Selmer stability in families of congruent Galois representations

Abstract

In this article I study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p\ge 5$. Motivated by analogies with Goldfeld's conjecture on ranks in quadratic twist families of elliptic curves, I investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg's local conditions under congruences of residual Galois representations. Let $X$ be a positive real number. Fix a residual representation $\bar{\rho }$ and a corresponding modular form $f$ of weight $2$ and optimal level. I count the number of level-raising modular forms $g$ of weight $2$ that are congruent to $f$ modulo $p$, with level $N_g\le X$, such that the $p$-rank of the Selmer groups of $g$ equals that of $f$. Under some mild assumptions on $\bar{\rho }$, I prove that this count grows at least as fast as $X(\log X{)}^{\alpha -1}$ as $X\to \infty$, for an explicit constant $\alpha >0$. The main result is a partial generalization of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

v2: minor corrections, accepted for publication in Math Proc. Cambridge Phil. Soc