{
  "id": "2108.13480",
  "version": "v1",
  "published": "2021-08-30T19:01:13.000Z",
  "updated": "2021-08-30T19:01:13.000Z",
  "title": "Arithmetic statistics for Galois deformation rings",
  "authors": [
    "Anwesh Ray",
    "Tom Weston"
  ],
  "comment": "21 pages, comments appreciated",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For a fixed elliptic curve without complex multiplication, it is shown that these deformation rings are unobstructed for all but finitely many primes. For a fixed prime $p$ and varying elliptic curve $E$, we relate the problem to the question of how often $p$ does not divide the modular degree. Heuristics due to M.Watkins based on those of Cohen and Lenstra indicate that this proportion should be $\\prod_{i\\geq 1} \\left(1-\\frac{1}{p^i}\\right)\\approx 1-\\frac{1}{p}-\\frac{1}{p^2}$. This heuristic is supported by computations which indicate that most elliptic curves (satisfying further conditions) have smooth deformation rings at a given prime $p\\geq 5$, and this proportion comes close to $100\\%$ as $p$ gets larger.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-30T19:01:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11F80",
      "11F11",
      "11F33"
    ],
    "keywords": [
      "arithmetic statistics",
      "galois deformation ring parametrizing lifts",
      "smooth deformation rings",
      "proportion comes close",
      "torsion group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}