Parity-induced Selmer Growth For Symplectic, Ordinary Families

TL;DR

This paper demonstrates that for certain Galois representations over quadratic extensions, Selmer ranks grow in a predictable manner over specific infinite extensions, extending previous methods to broader classes of arithmetic objects.

Contribution

It generalizes Mazur--Rubin's method to show Selmer rank growth in infinite extensions for various Galois representations, including abelian varieties and modular forms.

Findings

Selmer ranks are bounded below by extension degree in certain infinite towers.

Method applies to abelian varieties, modular forms, and Hida families.

Extends previous results to broader classes of Galois representations.

Abstract

Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\pm$ the maximal $\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of Nekov\'a\v{r}, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.