{
  "id": "1012.4825",
  "version": "v1",
  "published": "2010-12-21T22:26:26.000Z",
  "updated": "2010-12-21T22:26:26.000Z",
  "title": "Automorphic forms for elliptic function fields",
  "authors": [
    "Oliver Lorscheid"
  ],
  "comment": "26 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be the function field of an elliptic curve $X$ over $\\F_q$. In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over $F$. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of $\\P^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the $\\P^1$-bundles on $X$. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial $\\P^1$-bundle. Further, we determine the space of unramified $F'$-toroidal automorphic forms where $F'$ is the quadratic constant field extension of $F$. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke $L$-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series $E(\\blanc,s)$ where $s+1/2$ is a zero of the zeta function of $X$---with one possible exception in the case that $q$ is even and the class number $h$ equals $q+1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-21T22:26:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F41",
      "20C08",
      "05C75",
      "14D60",
      "14H52"
    ],
    "keywords": [
      "elliptic function fields",
      "hecke operator",
      "quadratic constant field extension",
      "contain non-trivial cusp forms",
      "unramified toroidal automorphic forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.4825L"
    }
  }
}