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Selmer stability in families of congruent Galois representations

Anwesh Ray

TL;DR

The paper analyzes how Selmer groups defined via Greenberg local conditions behave in families of modular Galois representations that are congruent modulo a fixed prime $p≥5$, drawing an analogy with Goldfeld’s conjecture on ranks. It combines Diamond–Taylor level-raising, Poitou–Tate duality, and Serre-density arguments to show that a substantial set of level-raising forms maintain the same $p$-Selmer rank as a fixed base form, yielding a lower bound $N_f(X) \gg X (\log X)^{α-1}$ with an explicit exponent $α=(p-3)/(p-1)^2$. The work extends rank-zero quadratic-twist results to modular forms and their Selmer groups, providing a concrete stability phenomenon for $p$-congruent families. The approach highlights how level-raising primes from a positive-density set govern the distribution of congruent forms with preserved Selmer structure, offering a modular-analytic analogue of classical arithmetic statistics results.

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 $\barρ$ 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 $\barρ$, I prove that this count grows at least as fast as $X (\log X)^{α- 1}$ as $X \to \infty$, for an explicit constant $α> 0$. 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.

Selmer stability in families of congruent Galois representations

TL;DR

The paper analyzes how Selmer groups defined via Greenberg local conditions behave in families of modular Galois representations that are congruent modulo a fixed prime , drawing an analogy with Goldfeld’s conjecture on ranks. It combines Diamond–Taylor level-raising, Poitou–Tate duality, and Serre-density arguments to show that a substantial set of level-raising forms maintain the same -Selmer rank as a fixed base form, yielding a lower bound with an explicit exponent . The work extends rank-zero quadratic-twist results to modular forms and their Selmer groups, providing a concrete stability phenomenon for -congruent families. The approach highlights how level-raising primes from a positive-density set govern the distribution of congruent forms with preserved Selmer structure, offering a modular-analytic analogue of classical arithmetic statistics results.

Abstract

In this article I study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime . 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 via Greenberg's local conditions under congruences of residual Galois representations. Let be a positive real number. Fix a residual representation and a corresponding modular form of weight and optimal level. I count the number of level-raising modular forms of weight that are congruent to modulo , with level , such that the -rank of the Selmer groups of equals that of . Under some mild assumptions on , I prove that this count grows at least as fast as as , for an explicit constant . 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.
Paper Structure (10 sections, 11 theorems, 68 equations)

This paper contains 10 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Main result
  4. Modular forms and their Selmer groups
  5. Notation
  6. Facts from Galois cohomology
  7. Modular Galois representations and level raising
  8. Selmer groups
  9. The residual Selmer group
  10. Main results

Key Result

Theorem A

Assume that no prime $\ell \equiv \pm 1 \pmod{p}$ divides $N_{\bar{\rho}}$. Then, with notation as above, where $\alpha := \frac{p - 3}{(p - 1)^2}$.

Theorems & Definitions (25)

  • Theorem A
  • Theorem 2.2: Carayol
  • Definition 2.3
  • Theorem 2.4: Diamond–Taylor diamond1994non
  • Definition 2.5: Greenberg Selmer group over $\mathbb{Q}$
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Proposition 2.9
  • ...and 15 more