{
  "id": "1904.05756",
  "version": "v1",
  "published": "2019-04-11T15:13:43.000Z",
  "updated": "2019-04-11T15:13:43.000Z",
  "title": "Non-vanishing theorems for central $L$-values of some elliptic curves with complex multiplication II",
  "authors": [
    "John Coates",
    "Yongxiong Li"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $q$ be any prime $\\equiv 7 \\mod 16$, $K = \\mathbb{Q}(\\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove the existence of a large explicit infinite family of quadratic twists of $A$ whose complex $L$-series does not vanish at $s=1$. This non-vanishing theorem is completely new when $q > 7$. Its proof depends crucially on the results established in our earlier paper for the Iwasawa theory at the prime $p=2$ of the abelian variety $B/K$, which is the restriction of scalars from $H$ to $K$ of the elliptic curve $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-11T15:13:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G40"
    ],
    "keywords": [
      "complex multiplication",
      "non-vanishing theorem",
      "gross elliptic curve",
      "hilbert class field",
      "abelian variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}