Explicit quadratic Chabauty over number fields

TL;DR

This paper extends explicit quadratic Chabauty methods to arbitrary number fields, enabling the computation of integral and rational points on certain hyperelliptic and bielliptic curves using p-adic heights and restriction of scalars.

Contribution

It generalizes quadratic Chabauty techniques to all number fields, combining classical Chabauty equations with p-adic height functions for broader applicability.

Findings

Successfully applied methods to several example curves.

Demonstrated practicality of the generalized techniques.

Extended the scope of explicit Chabauty methods.

Abstract

We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek's extension of classical Chabauty with equations defined in terms of p-adic heights attached to independent continuous idele class characters. We give several examples to show the practicality of our methods.