p-Adic sigma functions and heights on Jacobians of genus 2 curves

arXiv2302.03454MaRDI QIDQ6425711

Francesca Bianchi

Publication date: 7 February 2023

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 $h_{p}colonJ(overlinemathbbQ)omathbb{Q}_p$ by means of $p$-adic analogues of N'eron functions of divisor $2Theta$. 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 $lambd{a}_v$ 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 $lambd{a}_v$ and $h_p$ to local and global extended Coleman-Gross (and hence Nekov'av{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.

Has companion code repository: https://github.com/bianchifrancesca/padic_heights_g2

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