{
  "id": "1511.02212",
  "version": "v1",
  "published": "2015-11-06T19:57:21.000Z",
  "updated": "2015-11-06T19:57:21.000Z",
  "title": "How Large is $A_g(\\mathbb{F}_q)$?",
  "authors": [
    "Michael Lipnowski",
    "Jacob Tsimerman"
  ],
  "comment": "38 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties over the finite field of size $p.$ We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound $B(g,p)$ implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-06T19:57:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensional abelian varieties",
      "isomorphism classes",
      "lower bound",
      "fixed finite field",
      "large dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151102212L"
    }
  }
}