{
  "id": "math/0211394",
  "version": "v2",
  "published": "2002-11-26T18:52:51.000Z",
  "updated": "2003-12-22T18:29:10.000Z",
  "title": "Finiteness results for modular curves of genus at least 2",
  "authors": [
    "Matthew Baker",
    "Enrique Gonzalez-Jimenez",
    "Josep Gonzalez",
    "Bjorn Poonen"
  ],
  "comment": "53 pages; minor revisions made",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part of J_0(N) with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and each integer g at least 2, the set of genus g curves over k dominated by a Fermat curve is finite and computable.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-12-22T18:29:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G18",
      "14G35"
    ],
    "keywords": [
      "modular curve",
      "modular hyperelliptic curves",
      "similar finiteness results",
      "characteristic zero",
      "rational numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11394B"
    }
  }
}