Selmer stability in families of congruent Galois representations

Abstract In this paper, we 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, we 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. We 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 $\bar{\rho}$ , we prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$ , for an explicit constant $\alpha \gt 0$ . The main result is a partial generalisation of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.