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

TL;DR

This paper demonstrates that for certain Galois extensions and primes, the average size of p-Selmer groups over the extension fields remains bounded, revealing limitations of Galois descent methods for elliptic curves.

Contribution

It establishes bounds on the average p-Selmer group dimensions over specific Galois extensions, highlighting the failure of Galois descent in these cases.

Findings

Average p-Selmer group dimension is bounded over certain extensions.

Difference in Selmer group dimensions has bounded average across elliptic curves.

Provides bounds for representation-theoretic invariants of Mordell--Weil groups.

Abstract

We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/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/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: for each finite Galois extension K/F of number fields and prime number p, as E/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.