Selmer stability in families of congruent Galois representations

TL;DR

This paper investigates how Selmer groups vary in families of modular Galois representations congruent modulo a prime, showing that the count of forms with stable Selmer rank grows at least as fast as a specified function of level.

Contribution

It generalizes Ono and Skinner's results on quadratic twists to modular forms, analyzing Selmer group stability in families of congruent Galois representations.

Findings

Count of level-raising forms with stable Selmer rank grows at least as fast as X (log X)^{α - 1}

Proves a partial generalization of rank-zero quadratic twist theorems to modular forms

Establishes growth rate under mild residual representation assumptions

Abstract

In this article I study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$. 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 $N_g\leq X$, 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.