{
  "id": "2009.08622",
  "version": "v1",
  "published": "2020-09-18T04:19:18.000Z",
  "updated": "2020-09-18T04:19:18.000Z",
  "title": "Conjecture: 100% of elliptic surfaces over $\\mathbb{Q}$ have rank zero",
  "authors": [
    "Alex Cowan"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Based on an equation for the rank of an elliptic surface over $\\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-18T04:19:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J27",
      "11G05"
    ],
    "keywords": [
      "elliptic surface",
      "rank zero",
      "conjecture",
      "experimental evidence",
      "finite fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}