{
  "id": "math/0505571",
  "version": "v3",
  "published": "2005-05-26T14:02:41.000Z",
  "updated": "2005-09-20T20:15:43.000Z",
  "title": "Finite linear groups, lattices, and products of elliptic curves",
  "authors": [
    "Vladimir L. Popov",
    "Yuri G. Zarhin"
  ],
  "comment": "25 pages. Several examples are added",
  "journal": "J. Algebra 305 (2006), no. 1, 562--576.",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "Let $V$ be a finite dimensional complex linear space and let $G$ be an irreducible finite subgroup of $\\GL(V)$. For a $G$-invariant lattice $\\Lambda$ in $V$ of maximal rank, we give a description of structure of the complex torus $V/\\Lambda$. In particular, we prove that for a wide class of groups, $V/\\Lambda$ is isogenous to a self-product of an elliptic curve, and that in many cases $V/\\Lambda$ is isomorphic to a product of mutually isogenous elliptic curves with complex multiplication. We show that there are $G$ and $\\Lambda$ such that the complex torus $V/\\Lambda$ is not an abelian variety but one can always replace $\\Lambda$ by another $G$-invariant lattice $\\Delta$ such that $V/\\Delta$ is a product if elliptic curves with complex multiplication. We amplify these results with a criterion, in terms of the character and the Schur $\\mathbf Q$-index of $G$-module $V$, of the existence of a nonzero $G$-invariant lattice in $V$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-09-20T20:15:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K20",
      "14K22",
      "22E40",
      "32J18",
      "22E40"
    ],
    "keywords": [
      "elliptic curve",
      "finite linear groups",
      "invariant lattice",
      "finite dimensional complex linear space",
      "complex multiplication"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......5571P"
    }
  }
}