{
  "id": "2305.01956",
  "version": "v1",
  "published": "2023-05-03T08:16:24.000Z",
  "updated": "2023-05-03T08:16:24.000Z",
  "title": "Lower bounds for the number of number fields with Galois group $GL_2(\\mathbb{F}_\\ell)$",
  "authors": [
    "Anwesh Ray"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\ell\\geq 5$ be a prime number and $\\mathbb{F}_\\ell$ denote the finite field with $\\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\\mathbb{F}_\\ell)$ and absolute discriminant bounded above by $X$ is asymptotically at least $\\frac{X^{\\frac{1}{12\\ell(\\ell-1)^2}}}{\\log X}$. We also obtain a similar result for the number of surjective homomorphisms $\\rho:Gal(\\bar{\\mathbb{Q}}/\\mathbb{Q})\\rightarrow GL_2(\\mathbb{F}_\\ell)$ ordered by the prime to $\\ell$ part of the Artin conductor of $\\rho$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-03T08:16:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R32",
      "11R45"
    ],
    "keywords": [
      "number fields",
      "lower bounds",
      "galois group isomorphic",
      "galois extensions",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}