{
  "id": "1808.09544",
  "version": "v1",
  "published": "2018-08-28T20:58:16.000Z",
  "updated": "2018-08-28T20:58:16.000Z",
  "title": "Kolyvagin's result on the vanishing of $\\sha(E/K)[p^\\infty]$ and its consequences for anticyclotomic Iwasawa theory",
  "authors": [
    "Ahmed Matar",
    "Jan Nekovar"
  ],
  "comment": "32 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E$ be an elliptic curve defined over ${\\bf Q}$ and $K$ an imaginary quadratic field satisfying the Heegner hypothesis. A classical result of Kolyvagin states that, under suitable assumptions, if the basic Heegner point $y_K \\in E(K)$ is not divisible by an odd prime $p$, then the groups $E(K)/{\\bf Z} y_K$ and $\\sha(E/K)$ are finite and their orders are prime to $p$. In this article we develop the following themes: firstly, we discuss improvements of Kolyvagin's result, following Cha (2005) and Lawson and Wuthrich (2016). Secondly, we prove an abstract Iwasawa-theoretical result which allows us to deduce, under several additional assumptions, that similar vanishing holds for all layers in the anticyclotomic ${\\bf Z}_p$-extension of $K$. Analogous results hold for CM points on simple quotients of Jacobians of Shimura curves over totally real fields; this will be discussed in a separate article.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-28T20:58:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G18",
      "11G40"
    ],
    "keywords": [
      "anticyclotomic iwasawa theory",
      "kolyvagins result",
      "consequences",
      "basic heegner point",
      "totally real fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}