{
  "id": "2407.14264",
  "version": "v1",
  "published": "2024-07-19T12:40:46.000Z",
  "updated": "2024-07-19T12:40:46.000Z",
  "title": "Galois representations are surjective for almost all Drinfeld modules",
  "authors": [
    "Anwesh Ray"
  ],
  "comment": "Version 1: 16 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "This article advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let $F$ be the rational function field over a finite field. I establish that for Drinfeld modules of rank $r \\geq 2$, the $T$-adic Galois representation: $\\widehat{\\rho}_{\\phi, T}: Gal(F^{sep}/F) \\rightarrow GL_r(\\mathbb{F}_q[[T]])$ is surjective for a density $1$ set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-19T12:40:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11G09",
      "11R45"
    ],
    "keywords": [
      "drinfeld modules",
      "proof utilizes hilbert irreducibility",
      "surjective",
      "adic galois representation",
      "rational function field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}