{
  "id": "1405.7294",
  "version": "v1",
  "published": "2014-05-28T16:10:11.000Z",
  "updated": "2014-05-28T16:10:11.000Z",
  "title": "A converse to a theorem of Gross, Zagier, and Kolyvagin",
  "authors": [
    "Christopher Skinner"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E$ be a semistable elliptic curve over $\\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\\mathbb{Q})$ is one and the Tate-Shafarevich group of $E$ has finite order, then $\\mathrm{ord}_{s=1}L(E,s)=1$. We also prove the corresponding result for the abelian variety associated with a weight two newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\\mathrm{ord}_{s=1}L(f,s)=1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-28T16:10:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "11R23",
      "11G05",
      "11G07"
    ],
    "keywords": [
      "odd prime",
      "main theorem",
      "imaginary quadratic field",
      "non-split multiplicative reduction",
      "trivial character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.7294S"
    }
  }
}