{
  "id": "1110.0255",
  "version": "v4",
  "published": "2011-10-03T02:22:34.000Z",
  "updated": "2012-03-30T23:16:09.000Z",
  "title": "Determinants of Subquotients of Galois Representations Associated to Abelian Varieties",
  "authors": [
    "Eric Larson",
    "Dmitry Vaintrob"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\\ell$, the $\\ell^n$-torsion points of $A$ give rise to a representation $\\rho_{A, \\ell^n} : \\gal(\\bar{K} / K) \\to \\gl_{2g}(\\zz/\\ell^n\\zz)$. In particular, we get a mod-$\\ell$ representation $\\rho_{A, \\ell} : \\gal(\\bar{K} / K) \\to \\gl_{2g}(\\ff_\\ell)$and an $\\ell$-adic representation $\\rho_{A, \\ell} : \\gal(\\bar{K} / K) \\to \\gl_{2g}(\\zz_\\ell)$. In this paper, we describe the possible determinants of subrepresentations (or more generally, subquotients) of these two representation for $\\ell$ a prime number, as $A$ varies over all $g$-dimensional abelian varieties. Note that it is certainly not the case that any mod-$\\ell$ subquotient lifts to an $\\ell$-adic one. Nevertheless, the list of possible mod-$\\ell$ characters turns out to be remarkably similar to the list of possible $\\ell$-adic characters.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-03-30T23:16:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "14K02",
      "11G15",
      "14K15"
    ],
    "keywords": [
      "abelian variety",
      "galois representations",
      "determinants",
      "dimensional abelian varieties",
      "torsion points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.0255L"
    }
  }
}