Geometric quadratic Chabauty over number fields

Abstract

This article generalizes the geometric quadratic Chabauty method, initiated over 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. Contents 1. Introduction 2. The Poincaré biextension 3. Construction of the torsor T 4. The main theorem 5. The parametrization of Y t 6. End of proof and questions References