{
  "id": "0804.4276",
  "version": "v2",
  "published": "2008-04-27T10:49:18.000Z",
  "updated": "2008-05-12T09:18:08.000Z",
  "title": "On the non-existence of exceptional automorphisms on Shimura curves",
  "authors": [
    "Aristides Kontogeorgis",
    "Victor Rotger"
  ],
  "comment": "Title correction. 15 pages. This is an expanded version of the second half of version 1 of math/0604099",
  "doi": "10.1112/blms/bdn012",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is greater than or equal to 2, $\\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\\Aut (X_0(D, N)) = W_0(D, N)$ except for finitely many values of $(D, N)$ and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes $p\\mid D$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-05-12T09:18:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G18",
      "14G35"
    ],
    "keywords": [
      "shimura curves",
      "exceptional automorphisms",
      "non-existence",
      "indefinite rational quaternion algebra",
      "rigid analytic uniformization"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.4276K"
    }
  }
}