{
  "id": "2307.13134",
  "version": "v1",
  "published": "2023-07-24T21:25:06.000Z",
  "updated": "2023-07-24T21:25:06.000Z",
  "title": "Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory",
  "authors": [
    "Luca Mastella"
  ],
  "comment": "33 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article we prove a refinement of a theorem of Longo and Vigni in the anticyclotomic Iwasawa theory for modular forms. More precisely we give a definition for the ($\\mathfrak{p}$-part of the) Shafarevich-Tate groups $\\widetilde{\\mathrm{sha}}_{\\mathfrak{p}^\\infty}(f/K)$ and $\\widetilde{\\mathrm{sha}}_{\\mathfrak{p}^\\infty}(f/K_\\infty)$ of a modular form $f$ of weight $k >2$, over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\\mathbb{Z}_p$-extension $K_\\infty$ and we show that if the basic generalized Heegner cycle $z_{f, K}$ is non-torsion and not divisible by $p$, then $\\widetilde{\\mathrm{sha}}_{\\mathfrak{p}^\\infty}(f/K) = \\widetilde{\\mathrm{sha}}_{\\mathfrak{p}^\\infty}(f/K_\\infty) = 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-24T21:25:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11F11"
    ],
    "keywords": [
      "anticyclotomic iwasawa theory",
      "modular form",
      "shafarevich-tate group",
      "consequences",
      "imaginary quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}