{
  "id": "2012.15319",
  "version": "v1",
  "published": "2020-12-30T20:48:37.000Z",
  "updated": "2020-12-30T20:48:37.000Z",
  "title": "Vanishing of Dirichlet L-functions at the central point over function fields",
  "authors": [
    "Ravi Donepudi",
    "Wanlin Li"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field $\\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over $\\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over $\\mathbb{F}_q(t)$ whose $L$-functions vanish at the central point where $q=p^{4n}$ for any rational prime $p \\equiv 2 \\bmod 3$. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the $L$-functions of Dirichlet characters of other orders.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-30T20:48:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central point",
      "dirichlet l-functions",
      "geometric criterion",
      "supersingular superelliptic curves",
      "rational function field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}