{
  "id": "2309.13806",
  "version": "v1",
  "published": "2023-09-25T01:18:16.000Z",
  "updated": "2023-09-25T01:18:16.000Z",
  "title": "Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields",
  "authors": [
    "Aleksander Shmakov"
  ],
  "comment": "29 pages, comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "There is a natural probability measure on the set of isomorphism classes of principally polarized Abelian varieties of dimension $g$ over $\\mathbb{F}_q$, weighted by the number of automorphisms. The distributions of the number of $\\mathbb{F}_q$-rational points are related to the cohomology of fiber powers of the universal family of principally polarized Abelian varieties. To that end we compute the cohomology $H^i(\\mathcal{X}^{\\times n}_g,\\mathbb{Q}_\\ell)$ for $g=1$ using results of Eichler-Shimura and for $g=2$ using results of Lee-Weintraub and Petersen, and we compute the compactly supported Euler characteristics $e_\\mathrm{c}(\\mathcal{X}^{\\times n}_g,\\mathbb{Q}_\\ell)$ for $g=3$ using results of Hain and conjectures of Bergstr\\\"om-Faber-van der Geer. In each of these cases we identify the range in which the point counts $\\#\\mathcal{X}^{\\times n}_g(\\mathbb{F}_q)$ are polynomial in $q$. Using results of Borel and Grushevsky-Hulek-Tommasi on cohomological stability, we adapt arguments of Achter-Erman-Kedlaya-Wood-Zureick-Brown to pose a conjecture about the asymptotics of the point counts $\\#\\mathcal{X}^{\\times n}_g(\\mathbb{F}_q)$ in the limit $g\\rightarrow\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-25T01:18:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "principally polarized abelian varieties",
      "cohomological arithmetic statistics",
      "finite fields",
      "point counts",
      "natural probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}