{
  "id": "2204.07759",
  "version": "v1",
  "published": "2022-04-16T09:12:38.000Z",
  "updated": "2022-04-16T09:12:38.000Z",
  "title": "Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich groups for symmetric powers of elliptic curves",
  "authors": [
    "Naoto Dainobu"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For an elliptic curve $E$ over $\\mathbb{Q}$, putting $K=\\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\\mathrm{Cl}_K$ of $K$ as a $\\mathrm{Gal}(K/\\mathbb{Q})$-module. More precisely, for any $j$ with $1\\leqslant j \\leqslant p-2$, we give a condition that $\\mathrm{Cl}_K\\otimes \\mathbb{F}_p$ has the symmetric power $\\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\\mathrm{Gal}(K/\\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\\mathrm{Sym}^j V_p E$. Here $V_p E$ denotes the rational $p$-adic Tate module of $E$. This is a partial generalization of a result of Prasad and Shekhar for the case $j=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-16T09:12:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11G05",
      "11R34"
    ],
    "keywords": [
      "ideal class group",
      "bloch-katos tate-shafarevich group",
      "elliptic curve",
      "symmetric power",
      "number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}