Galois Representations and Galois Groups Over ℚ

  • Sara Arias-de-Reyna
  • Cécile Armana
  • Valentijn Karemaker
  • Marusia Rebolledo
  • Lara Thomas
  • Núria Vila
Conference paper

DOI: 10.1007/978-3-319-17987-2_8

Volume 2 of the book series Association for Women in Mathematics Series (AWMS)
Cite this paper as:
Arias-de-Reyna S., Armana C., Karemaker V., Rebolledo M., Thomas L., Vila N. (2015) Galois Representations and Galois Groups Over ℚ. In: Bertin M., Bucur A., Feigon B., Schneps L. (eds) Women in Numbers Europe. Association for Women in Mathematics Series, vol 2. Springer, Cham

Abstract

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 \(\ell\in [11,500,000]\).

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