{
  "id": "1509.07095",
  "version": "v1",
  "published": "2015-09-23T18:57:49.000Z",
  "updated": "2015-09-23T18:57:49.000Z",
  "title": "Endomorphism rings of reductions of elliptic curves and abelian varieties",
  "authors": [
    "Yuri G. Zarhin"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $\\Delta(v)$ of $End(E(v))$ is divisible by $N$ and the ratio $\\Delta(v)/N$ is relatively prime to $NM$. We also discuss similar questions for reductions of abelian varieties. The subject of this paper was inspired by an exercise in Serre's \"Abelian $\\ell$-adic representations and elliptic curves\" and questions of Mihran Papikian and Alina Cojocaru.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-23T18:57:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G07",
      "11G10",
      "14G25"
    ],
    "keywords": [
      "elliptic curve",
      "abelian varieties",
      "endomorphism ring",
      "nonarchimedean places",
      "relatively prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150907095Z"
    }
  }
}