:implementation_Neron_fcts}, we compute:\n\\begin{align*}\n\\lambda_p(v_1) &= 9 \\cdot 11 + 8 \\cdot 11^{2} + 8 \\cdot 11^{3} + 4 \\cdot 11^{4} + 5 \\cdot 11^{5} + 10 \\cdot 11^{6} + 8 \\cdot 11^{7} + 9 \\cdot 11^{8} + O(11^{9}),\\\\\n\\lambda_p(v_2) &= 2 \\cdot 11^{3} + 6 \\cdot 11^{4} + 10 \\cdot 11^{5} + 10 \\cdot 11^{6} + 9 \\cdot 11^{7} + 10 \\cdot 11^{8} + O(11^{9}),\\\\\n\\lambda_p(v_3) &=2 \\cdot 11 + 4 \\cdot 11^{2} + 2 \\cdot 11^{3} + 6 \\cdot 11^{4} + 2 \\cdot 11^{5} + 5 \\cdot 11^{6} + 3 \\cdot 11^{7} + 11^{8} + O(11^{9}).\n\\end{align*} \nWe deduce that\n\\begin{equation*}\n\\langle D_1,D_2\\rangle_{p,p}^N = 3 \\cdot 11 + 4 \\cdot 11^{2} + 4 \\cdot 11^{3} + 11^{4} + 11^{5} + 3 \\cdot 11^{6} + 3 \\cdot 11^{7} + 7 \\cdot 11^{8} + O(11^{9}),\n\\end{equation*}\nwhich agrees with the computation of \\cite{BBM0} up to $O(11^7)$, i.e.\\ up to the precision of the computation in \\emph{loc.\\ cit.}\n\nNext, we consider the prime $q = 2$. By Definition \\ref{def:Neron_fct_away_p}, we have\n\\begin{align*}\n\\lambda_2(v_1) &= -\\frac{2}{12^2}\\log_p\\left\\vert\\frac{T_1(12v_1)}{\\phi_{12}(v_1)} \\right\\vert_{2} = - \\frac{7}{6}\\log_p(2)\\\\\n\\lambda_2(v_2) &=  -\\frac{2}{3^2}\\log_p\\left\\vert\\frac{T_1(3v_2)}{\\phi_3(v_2)} \\right\\vert_{2} = - \\frac{2}{3}\\log_p(2)\\\\\n\\lambda_2(v_3) &=  -\\frac{2}{2^2}\\log_p\\left\\vert\\frac{T_1(2v_3)}{\\phi_2(v_3)} \\right\\vert_{2} = 0,\n\\end{align*}\nfrom which we conclude that\n\\begin{equation*}\n\\langle D_1,D_2\\rangle_{p,q}^N = \\frac{5}{6}\\log_p(2). \n\\end{equation*}\nFinally, for all $q\\not\\in\\{2,11\\}$, we have $\\lambda_q(v_1) = \\lambda_q(v_2) = \\lambda_q(v_3) = 0$, and hence $\\langle D_1,D_2\\rangle_{p,q}^N = 0$, which completes the numerical verification of Corollary \\ref{cor:lambda_eq_CG}. \n\n\nAs far as the verification of Corollary \\ref{cor:global_CG_same_as_this} is concerned, since $4[D_1] = -v_2$,\n\\begin{equation*}\nh_p([D_1]) = \\frac{h_p(v_2)}{16} = \\frac{1}{16}(\\lambda_p(v_2) + \\lambda_2(v_2)) = 8 \\cdot 11 + 2 \\cdot 11^{4} + 6 \\cdot 11^{5} + 6 \\cdot 11^{6} + 5 \\cdot 11^{7} + 9 \\cdot 11^{8} + O(11^{9}),\n\\end{equation*}\nwhich equals $\\sum_{q} \\langle D_1, D_2 \\rangle_{p,q}^N$.\n\n\\subsection{Integrals of differentials of the first, second and third kind}\\label{subsec:diff_first_2nd_third}\nLet now $p$ be any rational prime ($p=2$ is allowed) and let $C$ be defined over $\\mathbb{Q}_p$ and given, as usual, by a monic degree $5$ model with coefficients in $\\mathbb{Z}_p$. We make no assumptions on the reduction. By \\S \\ref{subsec:implementation_formal_gp_sigma}, we can compute the expansion of the strict formal group logarithm $\\mathcal{L} = (\\mathcal{L}_1,\\mathcal{L}_2)$. By Proposition \\ref{prop:exp_exp_log}, this converges on $J_1(\\mathbb{Q}_p)$. Let $u\\in J(\\mathbb{Q}_p)$. It was explained in \\S \\ref{subsec:implementation_Neron_fcts} how we can determine a positive integer $m$ such that $mu\\in J_1(\\mathbb{Q}_p)$. Then, for each $i\\in \\{1,2\\}$, the Colmez integral of $\\Omega_i$ satisfies\n\\begin{equation*}\n\\int_0^{u} \\Omega_i =\\frac{1}{m}\\mathcal{L}_i(mu). \n\\end{equation*}\nLet $\\omega_i = x^{i-1}\\frac{dx}{2y}$ and $P_1,P_2\\in C(\\mathbb{Q}_p)$. By Definition \\ref{def:basis_inv_diff_inv_der} and Theorem \\ref{thm:uniquetheory}\\thinspace{}\\ref{change_of_vars}, we have\n\\begin{equation*}\n\\int_{P_2}^{P_1} \\omega_i = \\int_{0}^{[P_1 - P_2]} \\Omega_i.\n\\end{equation*}\nIn this way, we can compute integrals of holomorphic differentials on $C$, without any assumption on the reduction. \n\\begin{rmk}\nFor genus $2$ curves described by $y^2 = f(x)$ where $f(x)$ is a sextic (or quintic) polynomial, the more general formal group description of Flynn \\cite{Flynn2, Flynn5} can be used to compute integrals of holomorphic differentials. See for example \\cite{flynn:flexible} for an application to Chabauty's method.\n\\end{rmk}\n\nA non-holomorphic closed differential of the second kind on $J$ is not translation-invariant, but it is semi-invariant \\cite[Theorem 2.8]{barsotti}. This means that it is linearly equivalent to its pullback under translation by any $u\\in J$, and hence, in principle, we can use the formal group machinery to compute the Colmez integral of such a differential. For example, for $i\\in\\{1,2\\}$, consider the differential (Lemma \\ref{lemma:explicit_subspace_H1dRJ})\n\\begin{equation*}\n\\eta_{i,J}^{(0)} = -X_{1i} \\Omega_1 - X_{i2}\\Omega_2.\n\\end{equation*}\nUsing Theorem \\ref{thm:uniquetheory}\\thinspace{}\\ref{change_of_vars} and \\cite[Proposition 4.10]{Uchida}, we see that the integral of $\\eta_{i,J}^{(0)}$ must satisfy\n\\begin{equation*}\n\\int_{mv}^{mu}\\eta_{i,J}^{(0)} = m\\int_{v}^{u}\\eta_{i,J}^{(0)} + \\frac{1}{m}\\frac{D_i(\\phi_m)}{\\phi_m}(u) - \\frac{1}{m}\\frac{D_i(\\phi_m)}{\\phi_m}(v) \\qquad (u,v\\in J(\\mathbb{Q}_p)).\n\\end{equation*}\nBy choosing $m$ appropriately, the computation of the integral on the left hand side can be reduced to solving a system of differential equations in the formal group (i.e.\\ over a power series ring); note that this is in fact an intermediate step in the computation of the $p$-adic sigma function. Pulling back to $C$ using \\eqref{eq:eta_iJ_eta_i} yields a formula for the integral of $\\eta_i^{(0)}$. The implementation of \\S \\ref{subsec:division_polynomials} does not immediately equip us with a way of computing $D_i(\\phi_m)$ (since we do not compute the division polynomials, but only their values); however, one could derive from the recurrence relations of Kanayama \\cite[Theorem 9 (corrected)]{Kanayama_corrections} and Uchida \\cite[Example 6.6]{Uchida} (cf.\\ \\S \\ref{subsec:division_polynomials}) an expression for $D_i(\\phi_m)(u)$ in terms of $\\phi_k(u)$ ($k\\leq m$), $D_i(\\phi_k)(u)$ ($k < m$) and higher order derivatives of $\\phi_k$ for $k\\leq 5$. \n\nFinally, Corollary \\ref{cor:lambda_eq_CG} provides a formula for the computation of differentials of the third kind on $C$ by means of $p$-adic N\\'eron functions, formula which we tested numerically in \\S \\ref{subsec:example_hts_Neron_fcts}.\n\nIn summary, our implementation, although primarily aimed at the computation of $p$-adic N\\'eron functions and heights, can be adapted to compute integrals of differentials of the first, second and third kind on $C$, and thereby offers an alternative (for the specific setting of a genus $2$ odd degree hyperelliptic curve) to Coleman integration algorithms in good reduction due to Balakrishnan--Bradshaw--Kedlaya and Balakrishnan--Besser \\cite{BBK09, Bes-Bal10} and Colmez--Vologodsky integration algorithms in  bad reduction due to Katz--Kaya and Kaya \\cite{Katz_Kaya, kaya_vologodskyII}.\n\n\n\\section{An application: quadratic Chabauty for bihyperelliptic curves} \\label{sec:application_bihyper}\nThe goal of this section is twofold. Balakrishnan--Besser--M\\\"uller \\cite{BBM0, BalakrishnanBesserMullerIntegralPoints} described a Chabauty-like method for computing integral points on odd degree hyperelliptic curves over $\\mathbb{Q}$ with genus equal to the Mordell--Weil rank of the Jacobian. This uses an extension of Coleman--Gross local height pairings to divisors with non-disjoint support, and the quadraticity of the global height. The technique is known as \\emph{quadratic Chabauty} (for integral points). The intermediate goal of the section is to rephrase this method in the genus 2 case in terms of our setup. In fact, we do not just translate, but rather re-prove, the results of \\cite{BBM0} in terms of N\\'eron functions (see Remark \\ref{rmk:comparison_not_applicable} below). For this we crucially use results of Stoll \\cite{Stoll} and M\\\"uller--Stoll \\cite{mueller-stoll}.\n\nOur main goal is to describe a quadratic Chabauty-like method for determining the rational points on certain genus 4 bihyperelliptic curves $X$. In particular, we assume that $X$ has two genus 2 quotients, each satisfying the assumptions of quadratic Chabauty for integral points. We then describe a $p$-adic locally analytic function on $X(\\mathbb{Q}_p)$, which,  when restricted to $X(\\mathbb{Q})$, takes values in a finite explicit subset of $\\mathbb{Q}_p$. This is a genus $4$ analogue of the quadratic Chabauty method for genus $2$ bielliptic curves of Balakrishnan--Dogra \\cite{BDQCI}; see also \\cite{Bianchi20, BP22}.\n\n\nLet $C$ and $K$ be as in Sections \\ref{sec:sigma_functions} and \\ref{sec:padic_hts}: that is, $K$ is a number field and $C$ is the curve over $K$ defined by an equation of the form $y^2 = f(x)$, where $f(x)$ is a monic polynomial of degree $5$ with coefficients in the ring of integers of $K$ and no repeated roots. Fix a prime $p$ and a non-trivial continuous $\\mathbb{Q}_p$-valued idele class character $\\chi$ for $K$. Given a non-archimedean place $v$ of $K$, let $\\lambda_v$ be the local $p$-adic N\\'eron function at $v$ (with respect to a choice of subspace of $H^1_{\\dR}(C\/K_v)$ if $\\chi$ is ramified at $v$).\n\nIf $\\chi$ is unramified at $v$, we saw (Lemma \\ref{lemma:equals_naive}) that there exists a finite index subgroup $H_v$ of $J(K_v)$ such that, for all $u\\in H_v\\setminus \\Supp(\\Theta)$, the local N\\'eron function $\\lambda_v$ only depends on the $v$-adic valuation of $X_{ij}(u)$, for $i,j\\in\\{1,2\\}$. More generally, we have the following. For $u\\in J_{\\Theta}(K_v)$, let\n\\begin{equation*}\n\\lambda_v^{\\naive}(u)= -\\frac{1}{n_v}\\chi_v(\\max_{i,j}\\{|X_{ij}(u)|_v,1\\})\n\\end{equation*}\nand let $\\mu_v\\colon J(K_v)\\to \\mathbb{Q}$ be the function of \\cite[Definition 3.1]{mueller-stoll}. \nThen the $p$-adic N\\'eron function at $v$ is obtained from $\\lambda_v^{\\naive}$ by adding a correction term:\n\\begin{equation*}\n\\lambda_v(u) = \\lambda_v^{\\naive}(u) +\\frac{1}{n_v}\\mu_v(u)\\chi_v(\\pi_v),\n\\end{equation*}\nwhere $\\pi_v$ is a uniformiser in $K_v$.\nDenote by $k_v$ the residue field of $K_v$, and let $\\Phi$ be the component group of the N\\'eron model of $J$ over $\\operatorname{Spec}(\\mathcal{O}_v)$, where $\\mathcal{O}_v$ is the ring of integers of $K_v$.\n\\begin{thm}[{\\hspace{1sp}\\cite[Theorems 3.10, 11.3, 7.4, Proposition 12.3, Lemma 12.5]{mueller-stoll}}] \\label{thm:properties_mu}\\leavevmode\n\\begin{enumerate}[label=(\\roman*)]\n\\item The set $H_v = \\{u\\in J(K_v):\\mu_v(u) = 0\\}$ is a subgroup of finite index of $J(K_v)$ containing $J_1(K_v)$.\n\\item The function $\\mu_v$ factors through $J(K_v)\/H_v$.\n\\item For every $u\\in J(K_v)$, we have $0\\leq \\mu_v(u) \\leq \\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(\\Delta)}{4}$, where $\\Delta$ is the discriminant of $C$. \n\\item Let $N$ be the exponent of $\\Phi(\\overline{k_v})$. Then, for every $u\\in J(K_v)$, we have $\\mu_v(u)\\in \\frac{1}{N}\\mathbb{Z}$. Moreover, $N\\leq \\max\\{2,\\lfloor{\\frac{\\mathop{\\mathrm{ord}}\\nolimits_v(\\Delta)^2}{3}\\rfloor}\\}$.\n\\item If our fixed equation for $C$ determines a model over $\\mathcal{O}_v$ with rational singularities, then $\\mu_v$ factors through $\\Phi(k_v)$.\n\\end{enumerate}\n\\end{thm}\n\nConsider now the function\n\\begin{equation*}\n\\nu_v\\colon \\{P\\in C(K_v): y(P)\\neq 0,\\infty \\}\\to \\mathbb{Q}_p, \\qquad \\nu_v(P) = \\lambda_v(2\\iota(P)) + \\frac{2}{n_v}\\chi_v(2y(P)).\n\\end{equation*}\n\nDenote by $C(\\mathcal{O}_v)$ the set of affine points with coordinates in $\\mathcal{O}_v$, with respect to our fixed equation for $C$.\n\n\\begin{lemma}\\label{lemma:nu_v}If $\\chi$ is unramified at $v$, the function $\\nu_v$ satisfies the following properties:\n\\begin{enumerate}[label=(\\roman*)]\n\\item\\label{lemma:nu_v_integral}  If $P\\in C(\\mathcal{O}_v)\\setminus \\{P:y(P) = 0\\}$, then\n\\begin{equation*}\n\\nu_v(P) = -\\frac{2}{n_v}\\chi_v(\\max\\{|f^{\\prime}(x(P))|_v, |2y(P)|_v\\}) +\\frac{1}{n_v}\\mu_v(2\\iota(P))\\chi_v(\\pi_v).\n\\end{equation*}\nIn particular, the set $\\Gamma_v\\colonequals\\nu_v(C(\\mathcal{O}_v)\\setminus\\{P:y(P) = 0\\})$ is finite, and there exists a computable finite set $\\Gamma_v^{\\prime}\\subset\\mathbb{Q}_p$ such that $\\Gamma_v\\subset\\Gamma_v^{\\prime}$. If $v$ is a prime of good reduction for our equation for $C$, we may take $\\Gamma_v^{\\prime} = \\{0\\}$.\n\\item \\label{lemma:nu_v_nonintegral} If $P\\not\\in C(\\mathcal{O}_v)$, then $\\nu_v(P) = \\frac{8}{n_v}\\chi_v(x(P))$. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor $P$ in the domain of $\\nu_v$, let \n\\begin{equation*}\n \\nu_v^{\\naive}(P)= \\lambda_v^{\\naive}(2\\iota(P)) + \\frac{2}{n_v}\\chi_v(2y(P)).\n\\end{equation*}\nIt follows from \\cite[\\S 2]{FlynnSmart} that \n\\begin{align*}\nX_{22}(2\\iota(P)) = 2x(P), \\quad X_{12}(2\\iota(P)) = -x(P)^2, \\\\\nX_{11}(2\\iota(P)) = \\frac{f^{\\prime}(x(P))^2 -(2y(P))^2(6x(P)^3+4b_1x(P)^2 + 2b_2x(P) + b_3)}{(2y(P))^2}.\n\\end{align*}\nTherefore, if $P\\in C(\\mathcal{O}_v)$, we have \n\\begin{equation*}\n\\nu_v^{\\naive}(P) = -\\frac{2}{n_v}\\chi_v(\\max\\{|f^{\\prime}(x(P))|_v, |2y(P)|_v\\}).\n\\end{equation*}\nThis can take finitely many values since $\\max\\{|f^{\\prime}(x(P))|_v, |2y(P)|_v\\}\\geq |2\\Disc(f)|_v$, where $\\Disc(f)$ is the discriminant of $f$. \n In particular, if $2\\Disc(f)$ is a $v$-adic unit, then $\\nu_v^{\\naive}(P)$ vanishes. \n\nIf $P\\not\\in C(\\mathcal{O}_v)$, the coordinates of $P$ satisfy $2\\mathop{\\mathrm{ord}}\\nolimits_v(y(P)) = 5\\mathop{\\mathrm{ord}}\\nolimits_v(x(P))$. Therefore, since the numerator of $X_{11}(2\\iota(P))$ is monic of degree $8$ as a polynomial in $x(P)$, we have\n\\begin{equation*}\n\\nu_v^{\\naive}(P) = \\frac{8}{n_v}\\chi_v(x(P)).\n\\end{equation*}\nIt remains to understand how $\\nu_v$ differs from $\\nu_v^{\\naive}$. For this, we apply Theorem \\ref{thm:properties_mu}.\n\\end{proof}\n\n\nAssume now for simplicity that $K=\\mathbb{Q}$. Recall from \\S \\ref{subsec:Colm_int} that every locally analytic group homomorphism $J(\\mathbb{Q}_p)\\to\\mathbb{Q}_p$ arises as an extension to $J(\\mathbb{Q}_p)$ of a formal group homomorphism to $\\mathbb{G}_a$. By abuse of notation, denote by\n\\begin{equation*}\n\\mathcal{L}=(\\mathcal{L}_1,\\mathcal{L}_2)\\colon J(\\mathbb{Q}_p)\\otimes \\mathbb{Q}_p\\to\\mathbb{Q}_p^{2}\n\\end{equation*}\nthe $\\mathbb{Q}_p$-linear map induced by the strict formal group logarithm.\n\\begin{ass}\\label{ass:QC_int}\nThe restriction of $\\mathcal{L}$ to $J(\\mathbb{Q})\\otimes \\mathbb{Q}_p$ is injective. \n\\end{ass}\n\n\\begin{rmk}\nIf the rank of $J(\\mathbb{Q})$ is at most $2$, Assumption \\ref{ass:QC_int} will often be satisfied: if $J$ is simple, see \\cite[Conjecture 1]{waldschmidt}.\n\\end{rmk}\nUnder Assumption \\ref{ass:QC_int}, any $\\mathbb{Q}_p$-linear map $J(\\mathbb{Q})\\otimes \\mathbb{Q}_p\\to \\mathbb{Q}_p$ is a linear combination of $\\mathcal{L}_1$ and $\\mathcal{L}_2$, and any quadratic form $J(\\mathbb{Q})\\otimes \\mathbb{Q}_p\\to\\mathbb{Q}_p$ is a linear combination of $\\mathcal{Q} = \\{\\mathcal{L}_1^2, \\mathcal{L}_1\\mathcal{L}_2,\\mathcal{L}_2^2\\}$. The latter applies in particular to the $p$-adic height $h_p$: there exist $\\alpha_1,\\alpha_2,\\alpha_3\\in \\mathbb{Q}_p$ such that\n\\begin{equation}\\label{eq:h_p_basis_qf}\n h_p(u) = \\alpha_1\\mathcal{L}_1^2 (u)+ \\alpha_2\\mathcal{L}_1(u)\\mathcal{L}_2(u) + \\alpha_3\\mathcal{L}_2^2(u) \\qquad \\text{for all }u\\in J(\\mathbb{Q}). \n \\end{equation} \n Restricting this equality to points of the form $2\\iota(P)$, where $P$ is a point on $C$, and writing $h_p$ as a sum of local N\\'eron functions, one can use Equation \\eqref{eq:h_p_basis_qf} and Lemma \\ref{lemma:nu_v}\\thinspace{}\\ref{lemma:nu_v_integral} to write down a locally analytic function $\\rho\\colon C(\\mathbb{Z}_p)\\to \\mathbb{Q}_p$ and a finite set $\\Gamma\\subset\\mathbb{Q}_p$ such that $\\rho(C(\\mathbb{Z}))\\subset \\Gamma$.  In other words, Lemma \\ref{lemma:nu_v} allows us to phrase the quadratic Chabauty method of \\cite{BBM0} in terms of $p$-adic N\\'eron functions, in place of local Coleman--Gross height pairings.\n\n\\begin{rmk}\\label{rmk:comparison_not_applicable}\nCorollary \\ref{cor:lambda_eq_CG} is a comparison result between the Coleman--Gross local height pairings and the $p$-adic N\\'eron functions. However, it only applies to divisors with disjoint support, while the method of \\cite{BBM0} crucially requires an extension of the Coleman--Gross local heights to arbitrary divisors. Lemma \\ref{lemma:nu_v} allows us to give a direct proof that suitable local N\\'eron functions can alternatively be used. \nOn the other hand, in \\cite{BKM22}, we prove a comparison result between these extended Coleman--Gross local height pairings and the N\\'eron functions, from which the rephrasing of quadratic Chabauty for $C(\\mathbb{Z})$ in terms of the $\\lambda_q$ is straightforward.\n\nWe also observe that the explicit examples of \\cite{BBM0, BalakrishnanBesserMullerIntegralPoints} rely, for the computation of $\\mathcal{L}(\\iota(P))$,  on algorithms for Coleman integration on $C$ based on Kedlaya's algorithm \\cite{kedlaya}. In our genus $2$ setting, we may replace this step with \\S \\ref{subsec:implementation_formal_gp_sigma} (see also \\S \\ref{subsec:diff_first_2nd_third}).\n\\end{rmk}\n\nWe now apply similar ideas to give an explicit quadratic-Chabauty-type method for determining the \\emph{rational} points on certain genus $4$ curves that admit degree $2$ maps to curves in the same form as $C$. This is an analogue of the explicit quadratic Chabauty method for the rational points on genus $2$ bielliptic curves due to Balakrishnan--Dogra \\cite{BDQCI}; our proof is based on the proofs thereof given in \\cite[Proposition 6.5]{Bianchi20} and \\cite[Theorem 2.3]{BP22}. \n\nIn particular, we give a proof that does not use Kim's theory \\cite{KimP1, Kimunipotent}, but only the properties of the local N\\'eron functions that we have described so far. As such, it is much more elementary in nature to the general framework of quadratic Chabauty for rational points of \\cite{BDQCI, SplitCartan}. \n \nUnlike the above discussion on integral points on $C$, to the author's knowledge, Proposition \\ref{prop:QC_bihyper} below is not a rephrasing of a  result involving Coleman--Gross local heights already appearing in the literature. On the other hand, using the comparison result of \\cite{BKM22} mentioned in  Remark \\ref{rmk:comparison_not_applicable}, it could be rephrased in terms of Coleman--Gross heights.\n\nSuppose that $X$ is a genus $4$ hyperelliptic curve over $\\mathbb{Q}$ given by an equation of the form\n\\begin{equation}\\label{eq:X}\nX\\colon y^2 = f_5x^{10} + f_4 x^8 + f_3 x^6 + f_2x^4 + f_1x^2 + f_0 , \\qquad f_i\\in \\mathbb{Z}.\n\\end{equation}\nThere are maps $\\varphi_1\\colon X\\to C_1$, $\\varphi_2\\colon X\\to C_2$, to the following curves $C_1$ and $C_2$:\n\\begin{align*}\nC_1\\colon y^2 = x^5 + f_4x^4 + f_3f_5x^3 + f_2f_5^2x^2 + f_1f_5^3x + f_0 f_5^4,\\qquad &\\varphi_1(x,y) = (f_5 x^2,f_5^2 y),\\\\\nC_2 \\colon y^2 = x^5 + f_1 x^4 + f_0f_2x^3 + f_0^2f_3x^2 + f_0^3f_4x + f_0^4 f_5,\\qquad &\\varphi_2(x,y) = (f_0x^{-2},f_0^2yx^{-5}).\n\\end{align*}\nDenote by $J_1$ and $J_2$ the Jacobian of $C_1$ and $C_2$, respectively, and by $\\mathcal{Q}_i$ the set of quadratic forms $\\{\\mathcal{L}_1^{2},\\mathcal{L}_1\\mathcal{L}_2,\\mathcal{L}_2^2\\}$ for $J_i$. Let $Z =  X(\\mathbb{Q}_p)\\setminus \\{z: x(z)\\in \\{0,\\infty\\}\\text{ or } y(z) = 0\\}$.\n\n\\begin{prop}\\label{prop:QC_bihyper}\nSuppose that $p$ is a prime of good reduction for the Equation \\eqref{eq:X}, and that $J_i$ satisfies Assumption \\ref{ass:QC_int} for every $i\\in\\{1,2\\}$,  so the global $p$-adic height on $J_i(\\mathbb{Q})$ is equal to the restriction to $J_i(\\mathbb{Q})$ of $\\sum_{Q\\in \\mathcal{Q}_i} \\alpha_{Q}^{(i)}Q$, for some $\\alpha_Q^{(i)}\\in\\mathbb{Q}_p$. The function $\\rho\\colon Z\\to \\mathbb{Q}_p$, defined by\n\\begin{align*}\n\\rho(z) &=\\lambda_p(2\\iota(\\varphi_1(z))) - \\lambda_p(2\\iota(\\varphi_2(z))) - 6\\chi_p(x(z))\\\\ \n&- \\sum_{Q\\in \\mathcal{Q}_1} \\alpha_{Q}^{(1)}Q(2\\iota(\\varphi_1(z))) + \\sum_{Q\\in \\mathcal{Q}_2} \\alpha_{Q}^{(2)}Q(2\\iota(\\varphi_2(z))),\n\\end{align*}\ncan be continued to a locally analytic function $\\tilde{\\rho}\\colon X(\\mathbb{Q}_p)\\to \\mathbb{Q}_p$. Moreover, there exists a finite and computable set $\\Upsilon\\subset\\mathbb{Q}_p$ such that\n\\begin{equation*}\n\\tilde{\\rho}(X(\\mathbb{Q}))\\subset \\Upsilon.\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nAt each point in $X(\\mathbb{Q}_p)\\setminus Z$, there are exactly two terms of $\\rho(z)$ that have a logarithmic singularity:\n\\begin{itemize}\n\\item if $x(z) = \\infty$, the terms are: $\\lambda_p(2\\iota(\\varphi_1(z)))$ and $-6\\chi_p(x(z))$;\n\\item if $x(z) = 0$, the terms are: $-\\lambda_p(2\\iota(\\varphi_2(z)))$ and $-6\\chi_p(x(z))$;\n\\item if $y(z) = 0$, the terms are: $\\lambda_p(2\\iota(\\varphi_1(z)))$ and $-\\lambda_p(2\\iota(\\varphi_2(z)))$.\n\\end{itemize}\nIn each of the three cases, we claim that the logarithmic singularities coming from the two terms cancel out. We prove this for $x(z) = \\infty$ and leave the other cases as an exercise to the reader. So assume $x(z) = \\infty$ and let $t$ be a uniformiser at $z$, reducing to a uniformiser modulo $p$: without loss of generality, we choose $t = x^{-1}$. Since $\\varphi_1$ is unramified at $z$, for any local coordinate $t_1$ at $\\varphi_1(z)$, we have $t_1(\\varphi_1(z(t))) = tu(t)$ for some unit power series $u(t)\\in \\mathbb{Z}_p[[t]]$. By definition of $\\lambda_p$, the logarithmic term of $\\lambda_p(2\\iota(\\varphi_1(z(t))))$ is the same as that of $-2\\chi_p(T_1(2\\iota(\\varphi_1(z(t)))))$. By Lemma \\ref{lemma:Ti_terms_t}, this is $-6\\chi_p(t) = 6\\chi_p(x(t))$. \n\nWe prove the existence of a finite computable set $\\Upsilon$ for $X(\\mathbb{Q})\\setminus Z$ (the limiting values will follow by continuity).\nIf $z\\in X(\\mathbb{Q})\\setminus Z$, we have\n\\begin{align*}\n\\rho(z)& =\\sum_{q\\neq p}(- \\lambda_q(2\\iota(\\varphi_1(z))) + \\lambda_q(2\\iota(\\varphi_2(z))) +6\\chi_q(x(z)))\\\\\n&= \\sum_{q\\neq p} (- \\nu_q(\\varphi_1(z)) + \\nu_q(\\varphi_2(z)) +4\\chi_q(f_5f_0^{-1}x(z)^4))=: \\sum_{q\\neq p} w_q(z).\n\\end{align*}\nLet $Z_q$ be defined analogously to $Z$, but for the prime $q$. We claim that there exists a finite computable set $\\Upsilon_q$ such that $w_q(Z_q)\\subset \\Upsilon_q$, and, moreover, that we can take $\\Upsilon_q = \\{0\\}$ at every prime $q$ of good reduction; the statement of the proposition then follows from this.  In order to prove the existence and properties of $\\Upsilon_q$, we analyse the $q$-adic valuation of $x(z)$ and, correspondingly, that of $x(\\varphi_1(z))$ and $x(\\varphi_2(z))$ and apply Lemma \\ref{lemma:nu_v}. We distinguish between two cases: \n\\begin{itemize}\n\\item $\\varphi_1(z)$ and $\\varphi_2(z)$ are both integral if and only if $-\\mathop{\\mathrm{ord}}\\nolimits_q(f_5)\\leq 2\\mathop{\\mathrm{ord}}\\nolimits_q(x(z))\\leq \\mathop{\\mathrm{ord}}\\nolimits_q(f_0)$.  In this case, each of the three terms in $w_q(z)$ takes values in a finite computable set (by Lemma \\ref{lemma:nu_v}\\thinspace \\ref{lemma:nu_v_integral} and the fact that $\\chi_q(x(z))$ only depends on the valuation of $x(z)$).\n\\item In the remaining cases, exactly one of $\\varphi_1(z)$ and $\\varphi_2(z)$ is integral. Say $\\varphi_i(z)$ is integral and $\\varphi_j(z)$ is not. Then we apply Lemma \\ref{lemma:nu_v}\\thinspace{}\\ref{lemma:nu_v_integral} to $\\nu_q(\\varphi_i(z))$; moreover, by Lemma \\ref{lemma:nu_v}\\thinspace{}\\ref{lemma:nu_v_nonintegral}, \n\\begin{equation*}\n(-1)^j \\nu_q(\\varphi_j(z)) + 4\\chi_q(f_5f_0^{-1}x(z)^4) = (-1)^{j}4\\chi_q(f_5f_0). \\qedhere\n\\end{equation*} \n\\end{itemize}\n\\end{proof}\n\n\\begin{rmk}\nWith Lemma \\ref{lemma:Ti_terms_t} and Lemma \\ref{lemma:nu_v} on hand, the proof of Proposition \\ref{prop:QC_bihyper} is very similar to the proof of the genus $2$ case in \\cite[Theorem 2.3]{BP22}, to which we refer the reader for more details.\n\\end{rmk}\n\n\n\\begin{example}\nConsider\n\\begin{equation*}\nX\\colon y^2 = x^{10} - x^6 +1\n\\end{equation*}\nwith corresponding genus $2$ curves\n\\begin{equation*}\nC_1\\colon y^2 = x^5 - x^3 + 1, \\qquad C_2\\colon y^2 = x^5 - x^2 + 1.\n\\end{equation*}\nLet $p=5$, let $\\chi = \\chi^{\\textup{cyc}}$ and consider the $p$-adic local N\\'eron functions on $C_1$ and $C_2$ with respect to their naive sigma functions. The rank of $J_i(\\mathbb{Q})$ is equal to $2$ for each $i$, and the assumptions of Proposition \\ref{prop:QC_bihyper} apply. \n\nThe discriminants of $C_1$ and $C_2$ are both equal to $2^8\\cdot 7\\cdot 431$. Therefore, with reference to the proof of the proposition, in order to determine a suitable set $\\Upsilon$ we have to compute $\\Upsilon_q$, for $q\\in\\{2,7,431\\}$, which reduces to computing $\\Gamma_q$ or a finite superset $\\Gamma_q^{\\prime}$ (as in Lemma \\ref{lemma:nu_v}\\thinspace{}\\ref{lemma:nu_v_integral}) for each of $C_1$ and $C_2$.  \nFor $q\\in\\{7,431\\}$, by \\cite[Proposition 5.2 and Remark below]{Stoll}, $\\mu_q$ is identically zero on $J_i(\\mathbb{Q}_q)$, for each $i$. Moreover, there is no point in $C_i(\\mathbb{Z}_q)$ for which the function $\\nu_q^{\\naive}$ defined in the proof of Lemma \\ref{lemma:nu_v} is non-trivial.\n Therefore, the set $\\Gamma_q$ is equal to $0$ for each of $C_1$ and $C_2$, and $\\Upsilon = \\Upsilon_2$. \n\nWe omit a proof of the following fact: we may take\\footnote{Rather than using Lemma \\ref{lemma:nu_v}\\thinspace{}\\ref{lemma:nu_v_integral}, this set was computed using the comparison results of \\cite{BKM22} mentioned in Remark \\ref{rmk:comparison_not_applicable} and the results in \\cite{BBM0, BalakrishnanBesserMullerIntegralPoints}. We thank Steffen M\\\"uller for computing that the set $T$ of \\cite[Theorem 3.1]{BBM0} is equal to $\\{0,-\\frac{2}{3}\\log_p(2)\\}$ for $C_1$, and to $\\{0,-\\frac{1}{2}\\log_p(2)\\}$ for $C_2$. The set $\\Upsilon$ was deduced from this.}\n\\begin{equation}\\label{eq:Gamma}\n\\Upsilon_2 = \\left\\{0, \\frac{8}{3}\\log_p(2), -2\\log_p(2)\\right\\}. \n\\end{equation}\n\nWe can compute the expansion of $\\tilde{\\rho}(z)$ in every residue disc of the reduction map $X(\\mathbb{Q}_p)\\to \\tilde{X}(\\mathbb{F}_p)$ and study the set\n\\begin{equation*}\n \\{z\\in X(\\mathbb{Q}_p):\\tilde{\\rho}(z) - \\upsilon,\\text{ for some } \\upsilon\\in \\Upsilon\\}\\supseteq X(\\mathbb{Q}).\n\\end{equation*}\n\n\nFor example, the points in $X(\\mathbb{Q}_p)$ reducing to $\\overline{(0,4)}\\in X(\\mathbb{F}_p)$ are parametrised by $t = \\frac{x}{p}\\in \\mathbb{Z}_p$ and \n\\begin{equation*}\n\\tilde{\\rho}(z(t)) = 2\\cdot 5 + 2\\cdot 5^2  +(2\\cdot 5^2 + 2\\cdot 5^3 )t^2 + (3\\cdot 5^3 )t^4  + O(5^{4}, t^6).\n\\end{equation*}\n\n\\begin{itemize}\n\\item If $\\upsilon =0$ or $\\upsilon = -2\\log_p(2) = 5 +O(5^2)$, the power series $\\rho(z(t)) - \\upsilon$ has no zeros in $\\mathbb{Z}_p$;\n\\item If $\\upsilon =\\frac{8}{3}\\log_p(2) = 2\\cdot 5 + 2\\cdot 5^2 + O(5^4)$, the power series $\\tilde{\\rho}(z(t)) - \\upsilon$  has a double root at $t=0$: this corresponds to the rational point $(0,-1)\\in X(\\mathbb{Q})$. The root has multiplicity two because $(0,-1)$ is fixed by the automorphism of $X$ mapping $(x,y)$ to $(-x,y)$, under which $\\tilde{\\rho}$ is invariant.\n\\end{itemize}\nThis example is intended as an informal illustration of Proposition \\ref{prop:QC_bihyper}, but upon writing down an argument for \\eqref{eq:Gamma} and more details on how we computed the expansion of $\\tilde{\\rho}$, it could be turned into a proof that $(0,-1)$ is the unique rational point reducing to $\\overline{(0,4)}$ modulo $5$. By repeating this procedure in every residue disc of $X(\\mathbb{Q}_p)$ (possibly in combination with the Mordell--Weil sieve), it should be possible to determine the full set of rational points $X(\\mathbb{Q})$.\n\\end{example}\n\n\\begin{rmk}\\label{rmk:QC_bihyper_gen}\nProposition \\ref{prop:QC_bihyper} is phrased in terms of $p$-adic local N\\'eron functions, which we defined in this article for genus $2$ curves and, more generally, we can compute on genus $\\leq 2$ curves. As previously mentioned, the proposition admits a version in terms of Coleman--Gross local height pairings of a divisor with itself. More generally, in terms of such Coleman--Gross local heights, the result can be extended to rational points on even genus $d-1\\geq 2$ hyperelliptic curves of the form:\n\\begin{equation*}\nX\\colon y^2 = \\sum_{n=0}^{d} f_n x^{2n}, \\qquad f_n\\in \\mathbb{Z}.\n\\end{equation*}\n\\end{rmk}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStreamer discharges are transient discharges that serve as precursors to other gas discharges such as sparks and lightning leaders. They are rapidly growing ionized channels that are characterized by a curved space charge layer around their plasma body, which screens the electric field in their interior and enhances it ahead of them~\\cite{Dhali1987, vitello_simulation_1994, Babaeva1997, Kulikovsky1998, Pancheshnyi_2005, ebert_2010, Nijdam_review_2020}. The enhanced field in the active zone at the streamer head exceeds the electric breakdown value, and the multiplication of electrons in this region drives the propagation of the streamer~\\cite{nijdam_probing_2010}. Streamers have multiple applications in various fields~\\cite{becker2004,Fridman_2005,bruggeman2013atmospheric,bruggeman2017foundations,Adamovich_2017} including, but not limited to, medicine~\\cite{fridman_applied_2008,graves2014,laroussi2014}, combustion~\\cite{popov2016,starikovskaia_plasma-assisted_2014}, and surface treatments~\\cite{bardos2010}.\n\nIn a recent paper~\\cite{francisco2021}, we have studied single positive streamers in dry air in a homogeneous background electric field of 15~kV\/cm, which is about half the breakdown field, at standard temperature and pressure. The radius and the velocity of the streamers increased with the streamer length, as observed by many authors before. When the electron attachment rate was artificially increased in regions below electric breakdown, we found that with increasing attachment rate, streamer velocities and radii could grow less, not at all, or even decrease.\nAdditionally, streamer heads could keep propagating even if the conductivity of the streamer channels was already negligible a short distance behind the streamer head. We did not specify gases where such dynamics could actually be observed.\n\nIn the current work, we show that the same variation of streamer dynamics can occur in ambient air by simply decreasing the homogeneous background electric field. We find that for a background field of about 4.675~kV\/cm, the streamer head propagates with a constant radius and velocity. The current that flows through the streamer channel is already negligible close behind the head - the electric field returns to the background field value at the back of an electrically isolated streamer head. If the background electric field is even smaller, the streamer velocity and radius decrease while the maximal electric field at the head rapidly increases, and this could go on until the streamer stops. Finding uniform streamer propagation in STP air confirms the old concept of the stability field~\\cite{phelps_fieldenchanced, griffiths_effectofairpressure, gallimberti_longspark} that is frequently used in high voltage engineering but had little support up to now from fundamental physical modeling.\n\nThe paper is structured as follows. Details about the numerical modeling are presented in section~\\ref{sec:model}, where the computational domain is described along with the initial conditions of the simulations in section~\\ref{sec:init}. Section~\\ref{sec:results} features and discusses the results of our simulations. In section~\\ref{sec:2cases}, we present the case of a uniformly translating streamer in ambient air together with the more familiar case of an accelerating streamer, and in section~\\ref{sec:behaviors}, we show how streamer behaviour more generally depends on the background electric field. We also include decelerating streamers in that section. Section~\\ref{sec:validation} has comparisons between our simulation results with experimental measurements, and we discuss there the original concept of the stability field and its connection to our solitary streamers.\nWe conclude in section~\\ref{sec:conclusion}, where we summarize our results and communicate ideas for future studies.\n\\section{Discharge Model\\label{sec:model}}\n\n\\subsection{Model equations and reactions}\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{c c c}\n        \\hline\n        1 & $e + {\\rm N}_2 \\xrightarrow{} 2 e + {\\rm N}_2^+$ & $k_1 \\left( E\/N \\right)$\\\\\n        2 & $e + {\\rm O}_2 \\xrightarrow{} 2 e + {\\rm O}_2^+$ & $k_2 \\left( E\/N \\right)$\\\\\n        3 & $e + {\\rm O}_2 + {\\rm O}_2 \\xrightarrow{} {\\rm O}_2^- + {\\rm O}_2$ & $k_3 \\left( E\/N \\right)$\\\\\n        4 & $e + {\\rm O}_2 \\xrightarrow{} {\\rm O} + {\\rm O}^-$ & $k_4 \\left( E\/N \\right)$\\\\\n        5 & ${\\rm M} + {\\rm O}_2^- \\xrightarrow{} e + {\\rm O}_2 + {\\rm M}$ & $k_5 \\left( E\/N \\right )$\\\\\n        6 & ${\\rm N}_2 + {\\rm O}^- \\xrightarrow{} e + {\\rm N}_2{\\rm O}$ & $k_6 \\left( E\/N \\right)$\\\\\n        7 & ${\\rm O}_2 + {\\rm O}^- \\xrightarrow{} {\\rm O}_2^- + {\\rm O}$ & $k_7 \\left ( E\/N \\right)$\\\\\n        8 & ${\\rm O}_2 + {\\rm O}^- + {\\rm M} \\xrightarrow{} {\\rm O}_3^- + {\\rm M}$ & $k_8 \\left ( E\/N \\right)$\\\\\n        9 & ${\\rm N}_2^+ + {\\rm N}_2 + {\\rm M} \\xrightarrow{} {\\rm N}_4^+ + {\\rm M}$ & $k_9$\\\\\n        10 & ${\\rm N}_4^+ + {\\rm O}_2 \\xrightarrow{} 2 {\\rm N}_2 + {\\rm O}_2^+$ & $k_{10}$\\\\\n        11 & ${\\rm O}_2^+ + {\\rm O}_2 + {\\rm M} \\xrightarrow{} {\\rm O}_4^+ + {\\rm M}$ & $k_{11}$ \\\\\n        12 & $e + {\\rm O}_4^+ \\xrightarrow{} 2 {\\rm O}_2$ & $k_{12} \\left ( E\/N \\right)$ \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{List of reactions included in the model. M stands for both ${\\rm O}_2$ and ${\\rm N}_2$, and $E\/N$ is the reduced electric field calculated from the electric field $E$ and the gas density $N$. The electron impact reactions $1-4$ have reaction rate coefficients calculated with Bolsig+~\\cite{hagelaar_solving_2005} while the reaction rate coefficients of the ion reactions $5-11$ were taken from \\cite{Pancheshnyi_effective_2013, Aleksandrov_ionization_1999}. The reaction rate coefficient of reaction 12 is calculated~\\cite{kossyi_kinetic_1992} from the mean electron energy calculation of Bolsig+.}\n    \\label{tab:rxns}\n\\end{table}\n\nWe used a plasma fluid model with local field approximation to simulate positive streamers in artificial dry air at standard temperature and pressure at different homogeneous background electric fields. The model equations, transport coefficients, and included reactions and reaction rate coefficients are the same as in our earlier paper~\\cite{francisco2021}. \n\nThe electron density $n_e$ evolves in time according to the equation\n\\begin{equation}\n    \\frac{\\partial{n_e}}{\\partial{t}} = \\nabla \\cdot \\left( n_e \\mu_e \\textbf{E} + D_e \\nabla n_e  \\right) + S_i - S_\\eta + S_{ph} + S_{ion},\n\\end{equation}\nwhere $\\mu_e$ is the electron mobility, $\\textbf{E}$ is the electric field, $D_e$ is the electron diffusion coefficient, $S_i$ is the impact ionization source term, $S_\\eta$ is the electron attachment source term, $S_{ph}$ is the non-local photoionization source term, and $S_{ion}$ is the source term for electron detachment reactions minus the electron-ion recombination reaction. Table~\\ref{tab:rxns} summarizes the reactions incorporated in the model.\n\nNearly all reaction rate coefficients in Table~\\ref{tab:rxns} are a function of the reduced electric field, and only reactions~9-11 have constant reaction rate coefficients. The electron Boltzmann equation solver Bolsig+~\\cite{hagelaar_solving_2005} was utilized to calculate the reaction rate coefficients for the electron impact reactions and the transport coefficients $\\mu_e$ and $D_e$ using electron-neutral scattering cross sections obtained from the Phelps database~\\cite{phelps_anisotropic_1985,phelps_data} retrieved in March 2019. \n\nThe source terms for impact ionization, electron attachment, and electron detachment minus electron-ion recombination are computed using\n\\begin{equation}\n    S_i = k_1 n_e \\left[ {\\rm N}_2 \\right] + k_2 n_e \\left[ {\\rm O}_2 \\right], \n\\end{equation}\n\\begin{equation}\n    S_\\eta = k_3 n_e \\left [ {\\rm O}_2 \\right]^2 + k_4 n_e \\left[ {\\rm O}_2 \\right].\n\\end{equation}\n\\begin{equation}\n    S_{ion} = k_5 \\left [ {\\rm M} \\right]\\left [ {\\rm O}_2^{-} \\right] + k_6  \\left[ {\\rm N}_2 \\right]\\left[ {\\rm O}^{-} \\right] - k_{12}  n_e \\left[ {\\rm O}_4^{+} \\right],\n\\end{equation}\nwhere [Z$_i$] stands for the density of the species Z$_i$, and [M] = [N$_2$] + [O$_2$]. [N$_2$] and [O$_2$] are assumed to be constant in our simulations as the degree of ionization within streamers at standard temperature and pressure is small.\n\nThe photoionization source term is given by\n\\begin{equation}\n    S_{ph}({ \\bf r})=\\int d^3 r'\\;\\frac{I( {\\bf r'}) f(|{\\bf r}-{\\bf r'}|)}{4 \\pi |{\\bf r} - {\\bf r'}|^2}\n\\label{equ:photo}\n\\end{equation}\nwhere $I\\left({\\bf r}\\right)$ is the source of ionizing photons, $f(r)$ is the absorption function, and $4\\pi |{\\bf r} - {\\bf r'}|^2$ is a geometric factor. Following Zheleznyak's model~\\cite{zheleznyak_photoionization_1982}, the photon source term $I\\left({\\bf r}\\right)$ is calculated using\n\\begin{equation} \\label{eq:photoionization}\nI\\left(\\textbf{r}\\right) = \\frac{ p_q}{p + p_q}\\xi S_i\\left(\\textbf{r}\\right)\n\\end{equation}\nwhere $p$ is the actual gas pressure, $p_q$ is a gas-specific quenching pressure, and $\\xi$ is a proportionality factor. In principle, this proportionality factor is field-dependent~\\cite{zheleznyak_photoionization_1982}, but in this paper, we set it to $\\xi = 0.075$. Furthermore, we use a quenching\npressure of $p_q = 40 \\, \\mathrm{mbar}$.\nIn Zheleznyak's model, $f(r)$ is an effective function for the absorption of photons in the wave length range of\n$98$ to $102.5$~nm. It is obtained with\n \\begin{equation}\nf(r)=\\frac{\\exp(-\\chi_{\\mathrm{min}}p_{O_2}r)-\\exp({-\\chi_{\\mathrm{max}}}p_{O_2}r)}{r\\ln(\\chi_{\\mathrm{max}}\/\\chi_{\\mathrm{min}})},\n\\label{equ:absorption-function}\n\\end{equation}\nwhere $\\chi_{\\mathrm{max}}\\approx1.5\\times10^2\/(\\textnormal{mm bar})$,\n$\\chi_{\\mathrm{min}}\\approx2.6\/(\\textnormal{mm bar})$, and $p_{O_2}$ is the\npartial pressure of oxygen.\nWe used a set of Helmholtz differential equations~\\cite{bourdon_efficient_2007,luque_photoionization_2007} with Bourdon's three-term parameters~\\cite{bourdon_efficient_2007} to evaluate the photoionization integral.\n\nThe charged species ${\\rm N}_2^+$, ${\\rm N}_4^+$, ${\\rm O}_2^+$, ${\\rm O}_4^+$, ${\\rm O}^-$, ${\\rm O}_2^-$, and ${\\rm O}_3^-$, and the neutral species O and ${\\rm N}_2{\\rm O}$ evolve in time according to the continuity equation\n\\begin{equation}\n\\frac{\\partial \\left[ Z_i \\right ]}{\\partial t} =  - s_i \\nabla \\cdot \\left( \\left[ Z_i \\right ] \\mu_i \\textbf{E} \\right) + S_{Z_i}\n\\label{eq:ion_cont}\n\\end{equation}\nwhere $s_i = \\pm 1$ is the sign of the electric charge of species $i$ and $\\mu_i$ is their mobility. Since ion mobilities are typically about two orders of magnitude lower than electron mobilities, we for simplicity neglect ion motion in most of this paper. However, we investigate the effect of ion motion in section~\\ref{sec:ion_motion}, in which all ion mobilities are set to 2.2 $\\times 10^{-4}$ m$^2\/$V s \\cite{Tochikubo_2002}. Finally, neutral species are always immobile in our simulations.\n\nCalculations for the electric potential $\\phi$ and the electric field use the equations\n\\begin{equation}\n\\nabla^2 \\phi = - \\frac{\\rho}{\\epsilon_0},\\quad \\textbf{E} = - \\nabla \\phi,\n\\end{equation}\nwhere $\\rho$ is the space charge density and $\\epsilon_0$ is the vacuum permittivity. The space charge density is calculated using $\\rho = e \\left ( n_i - n_e \\right)$ where $e$ is the elementary charge and $n_i$ is the density of all positive ions minus the density of all negative ions. \n\n\\subsection{Computational method and domain \\label{sec:domain}}\n\nThe simulations were run using Afivo-streamer~\\cite{teunissen_simulating_2017, Teunissen_afivo_2018}, a simulation tool for plasma fluid models that uses geometric multigrid techniques, an octree-based adaptive mesh refinement system, and OpenMP parallelization. The present results are for single streamers, and these assume that they are cylindrically symmetric. This allows the calculation to be performed effectively in just the two coordinates $r$ and $z$.\n\nOur computational domain in this study is cylindrically symmetric and has a length of $40$~mm and a radius of $20$~mm. To disregard boundary effects, the simulation is set to end once the streamer head is within $10$~mm from the opposite end of the domain. The streamer head position is identified as the point where the electric field is maximum in the domain.\n\nThe electric potential was fixed at $z = 0$~mm and $z = 40$~mm to achieve a homogeneous background electric field pointing in the $-\\hat{z}$ direction. At $r =20$~mm, Neumann zero boundary conditions ($\\partial_r \\phi = 0$) were applied on the electric potential, and for $r = 0$~mm, the boundary condition follows from cylindrical symmetry. Neumann zero boundary conditions are applied for the electron density at all boundaries, and no background ionization was introduced into the domain.\n\nWe used the same refinement criteria as described in ~\\cite{francisco2021}: \nAdaptive mesh refinement is employed with the grid set to have a minimum size of 2.4~$\\mu$m.\nThe refinement and derefinement criteria are based on the local electric field value as in \\cite{teunissen_simulating_2017} with an additional criterion based on the charge density: refine if $\\alpha(1.2\\times E) \\Delta x > 0.5 $ and derefine if both $\\alpha(1.2\\times E)\\Delta x < 7.5 \\times 10^{-2}$ and $|\\rho|\/\\epsilon_0 < 9.0 \\times 10^{10}~\\rm{V\/m^2}$, where $\\alpha(E)$ is the field-dependent ionization coefficient, $E$ is the electric field strength, and $\\Delta x$ is the grid spacing.\nTo obtain a clearer picture of the equipotential lines in the regions behind the streamer head, we modified our derefinement criterion for the streamers with a background field of $4.65$~kV\/cm and below so that derefinement stops when the cell width gets to $4~\\mu$m.\n\n\\subsection{Initial conditions \\label{sec:init}}\n\n{\\bf For homogeneous background electric fields of at least $14$~kV\/cm}, streamers easily initiate and propagate from a neutral seed of equal electron and positive ion densities, which we placed on the upper boundary of the domain, along the axis of symmetry. Another neutral seed is placed below the first seed to provide an initial source of electrons.\nThe first seed is $0.25$~mm wide, $1$~mm long, and has $2.25 \\times 10^{20} \\rm{\/m}^3$ electrons and positive ions while the second seed is $0.2$~mm wide, $2$~mm long, and has $10^{17} \\rm{\/m}^3$ electrons and positive ions. Both seeds decay with a Gaussian profile. This set-up is illustrated in the left-most panel of figure~\\ref{fig:initial_conditions}.\n\n{\\bf Single streamers are more difficult to obtain in lower background electric fields} because either the field enhancement proves to be insufficient for streamers to initiate or the streamer branches after propagating a short length. Branching breaks the cylindrical symmetry of a single streamer channel, and thus cylindrically symmetric simulations are not appropriate to describe such phenomena~\\cite{luque_density_2012}. To investigate low-field streamers, a streamer is first initiated and allowed to propagate for some time in a higher background field before the background electric field is instantaneously reduced to a lower value. This approach allows us to study single continuously propagating and non-branching streamers in fields lower than $14$~kV\/cm.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure1.eps}\n    \\caption{Initial conditions for the different background electric fields used in this paper. Shown are contour plots for the electron number densities together with purple equipotential lines. The leftmost panel is for streamers in background fields of $14$~kV\/cm and higher, the middle panel is for streamers with background fields below $14$~kV\/cm to $9$~kV\/cm, and the rightmost panel is for streamers in fields below $9$~kV\/cm. Note that the computational domain extends from $0$ to $40$~mm in the $z$ direction and $0$ to $20$~mm in the $r$ direction, and only a part of the domain is shown in this figure.}\n    \\label{fig:initial_conditions}\n\\end{figure}\n\n{\\bf For electric fields from $9$~kV\/cm to $12$~kV\/cm}, a streamer was first initiated in a field of $14$~kV\/cm and allowed to grow for $20$~ns before instantaneously reducing the background electric field. Thus, the low-field streamers grow from a streamer with a $53.5~\\mu$m radius and head at $z = 37.6$~mm as shown in the middle panel of figure~\\ref{fig:initial_conditions}.\n\n{\\bf For even lower fields}, this approach still encounters the same initiation and branching problems that were previously stated.\nThus, for streamers in background electric fields below $9$~kV\/cm, we used the $9$~kV\/cm streamer after $40$~ns of propagation as the initial condition, i.e. the field was reduced twice. First, the field was changed from $14$ to $9$~kV\/cm after $20$~ns, and then it was modified further to the final electric field after $40$~ns. This gives a $155~\\mu$m wide streamer with its head at $z = 33.4$~mm as the starting state for these lower field simulations. This initial condition can be seen in the right-most panel of figure~\\ref{fig:initial_conditions}, which matches the left-most panel of figure~\\ref{fig:electron-density}. This last approach allowed us to simulate single streamers in background electric fields as low as $4.5$~kV\/cm.\n\n\\subsection{Calculation of optical radii}\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{c c c}\n        \\hline\n        $1$  & ${\\rm e} + {\\rm N}_2 \\xrightarrow{} {\\rm e} + {\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right)$ & $k_{ex}\\left( E\/N \\right)$  \\\\\n        $2$  & ${\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right) + {\\rm N}_2 \\xrightarrow{} 2{\\rm N}_2$  & $k_q^{{\\rm N}_2}$ \\\\\n        $3$  & ${\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right) + {\\rm O}_2 \\xrightarrow{} {\\rm N}_2 + {\\rm O}_2$  & $k_q^{{\\rm O}_2}$ \\\\\n        $4$  & ${\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right) \\xrightarrow{} {\\rm N}_2\\left( {\\rm B} ^3\\Pi_g\\right) + h\\nu$  & $1\/\\tau_0$ \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{Reactions to calculate the optical emission of streamers. Bolsig+~\\cite{hagelaar_solving_2005} with the Phelps database~\\cite{phelps_anisotropic_1985, phelps_data} was used to calculate for $k_{ex}\\left( E\/N \\right)$, while $k_q^{{\\rm N}_2} = 0.13 \\times 10^{-10}$ cm$^3\/{\\rm s}$, $k_q^{{\\rm O}_2} = 3.0 \\times 10^{-10}$ cm$^3\/{\\rm s}$, and $\\tau_0 = 42$~ns are from \\cite{Pancheshnyi_2000}. Reaction~$4$ leads to the emission of optical photons with wavelength $337.1$~nm~\\cite{Pancheshnyi_2000} or energy $3.7$~eV.}\n    \\label{tab:optic_rxns}\n\\end{table}\n\nAll radii given in the present paper are optical radii, as they would be measured experimentally. More precisely, this optical radius is half of the full width at half maximum (FWHM) of the calculated optical emission, in contrast to the definition of the streamer radius as the location of the maximum of the radial component of the electric field in previous papers~\\cite{francisco2021, bagheriSimulationPositiveStreamers2020}. Four additional reactions were added to our model to incorporate the density of ${\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right)$, the excited state of $N_2$ responsible for most radiation in the visible spectral region~\\cite{Pancheshnyi_2005}. These reactions are listed in Table~\\ref{tab:optic_rxns} with their corresponding reaction rate coefficients.\n\nWe compute the optical radius from the density of ${\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right)$. A forward Abel transform was done on $\\left[{\\rm N}_2\\left( {\\rm C} ^3\\Pi_u\\right)\\right]$ in cylindrical coordinates to get its 2D projection in Cartesian coordinates. From 2D projection we only considered the area below $z = 33$~mm to disregard the effects of the seeds used for initiation. The densities were normalized and summed along the $z$ axis, producing a 1D profile along the $y$ axis from where we searched for the maximum density. From the point of maximum density, the farthest $y$ coordinates where the density was at least half of the maximum density were identified, and the distance between these two identified points was regarded as the head diameter. Half of that value is the optical radius we report.\n\n\\section{Simulation results \\label{sec:results}}\n\nFirst, in section~\\ref{sec:2cases}, we will discuss the particular cases of single streamers in a background field of 4.65 kV\/cm and 14~kV\/cm which are examples of solitary and accelerating streamers. Then we will look at streamer behavior as a function of the background field in section~\\ref{sec:behaviors}.\n\n\\subsection{Solitary streamers and accelerating streamers \\label{sec:2cases}}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure2.eps}\n    \\caption{Time evolution of the electron density of streamers in air at different background electric fields. Purple equipotential lines are included. The panels of the $4.65$~kV\/cm streamer differ by time steps of $100$~ns, while the $14$~kV\/cm streamer is shown in time steps of $17.5$~ns. The full $z$ axis is shown, but the figure zooms into the radial region $r\\le 3$~mm, while the full simulation domain extends up to $r = 20$~mm.}\n    \\label{fig:electron-density}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth, keepaspectratio]{Figure3.eps}\n    \\caption{Axial profiles, line charge density, and current of the streamers in the same background fields of $14$ and $4.65$~kV\/cm as in figure~\\ref{fig:electron-density}. Here they are shown  when their maximal electric field is at $z = 15$, $20$, and $25$~mm. The panels show, from top to bottom, as a function of the axis coordinate $z$:\n    (a) the electric potential $\\phi$ on axis, (b) the electric field profile $E$ on axis, (c) the electron number density $n_e$ on axis, (d) the line charge density $\\lambda$, which is the charge density integrated over the radial cross section (where the dashed lines represent negative values), (e) the electric current $I$, which is the current density also integrated over the radial cross section. The legend on the first panel applies to all panels.} \n    \\label{fig:axis}\n\\end{figure}\n\nFigure~\\ref{fig:electron-density} shows the evolution of streamers in background electric fields of $4.65$ and $14$~kV\/cm. The panels show the color-coded electron density together with equipotential lines in purple. For the lower field, the streamer is shown in time steps of $100$~ns, while for the higher field, in time steps of $17.5$~ns. The same streamers are presented in figure~\\ref{fig:axis} showing the electric potential, the electric field and the electron density along the streamer axis, and the line charge density and the electric current. The last two are obtained by integrating across the streamer cross section the charge density and the current density, respectively. The integration was done up to $r = 5$~mm. Several basic differences can be noted between the two streamers as they propagate through the 40~mm gap.\n\n{\\bf The solitary streamer.} $\\quad$ The streamer in the $4.65$~kV\/cm field grows by about an equal length within each time step of $100$~ns. The electron density is strongly reduced about $10$~mm behind the streamer head, and the electric field returns to its background value in this region and further behind, as can be seen from the straight and equidistant equipotential lines. Overall, the pattern of electron density and deflected equipotential lines is transported almost uniformly, without changes in shape. The streamer transports a constant amount of positive charge within its finite length, and there is no negative charge visible in the line charge density in Fig.~\\ref{fig:axis}. We will call this streamer a solitary streamer or a uniformly translating streamer.\n\n{\\bf The accelerating streamer.} $\\quad$ The streamer in the $14$~kV\/cm field is shown in time steps of $17.5$~ns in Fig.~\\ref{fig:electron-density}. It clearly accelerates, and its head radius increases. The electron density varies little along the whole channel for all time steps. There is electric current flowing in the order of $10^{-1}$~A along the whole channel, and the back part charges negatively while the front part accumulates positive charge - there is electric polarization along the whole channel. This is visible in the line charge density as well as in the field distortion along the whole body of the streamer channel. We will call this streamer an accelerating streamer.\n\nLater in section~\\ref{sec:behaviors} we will also discuss decelerating streamers and the fact that the solitary streamers exist only on the borderline between accelerating and decelerating streamers.\n\nThe lowest electric field inside the accelerating streamer is $4.7$~kV\/cm, located around the middle section of the streamer channel. For the solitary streamer in the 4.65 kV\/cm background field, the electric field right behind the ionization front is as low as $0.5$~kV\/cm and rises to the background value behind the solitary structure.\nThe attachment times are $25$~ns and $92$~ns in these two fields respectively, and the solitary streamer propagates with a velocity of $7\\times 10^4$~m\/s. According to these times and velocities, the electron density is substantially reduced about $1.7$ to $6.4$~mm behind the streamer head, in accordance with the simulations. The slow propagation of the solitary streamer also gives electrons sufficient time to recombine with O$_4^+$ molecules. Recombination times of $25$ to $57$~ns were calculated for the electric field range in the solitary streamer's channel. These times were computed using the [O$_4^+$] values at the locations corresponding to the electric fields given above.\nThe accelerating streamer propagates much faster, with a higher internal field, leaving no time for electron attachment.\nWe see in the third panel of Figure~\\ref{fig:axis} that the electron density of the solitary streamer decays behind the ionization front by several orders of magnitude while the electron density in the channel of the accelerating streamer is essentially constant.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure4.eps}\n    \\caption{Plots of the $4.65$~kV\/cm streamer zoomed into the streamer head when it is at $z = 20$ mm. From left to right: (1) electric field with white equipotential lines, (2) space charge density, (3) electron density, and (4) negative ion density.}\n    \\label{fig:head}\n\\end{figure*}\n\nIn Figure~\\ref{fig:head}, several plots zooming in on the head of the solitary streamer are presented. The electric field inside the channel of the solitary streamer is screened to a low value, represented by the widely separated horizontal equipotential lines. Almost all the net charge is located on the streamer surface - the positive space charge layer shown in the second panel of Fig.~\\ref{fig:head}. The low electric field in the neutral region leads to fast electron attachment as discussed above, and this is evident in the electron density contour plot, where the electron density reduces in magnitude behind the streamer head. Electron attachment produces negative ions. The density of negative ions at the back end of the channel is lower than the electron density at the streamer head due to to electron-ion recombination.\n\n\\subsection{Propagation modes as a function of the field \\label{sec:behaviors}}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure5.eps}\n    \\caption{Properties of positive streamers as a function of length in different background fields. Top panels show the streamer velocity, middle panels the optical radius, and bottom panels the maximum electric field. Plots on the right are on a different y-axis scale and focus on the streamers in fields of $5$~kV\/cm and lower. Radius measurements have uncertainties of $\\pm 1.2~\\mu$m.}\n    \\label{fig:properties}\n\\end{figure*}\n\nThree parameter regimes can be identified in Figure~\\ref{fig:properties}, which has the velocity, optical radius, and maximum electric field of the streamers as a function of length. First, there are the accelerating streamers that speed up as they lengthen, and their radius increases as they accelerate. This is the case for streamers in background electric fields above $4.65$~kV\/cm. This is also the case most frequently reported and commonly observed in streamer simulations.\n\nSecond, there are uniformly propagating streamers, in a background field of $4.65$~kV\/cm. They exist as a limit between accelerating and decelerating streamers, and they maintain a nearly uniform velocity. Other streamer properties such as the head radius and enhanced electric field do not change in time either. For the streamer in our simulation, the radius remained at $65~\\mu$m while it was uniformly propagating. These solitary streamers can maintain their shape because they have a finite and constant length where the electron density is relevant and the electric field is modified. They carry a fixed amount of positive charge over a finite length, and therefore act as a point charge from a sufficiently far distance. The streamer is able to propagate indefinitely in this background field. This behavior can be related to the old concept of the streamer stability field, which we discuss further in section~\\ref{sec:stability_field}.\n\nThird and last, there are the decelerating streamers. We find them in fields below  $4.65$~kV\/cm. Streamers in such fields slow down as they lengthen, and their head radius decreases in time while the maximum electric field increases. This happens because the electric screening of the streamer interior improves when the ionization front slows down. The decreasing radii of the decelerating streamers can be explained by the decreasing potential in the streamer head due to voltage lost in the streamer channel~\\cite{starikovskiy2021}. Some of our simulated decelerating streamers do not manage to cross the domain, as shown by the case of the streamer with a background field of $4.5$~kV\/cm. The streamer decelerated and eventually stagnated with a streamer radius of $49~\\mu$m. This stagnating behavior was described earlier in~\\cite{starikovskiy2021, Pancheshnyi_2004, Starikovskiy_2020}. Numerically, we observe that the simulation time steps, which are usually in picoseconds, drop by two orders of magnitude because the maximum electric field values suddenly increase to magnitudes greater than $300$~kV\/cm in a very small region ahead of the ionization front. One reason for this numerical instability may be the artificial diffusion of electrons from the channel to the high-field region ahead of the streamer tip~\\cite{Teunissen_2020}. The physical process of streamer stagnation was always accompanied by such numerical instabilities in our simulations.\n\n\\subsection{Nonlinear dependence of field enhancement and plasma chemistry on the background field}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure6.eps}\n    \\caption{Maximal (top) and minimal (bottom) electric field in the streamer as a function of the background electric field. The maximal field is measured at the streamer head while the minimal field is from behind the streamer ionization front. The values were acquired when the streamer heads were at position $z = 20$~mm.}\n    \\label{fig:fields}\n\\end{figure}\n\nThe streamer dynamics nonlinearly depend on the background electric field $E_\\textrm{back}$. In the top panel of Figure~\\ref{fig:fields}, we see the maximal field $E_\\textrm{max}$ as a function of the background field $E_\\textrm{back}$, evaluated at the moment when the streamer heads are at $z=20$~mm. The curve has a minimum of about $E_\\textrm{max}=120$~kV\/cm for a background electric field of about $E_\\textrm{back}=12$~kV\/cm. For $E_\\textrm{back}$ increasing up to 26~kV\/cm, the maximal field increases up to 140~kV\/cm, while below 10~kV\/cm the maximal electric field increases rapidly, until it diverges for $E_\\textrm{back}=4.5$~kV\/cm. As the electron energy distribution and the induced plasma chemistry depend on the electric field configuration, we conclude that the plasma chemistry could also depend nonlinearly on the background electric field. This observation requires further investigation in the future. \n\nThe minimum electric field behind the ionization front of the streamers as a function of the background field is presented in the bottom panel of Figure~\\ref{fig:fields}. We found that the minimum electric field inside the streamer channel depends almost linearly on the background electric field. It vanishes for the stagnating streamer, and it reaches $9$~kV\/cm for $E_\\textrm{back}=26$~kV\/cm.\n\n\\subsection{Ion motion \\label{sec:ion_motion}}\n\nAs electrons attach to oxygen and form negative ions in the channel, we briefly explore the effect of ion motion on streamer behavior. Incorporating ion motion in streamer simulations with $14$~kV\/cm and $9$~kV\/cm background fields did not visibly change anything in the results. For these cases, the streamer still propagates fast enough that ion motion has negligible effects. We only start to observe effects in low background electric fields, when enough time is available to deplete the electron density through attachment and recombination.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure7.eps}\n    \\caption{Charge density of the $4.65$~kV\/cm streamer without ion motion (left) and with ion motion (right), including green equipotential lines.}\n    \\label{fig:ion_contour}\n\\end{figure}\n\nFigure~\\ref{fig:ion_contour} shows the total charge density of the solitary streamer with and without ion motion. We see that the channel of the streamer with ion motion is wider at the back. The space charge layer of these streamers is made up of positive ions, and without ion motion they remain fixed in space, and only reactions can change the densities of these ions in time. With the inclusion of ion motion, these ions are now moving radially outward in response to the electric field they are subjected to. The ion drift in the local field also causes the streamer head to lose some focus, leading to slower propagation. We observe similar phenomena in negative streamers, whose space charge layers are made up of the very mobile electrons.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figure8.eps}\n    \\caption{Velocity (top panel) and maximum electric field (bottom panel) of streamers against streamer head position for simulations with and without ion motion. Broken lines stand for simulations with ion motion. The legend on the top panel applies to the bottom panel as well.}\n    \\label{fig:ion_motion}\n\\end{figure}\n\nWhen ion motion is included, the streamer propagates more slowly. This can be observed in the upper panel of Figure~\\ref{fig:ion_motion}, where streamers in the same background field with and without ion motion are presented. The previously discovered uniformly translating streamer at $4.65$~kV\/cm decelerates when ion motion is included in the simulation. A new background electric field for uniform translation was found at $4.675$~kV\/cm, only slightly higher than the previous background field, with a slightly lower uniform velocity of $0.66 \\times 10^{5}$~m\/s. Thus, the effect of ion motion on streamer dynamics does not appear to be strong in this case. We will be using this new uniformly translating streamer for our comparisons in section~\\ref{sec:validation}.\n\nThe maximal electric field of the streamers in fields of $4.65$ to $4.7$~kV\/cm with and without ion motion is plotted as a function of the streamer head position in the lower panel of Figure~\\ref{fig:ion_motion}. When ion motion is included, the maximal electric fields at the same background electric field are higher, which is consistent with the smaller head radii. \n\nFinally, we compare the maximal electron drift velocity with the velocity of the uniformly translating streamer with ion motion. The maximal electric field at the streamer head has a constant value of $171$~kV\/cm, which gives us an electron drift velocity of $5.3 \\times 10^{5}$~m\/s, while the streamer velocity is $6.6\\times 10^4$~m\/s - almost an order of magnitude smaller than the drift velocity. This is possible for positive streamers, where these velocities are directed in opposite directions, but not for negative streamers.\n\n\\section{Comparison with experiments} \\label{sec:validation}\n\n\\subsection{The stability field \\label{sec:stability_field}}\n\nRecently, the concept of the streamer stability field has been more commonly used in association with streamers propagating in inhomogeneous electric fields. It relates the maximum length a streamer could gain to the applied voltage~\\cite{allen1991, Allen_1995, Babaeva_1996, seeger_2018, Veldhuizen_2002}. An older definition used the term stability field to mean the homogeneous electric field in which a streamer would propagate in a stable manner - without changes in velocity and shape~\\cite{phelps_fieldenchanced, griffiths_effectofairpressure, gallimberti_longspark}.\n\nIf we only consider the streamer channel length as the length behind the streamer head with substantial electron density, we observe the solitary streamer to have a constant length as it propagates. The solitary streamer streamer has a uniform shape, and it follows, using the older definition of the stability field, that the solitary streamer is propagating in the stability field of STP dry air at $4.675$~kV\/cm. This value agrees with the measured stability field of $4$~kV\/cm in experiments~\\cite{griffiths_effectofairpressure, gallimberti_longspark} for the original definition.\nWith the newer definition, the stability field is reported to be between 4.5-5~kV\/cm~\\cite{razer1991, Allen_1995}.\n\n\\subsection{Radius and velocity of solitary and minimal streamers.}\n\nIn the pin-to-plate experiments of \\cite{Briels_2006}, it was found that after several branching events or in a quite weak field, streamers would approach a minimal diameter, and they were called minimal streamers. The solitary streamers are essentially the thinnest streamers that we found in our simulations as the stagnating streamers are not much thinner and hardly emit any light. Therefore we now compare their properties.\n\nThe simulated solitary streamer, that includes ion motion, has a radius of $55~\\mu$~m and this value is not far from the experimental findings in \\cite{nijdam_probing_2010}, which give $65~\\mu$m as the minimal streamer diameter in $1$~bar air. The uniform velocity of our solitary streamer is $0.7 \\times 10^{5}$~m\/s, which falls in the range of the measured velocity of $\\left(0.5 - 1\\right) \\times 10^5$~m\/s of minimal streamers. Therefore we can conclude that the simulations match the experiments within 20\\%.\n\n\\section{Conclusions and Outlook \\label{sec:conclusion}}\n\nWe simulated single positive streamers in air at standard temperature and pressure in homogeneous background fields ranging from $4.5$~kV\/cm to $26$~kV\/cm in a 4~cm gap, and we came to the following conclusions:\n\\begin{itemize}\n    \\item[1.] The solitary streamer (or uniformly translating streamer) with dominant electron attachment behind the head lays a theoretical basis for the much used concept of a stability field. Streamers in higher fields increase in radius and velocit\n    , while the solitary streamer transports a fixed amount of positive charge that is substantial only over a finite length.\n    \\item[2.] The solitary streamer motion explains how a streamer can propagate over distances in meter length-scales though the conductivity of the back part of the channel disappearing due to attachment and recombination. The velocity of such a streamer can be an order of magnitude smaller than the electron drift velocity in its maximal electric field.\n    \\item[3.] The value of the stability field of 4.675~kV\/cm in our simulations in STP air agrees well experimentally measured values.\n    \\item[4.] Minimal streamers are the thinnest and slowest streamers that have been experimentally observed~\\cite{Briels_2006}. Our values for the optical radius and velocity of solitary streamers agree well with measurements of these so-called minimal streamers. Even better agreement could possibly be found if for example humidity, repetition rate, and fluid model limitations were taken into account.\n    \\item[5.] Ion motion plays a minor role for solitary streamers, but its effect increases as streamers slow down.\n    \\item[6.] The maximal electric field at the streamer head is not a monotonic function of the background field, but it has a minimum for a background field of about 12~kV\/cm. The implications of this on the electron energy distribution and on the optimization of the plasma chemistry will need to be investigated.\n\\end{itemize}\n\nFuture research could look into model reduction based on the solitary streamer, as it does not depend on time in a co-moving frame. How our current findings translate to other gases with different plasma-chemical reactions and photoionization rates also merits further investigation. There is an avenue for exploring the behavior of accelerating streamers on longer timescales, and the existence of the solitary positive streamer also raises the question of whether the solitary mode of propagation could also be observed in negative streamers.\nFinally, another open question is how and when solitary streamers form in background fields with a spatial gradient, as is common in experiments.\n\n\\section{Acknowledgments}\nWe thank Andy Martinez and Xiaoran Li for the help with post-processing methods.\nH.F. was funded by the European Union's Horizon 2020 research and innovation\nprogramme under the Marie Sk\u0142odowska-Curie grant agreement SAINT 72233.\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}