{
  "id": "math/0701305",
  "version": "v2",
  "published": "2007-01-10T19:27:26.000Z",
  "updated": "2007-05-30T23:19:17.000Z",
  "title": "Computing endomorphism rings of Jacobians of genus 2 curves over finite fields",
  "authors": [
    "David Freeman",
    "Kristin Lauter"
  ],
  "comment": "Revised version, with minor corrections and incorporating reader comments. Proposition 3.7 and Lemma 6.5 are new. To appear in Proceedings of SAGA 2007, Tahiti",
  "categories": [
    "math.NT"
  ],
  "abstract": "We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups $J[\\ell^d]$ for prime powers $\\ell^d$. We use these algorithms to create the first implementation of Eisentr\\\"ager and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \\cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute $p^3$ curves for many small primes $p$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-05-30T23:19:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G25",
      "11G15",
      "14G50"
    ],
    "keywords": [
      "computing endomorphism rings",
      "finite field",
      "computing igusa class polynomials",
      "chinese remainder theorem",
      "quartic cm field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1305F"
    }
  }
}