Galois Representations and Galois Groups Over ℚ

  • Sara Arias-de-Reyna
  • Cécile Armana
  • Valentijn Karemaker
  • Marusia Rebolledo
  • Lara Thomas
  • Núria Vila
Part of the Association for Women in Mathematics Series book series (AWMS, volume 2)


In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let \(C/\mathbb{Q}\) be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety, and let \(\bar{\rho }_{\ell}: G_{\mathbb{Q}} \rightarrow \mathrm{ GSp}(J(C)[\ell])\) be the Galois representation attached to the \(\ell\)-torsion of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes \(\ell\) (if they exist) such that \(\bar{\rho }_{\ell}\) is surjective. In particular we realize \(\mathrm{GSp}_{6}(\mathbb{F}_{\ell})\) as a Galois group over \(\mathbb{Q}\) for all primes \(\ell\in [11,500,000]\).


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sara Arias-de-Reyna
    • 1
  • Cécile Armana
    • 2
  • Valentijn Karemaker
    • 3
  • Marusia Rebolledo
    • 4
  • Lara Thomas
    • 5
  • Núria Vila
    • 6
  1. 1.Mathematical Research UnitUniversity of LuxembourgLuxembourgLuxembourg
  2. 2.Laboratory of MathematicsUniversity of Franche-ComteéBesançonFrance
  3. 3.Mathematical InstituteUtrecht UniversityUtrechtThe Netherlands
  4. 4.Laboratory of MathematicsBlaise Pascal UniversityClermont-Ferrand, AubièreFrance
  5. 5.Pure and Applied Mathematics UnitENS LyonLyonFrance
  6. 6.Department of Algebra and GeometryUniversity of BarcelonaBarcelonaSpain

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