A naive p-adic height on the Jacobians of curves of genus 2

Abstract

Consider a genus 2 curve defined over $\mathbb{Q}$ given by an affine equation of the form $y^2 = f(x)$ for some polynomial $f$ of degree 5, and let $p$ be an odd prime. Extending work of Perrin-Riou for elliptic curves, we construct a naive $p$-adic height function on a finite index subgroup of the Jacobian $J$ of this curve, using the explicit embedding of $J$ in $\mathbb{P}^8$ and the associated formal group described by Grant. We use the naive height to construct a global height $h_p: J(\mathbb{Q}) \rightarrow \mathbb{Q}_p$ using a limit construction analogous to Tate's construction of the N\'{e}ron-Tate height, and show that it is quadratic. We then compare $h_p$ to a $p$-adic height constructed in a different way by Bianchi and show that they are equal.