{
  "id": "2505.03070",
  "version": "v1",
  "published": "2025-05-05T23:30:58.000Z",
  "updated": "2025-05-05T23:30:58.000Z",
  "title": "Selmer stability in families of congruent Galois representations",
  "authors": [
    "Anwesh Ray"
  ],
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-05T23:30:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11R32",
      "11R45"
    ],
    "keywords": [
      "congruent galois representations",
      "selmer stability",
      "selmer groups",
      "modular galois representations",
      "greenbergs local conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}