Integral Galois module structure of Mordell--Weil groups

Abstract: Let E/Q be an elliptic curve, G a finite group and V a fixed finite dimensional rational representation of G. As we run over G-extensions F/Q with E(F)⊗Q isomorphic to V , how does the Z[G]-module structure of E(F) vary from a statistical point of view? I will report on joint work with Alex Bartel in which we propose a heuristic giving a conjectural answer to an instance of this question, and make progress towards its proof. In the process I will relate the question to quantifying the failure of the Hasse principle in certain families of genus 1 curves, and explain a close analogy between these heuristics and Stevenhagen's conjecture on the solubility of the negative Pell equation.

algebraic geometryalgebraic topologygroup theorynumber theoryrepresentation theory

Audience: researchers in the discipline

Queen Mary University of London Algebra and Number Theory Seminar