{
  "id": "2102.03087",
  "version": "v1",
  "published": "2021-02-05T10:24:47.000Z",
  "updated": "2021-02-05T10:24:47.000Z",
  "title": "Analytic ranks of automorphic L-functions and Landau-Siegel zeros",
  "authors": [
    "Hung M. Bui",
    "Kyle Pratt",
    "Alexandru Zaharescu"
  ],
  "comment": "50 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $\\leq 1$. We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank $\\leq 2$. In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ we show the rank of $J_0(q)$ is asymptotically equal to the rank predicted by the Brumer-Murty conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-05T10:24:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F41",
      "11F66",
      "11M20",
      "11F11"
    ],
    "keywords": [
      "landau-siegel zeros",
      "automorphic l-functions",
      "analytic rank equal",
      "sparse set",
      "large primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}