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Mordell–Weil groups of elliptic curves — beyond ranks

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If E/Q is an elliptic curve, and F/Q is a finite Galois extension, then E(F) is not merely a finitely generated abelian group, but also a Galois module. If we fix a finite group G, and let F vary over all G-extensions, then what can we say about the statistical behaviour of E(F) as a Z[G]-module? In this talk I will report on joint work with Adam Morgan, in which we investigate a special case of this very general question. Our work has surprising connections to questions about frequency of failure of the Hasse principle for genus 1 hyperelliptic curves, as well as to Stevenhagen’s conjecture on the solubility of the negative Pell equation.

This talk is part of the Number Theory Seminar series.

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