Computing p-adic heights on hyperelliptic curves

TL;DR

This paper presents a faster, simpler algorithm for computing p-adic heights on hyperelliptic curves, applicable to both odd and even degrees, enabling new applications in number theory.

Contribution

The authors develop an improved algorithm for p-adic height computation on hyperelliptic curves that surpasses previous methods in speed and simplicity, and works for both odd and even degrees.

Findings

Algorithm is significantly faster than previous methods.

Works for both odd and even degree hyperelliptic curves.

Enables new numerical experiments related to the Birch and Swinnerton-Dyer conjecture.

Abstract

We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow their general strategy, our algorithm is significantly faster and simpler and works for both odd and even degree. We discuss a precision analysis and an implementation in SageMath. Our work has several applications, also discussed in this article. These include various versions of the quadratic Chabauty method, and numerical evidence for a p-adic version of the conjecture of Birch and Swinnerton-Dyer in cases where this was not previously possible.