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The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Part of: Multiplicative number theory Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 September 2024

ROSS PATERSON
ROSS PATERSON*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ. e-mail: [email protected]
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Abstract

We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$, and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F.

The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11G07: Elliptic curves over local fields 11N45: Asymptotic results on counting functions for algebraic and topological structures 14H52: Elliptic curves
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 34
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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