{
  "id": "2201.08777",
  "version": "v1",
  "published": "2022-01-21T16:51:51.000Z",
  "updated": "2022-01-21T16:51:51.000Z",
  "title": "Generalizations of results of Friedman and Washington on cokernels of random $p$-adic matrices",
  "authors": [
    "Gilyoung Cheong",
    "Nathan Kaplan"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let $p$ be prime and $X$ be a Haar-random $n \\times n$ matrix over $\\mathbb{Z}_{p}$, the ring of $p$-adic integers. Let $P_{1}(t), \\dots, P_{l}(t) \\in \\mathbb{Z}_{p}[t]$ be monic polynomials of degree at most $2$ whose images modulo $p$ are distinct and irreducible in $\\mathbb{F}_{p}[t]$. For each $j$, let $G_{j}$ be a finite module over $\\mathbb{Z}_{p}[t]/(P_{j}(t))$. We show that as $n$ goes to infinity, the probabilities that $\\mathrm{cok}(P_{j}(X)) \\simeq G_{j}$ are independent, and each probability can be described in terms of a Cohen-Lenstra distribution. We also show that for any fixed $n$, the probability that $\\mathrm{cok}(P_{j}(X)) \\simeq G_{j}$ for each $j$ is a constant multiple of the probability that that $\\mathrm{cok}(P_{j}(\\bar{X})) \\simeq G_{j}/pG_{j}$ for each $j$, where $\\bar{X}$ is an $n \\times n$ uniformly random matrix over $\\mathbb{F}_{p}$. These results generalize work of Friedman and Washington and prove new cases of a conjecture of Cheong and Huang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-21T16:51:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic matrices",
      "washington",
      "generalizations",
      "probability",
      "finite module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}