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

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.