The distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families II

TL;DR

This paper studies the distribution of ll^{}}-Selmer groups in degree ll twist families of Galois modules over number fields, providing conditions for explicit distribution computation and showing the average rank is bounded.

Contribution

It extends previous work by identifying conditions to compute ll^{}}-Selmer group distributions and demonstrates the boundedness of average ranks in quadratic twist families.

Findings

Derived conditions for computing ll^{}}-Selmer group distributions.

Proved the average rank in quadratic twist families is bounded.

Extended understanding of Selmer groups in number field contexts.

Abstract

We continue the investigation of the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields begun in the previous paper. Building off the work on higher Selmer groups in that part, we find conditions under which we can compute the distribution of the $\ell^{\infty}$-Selmer groups for a given degree $\ell$ twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.