{
  "id": "2411.19318",
  "version": "v1",
  "published": "2024-11-28T18:43:36.000Z",
  "updated": "2024-11-28T18:43:36.000Z",
  "title": "On the Distribution of Class Groups of Abelian Extensions",
  "authors": [
    "Yuan Liu"
  ],
  "comment": "with an appendix by Peter Koymans",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a finite abelian group $\\Gamma$, we study the distribution of the $p$-part of the class group $\\operatorname{Cl}(K)$ as $K$ varies over Galois extensions of $\\mathbb{Q}$ or $\\mathbb{F}_q(t)$ with Galois group isomorphic to $\\Gamma$. We first construct a discrete valuation ring $e\\mathbb{Z}_p[\\Gamma]$ for each primitive idempotent $e$ of $\\mathbb{Q}_p[\\Gamma]$, such that 1) $e\\mathbb{Z}_p[\\Gamma]$ is a lattice of the irreducible $\\mathbb{Q}_p[\\Gamma]$-module $e\\mathbb{Q}_p[\\Gamma]$, and 2) $e\\mathbb{Z}_p[\\Gamma]$ is naturally a quotient of $\\mathbb{Z}_p[\\Gamma]$. For every $e$, we study the distribution of $e\\operatorname{Cl}(K):=e\\mathbb{Z}_p[\\Gamma] \\otimes_{\\mathbb{Z}_p[\\Gamma]} \\operatorname{Cl}(K)[p^{\\infty}]$, and prove that there is an ideal $I_e$ of $e\\mathbb{Z}_p[\\Gamma]$ such that $e\\operatorname{Cl}(K) \\otimes (e\\mathbb{Z}_p[\\Gamma]/I_e)$ is too large to have finite moments, while $I_e \\cdot e\\operatorname{Cl}(K)$ should be equidistributed with respect to a Cohen--Lenstra type of probability measure. We give conjectures for the probability and moment of the distribution of $I_e\\cdot e\\operatorname{Cl}(k)$, and prove a weighted version of the moment conjecture in the function field case. Our weighted-moment technique is designed to deal with the situation when the function field moment, obtained by counting points of Hurwitz spaces, is infinite; and we expect that this technique can also be applied to study other bad prime cases. Our conjecture agrees with the Cohen--Lenstra--Martinet conjecture when $p\\nmid |\\Gamma|$, and agrees with the Gerth conjecture when $\\Gamma=\\mathbb{Z}/p\\mathbb{Z}$. We also study the kernel of $\\operatorname{Cl}(K) \\to \\bigoplus_e e\\operatorname{Cl}(K)$, and show that the average size of this kernel is infinite when $p^2\\mid |\\Gamma|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-28T18:43:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class group",
      "abelian extensions",
      "distribution",
      "function field case",
      "function field moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}