Geometric quadratic Chabauty over number fields

TL;DR

This paper extends the geometric quadratic Chabauty method from rational numbers to arbitrary number fields, providing a conditional bound on rational points for certain curves, and offers a more direct generalization approach.

Contribution

It generalizes the quadratic Chabauty method to number fields and establishes a conditional bound on rational points for specific curves.

Findings

Provides a conditional bound on rational points over number fields.

Generalizes the quadratic Chabauty method to arbitrary number fields.

Offers a more direct approach to the method's generalization.

Abstract

This article generalizes the geometric quadratic Chabauty method, initiated over $\mathbb{Q}$ by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell-Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.