{
  "id": "1403.7168",
  "version": "v2",
  "published": "2014-03-27T18:46:06.000Z",
  "updated": "2016-05-03T19:24:51.000Z",
  "title": "P-torsion monodromy representations of elliptic curves over geometric function fields",
  "authors": [
    "Jacob Tsimerman",
    "Benjamin Bakker"
  ],
  "comment": "Comments Welcome! v2: Many improvements to the exposition and some proofs, based on suggestions of the referee. To appear in Ann. of Math",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Given a complex quasiprojective curve $B$ and a non-isotrivial family $\\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\\mathcal{E}[p]$ yields a monodromy representation $\\rho_\\mathcal{E}[p]:\\pi_1(B)\\rightarrow \\mathrm{GL}_2(\\mathbb{F}_p)$. We prove that if $\\rho_{\\mathcal E}[p]\\cong \\rho_{\\mathcal E'}[p]$ then $\\mathcal{E}$ and $\\mathcal E'$ are isogenous, provided $p$ is larger than a constant depending only on the gonality of $B$. This can be viewed as a function field analog of the Frey--Mazur conjecture, which states that an elliptic curve over $\\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p> 17$. The proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-27T18:46:06.000Z",
      "title": "p-torsion monodromy representations of elliptic curves over geometric function fields",
      "abstract": "Given a complex algebraic curve $C$ and a non-isotrivial family $\\mathcal{E}$ of elliptic curves over $C$, the $p$-torsion $\\mathcal{E}[p]$ yields a monodromy representation $\\rho_\\mathcal{E}[p]:\\pi_1(C)\\rightarrow \\mathrm{GL}_2(\\mathbb{F}_p)$. We prove that if $\\rho_{\\mathcal E}[p]\\cong \\rho_{\\mathcal E'}[p]$ then $\\mathcal{E}$ and $\\mathcal E'$ are isogenous, provided $p$ is larger than a constant depending only on the gonality of $C$. This can be viewed as a function field analog of the Frey--Mazur conjecture, which states that an elliptic curve over $\\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\\geq 17$. The proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0.",
      "comment": "Comments Welcome!",
      "journal": null,
      "doi": null,
      "authors": [
        "Jacob Tsimerman",
        "Ben Bakker"
      ]
    },
    {
      "version": "v2",
      "updated": "2016-05-03T19:24:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H52",
      "14G35",
      "14K02"
    ],
    "keywords": [
      "elliptic curve",
      "p-torsion monodromy representations",
      "geometric function fields",
      "function field analog",
      "complex algebraic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7168T"
    }
  }
}