{
  "id": "1911.07215",
  "version": "v1",
  "published": "2019-11-17T12:07:52.000Z",
  "updated": "2019-11-17T12:07:52.000Z",
  "title": "On Landau-Siegel zeros and heights of singular moduli",
  "authors": [
    "Christian Táfula"
  ],
  "comment": "43 pages, 1 figure. For the scripts used to generate the figure, see https://github.com/tafula/loght_jtd",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $D < 0$ be a fundamental discriminant, $\\chi_D$ the Dirichlet character associated to $\\mathbb{Q}(\\sqrt{D})$, and $\\tau_D := i\\sqrt{|D|}/2$ if $D \\equiv 0 \\,(\\mathrm{mod}~4)$ or $\\tau_D := (-1+i\\sqrt{|D|})/2$ if $D \\equiv 1 \\,(\\mathrm{mod}~4)$. Based on the work of Granville-Stark [DOI:10.1007/s002229900036], and a theorem of Duke [DOI:10.1007/BF01393993] on the uniform distribution of Heegner points, we show that \\[ \\mathrm{ht}(j(\\tau_D)) = 6\\, \\Bigg(\\sum_{\\substack{0 < \\mathrm{Re}(\\varrho) < 1 \\\\ L(\\varrho,\\chi_D)=0}} \\frac{1}{\\varrho} \\Bigg) + C + o(1) \\] as $D\\to -\\infty$, where $j$ is the $j$-invariant function, $\\mathrm{ht}$ is the absolute logarithmic Weil height, and $C \\approx 11.511550\\ldots$. From that, we measure the effect of the largest real zero of $L(s,\\chi_D)$ (say, $\\beta_D$) to the growth of $\\mathrm{ht}(j(\\tau_D))$, allowing us to obtain, from the uniform $abc$-conjecture for number fields, the estimate \\[ \\beta_D \\leq 1 - \\frac{10/(5-\\sqrt{5}) + o(1)}{\\log(|D|)}, \\] where $10/(5-\\sqrt{5}) \\approx 3.618033\\ldots$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-17T12:07:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M20",
      "11G50",
      "11N37"
    ],
    "keywords": [
      "landau-siegel zeros",
      "singular moduli",
      "absolute logarithmic weil height",
      "largest real zero",
      "number fields"
    ],
    "tags": [
      "github project"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}