{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/p-adic-sigma-functions-and-heights-on","title":"p-Adic sigma functions and heights on Jacobians of genus 2 curves","arxiv_id":"2302.03454","date":"2023-02-07","proceeding":null,"authors":["Francesca Bianchi"],"abstract":"Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\\infty$ at infinity, let $J$ be its Jacobian, let $\\Theta$ be the theta divisor with respect to $\\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\\colon J(\\overline{\\mathbb{Q}})\\to \\mathbb{Q}_p$ by means of $p$-adic analogues of N\\'eron functions of divisor $2\\Theta$. We define such N\\'eron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic N\\'eron function $\\lambda_v$ at a non-archimedean place $v$ of $K$ is the image, under a suitable trace map, of a symmetric $v$-adic Green function of divisor $\\Theta$ \\`a la Colmez. We use this to relate $\\lambda_v$ and $h_p$ to local and global extended Coleman-Gross (and hence Nekov\\'a\\v{r}) $p$-adic height pairings. We provide examples of our implementation, including one for a prime $p$ greater than $10^6$, and explain how similar techniques can be used to compute $p$-adic integrals of differentials of the first, second and third kind on $C$ independently of the reduction type. As an application, we also give an explicit quadratic Chabauty function vanishing on the rational points on certain genus $4$ bihyperelliptic curves.","url_abs":"https://arxiv.org/abs/2302.03454v1","url_pdf":"https://arxiv.org/pdf/2302.03454v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"p-adic-sigma-functions-and-heights-on","repo_url":"https://github.com/bianchifrancesca/padic_heights_g2","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}