{
  "id": "2312.04666",
  "version": "v1",
  "published": "2023-12-07T19:57:58.000Z",
  "updated": "2023-12-07T19:57:58.000Z",
  "title": "The algebra $\\mathbb{Z}_\\ell[[\\mathbb{Z}_p^d]]$ and applications to Iwasawa theory",
  "authors": [
    "Andrea Bandini",
    "Ignazio Longhi"
  ],
  "comment": "(Very) Preliminary version. Comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\ell$ and $p$ be distinct primes, and let $\\Gamma$ be an abelian pro-$p$-group. We study the structure of the algebra $\\Lambda:=\\mathbb{Z}_\\ell[[\\Gamma]]$ and of $\\Lambda$-modules. In the case $\\Gamma\\simeq \\mathbb{Z}_p^d$, we consider a $\\mathbb{Z}_p^d$-extension $K/k$ of a global field $k$ and use the structure theorems to provide explicit formulas for the orders and $\\ell$-ranks of certain Iwasawa modules (namely $\\ell$-class groups and $\\ell$-Selmer groups) associated with the finite subextensions of $K$. We apply this new approach to provide different proofs and generalizations of results of Washington and Sinnott on $\\ell$-class groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-07T19:57:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11R29"
    ],
    "keywords": [
      "iwasawa theory",
      "applications",
      "class groups",
      "distinct primes",
      "structure theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}