{
  "id": "math/0406012",
  "version": "v1",
  "published": "2004-06-01T11:29:53.000Z",
  "updated": "2004-06-01T11:29:53.000Z",
  "title": "Vanishing of L-functions of elliptic curves over number fields",
  "authors": [
    "Chantal David",
    "Jack Fearnley",
    "Hershy Kisilevsky"
  ],
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $E$ be an elliptic curve over $\\mathbb{Q}$, with L-function $L_E(s)$. For any primitive Dirichlet character $\\chi$, let $L_E(s, \\chi)$ be the L-function of $E$ twisted by $\\chi$. In this paper, we use random matrix theory to study vanishing of the twisted L-functions $L_E(s, \\chi)$ at the central value $s=1$. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that $L_E(1, \\chi)=0$, but that for any fixed prime $k \\geq 7$, there are only finitely many character of order $k$ such that $L_E(1, \\chi)$ vanishes. With the Birch and Swinnerton-Dyer Conjecture, those conjectures can be restated to predict the number of cyclic extensions $K/\\mathbb{Q}$ of prime degree such that $E$ acquires new rank over $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-06-01T11:29:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40"
    ],
    "keywords": [
      "elliptic curve",
      "number fields",
      "l-function",
      "random matrix theory predicts",
      "central value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6012D"
    }
  }
}