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

TL;DR

This paper constructs and compares a naive p-adic height on Jacobians of genus 2 curves over rationals, extending elliptic curve methods and establishing its quadraticity and equivalence to other p-adic heights.

Contribution

It introduces a new naive p-adic height on genus 2 Jacobians, extending Perrin-Riou's work and comparing it to existing p-adic height constructions.

Findings

The naive p-adic height is quadratic.

The constructed height coincides with Bianchi's p-adic height.

The height is explicitly defined via the Jacobian's embedding and formal group.

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.