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


Modular Form Characteristic Polynomial Galois Group Abelian Variety Galois Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors would like to thank Marie-José Bertin, Alina Bucur, Brooke Feigon, and Leila Schneps for organizing the WIN-Europe conference which initiated this collaboration. Moreover, we are grateful to the Centre International de Rencontres Mathématiques, the Institut de Mathématiques de Jussieu, and the Institut Henri Poincaré for their hospitality during several short visits. The authors are indebted to Irene Bouw, Jean-Baptiste Gramain, Kristin Lauter, Elisa Lorenzo, Melanie Matchett Wood, Frans Oort, and Christophe Ritzenthaler for several insightful discussions. We also want to thank the anonymous referee for her/his suggestions that helped us to improve this paper.

S. Arias-de-Reyna and N. Vila are partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain, C. Armana by a BQR 2013 Grant from Université de Franche-Comté and M. Rebolledo by the ANR Project Régulateurs ANR-12-BS01-0002. L. Thomas thanks the Laboratoire de Mathématiques de Besançon for its support.


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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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