Advertisement

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)

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

Keywords

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.

Notes

Acknowledgements

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.

References

  1. Arias-de-Reyna, S., Dieulefait, L., Shin, S.-W., Wiese, G.: Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties. Math. Ann. 361(3), 909–925 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. Arias-de-Reyna, S., Dieulefait, L, Wiese, G.: Classification of subgroups of symplectic groups over finite fields containing a transvection. Demonstratio Math. (2014, preprint)Google Scholar
  3. Arias-de-Reyna, S., Kappen, C.: Abelian varieties over number fields, tame ramification and big Galois image. Math. Res. Lett. 20(1), 1–17 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. Arias-de-Reyna, S., Vila, N.: Tame Galois realizations of \(\mathrm{GSp}_{4}(\mathbb{F}_{\ell})\) over \(\mathbb{Q}\). Int. Math. Res. Not. IMRN 9, 2028–2046 (2011)MathSciNetGoogle Scholar
  5. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997). Computational algebra and number theory (London, 1993)Google Scholar
  6. Dieulefait, L.V.: Explicit determination of the images of the Galois representations attached to abelian surfaces with \(\mathrm{End}(A) = \mathbb{Z}\). Exper. Math. 11(4), 503–512 (2002a)MathSciNetCrossRefMATHGoogle Scholar
  7. Dieulefait, L.V.: On the images of the Galois representations attached to genus 2 Siegel modular forms. J. Reine Angew. Math. 553, 183–200 (2002b)MathSciNetMATHGoogle Scholar
  8. Dettweiler, M., Kühn, U., Reiter, S.: On Galois representations via Siegel modular forms of genus two. Math. Res. Lett. 8(4), 577–588 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. Dieulefait, L., Vila, N.: Projective linear groups as Galois groups over \(\mathbb{Q}\) via modular representations. J. Symb. Comput. 30(6), 799–810 (2000). Algorithmic methods in Galois theoryGoogle Scholar
  10. Dieulefait, L., Vila, N.: On the images of modular and geometric three-dimensional Galois representations. Am. J. Math. 126(2), 335–361 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. Dieulefait, L., Vila, N.: Geometric families of 4-dimensional Galois representations with generically large images. Math. Z. 259(4), 879–893 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. Dieulefait, L., Vila, N.: On the classification of geometric families of four-dimensional Galois representations. Math. Res. Lett. 18(4), 805–814 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. Dieulefait, L., Wiese, G.: On modular forms and the inverse Galois problem. Trans. Am. Math. Soc. 363(9), 4569–4584 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. Gramain, J.-B.: On defect groups for generalized blocks of the symmetric group. J. Lond. Math. Soc. (2) 78(1), 155–171 (2008)Google Scholar
  15. Hall, C.: Big symplectic or orthogonal monodromy modulo l. Duke Math. J. 141(1), 179–203 (2008)Google Scholar
  16. Hall, C.: An open-image theorem for a general class of abelian varieties. Bull. Lond. Math. Soc. 43(4), 703–711 (2011). With an appendix by Emmanuel KowalskiGoogle Scholar
  17. James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16. Addison-Wesley, Reading (1981). With a foreword by P. M. Cohn, With an introduction by Gilbert de B. RobinsonGoogle Scholar
  18. Khare, C., Larsen, M., Savin, G.: Functoriality and the inverse Galois problem. Compos. Math. 144(3), 541–564 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. Lang, S.: Abelian Varieties. Interscience Tracts in Pure and Applied Mathematics. No. 7. Interscience, New York/London (1959)Google Scholar
  20. Le Duff, P.: Représentations galoisiennes associées aux points d’ordre \(\ell\) des jacobiennes de certaines courbes de genre 2. Bull. Soc. Math. France 126(4), 507–524 (1998)MathSciNetMATHGoogle Scholar
  21. Liu, Q.: Courbes stables de genre 2 et leur schéma de modules. Math. Ann. 295(2), 201–222 (1993)MathSciNetCrossRefMATHGoogle Scholar
  22. Lockhart, P.: On the discriminant of a hyperelliptic curve. Trans. Am. Math. Soc. 342(2), 729–752 (1994)MathSciNetCrossRefMATHGoogle Scholar
  23. Mumford, D.: A note of Shimura’s paper “Discontinuous groups and abelian varieties”. Math. Ann. 181, 345–351 (1969)MathSciNetCrossRefMATHGoogle Scholar
  24. Ribet, K.A.: On \(\ell\)-adic representations attached to modular forms. Invent. Math. 28, 245–275 (1975)MathSciNetCrossRefMATHGoogle Scholar
  25. Reverter, A., Vila, N.: Some projective linear groups over finite fields as Galois groups over \(\mathbb{Q}\). In: Recent Developments in the Inverse Galois Problem (Seattle, WA, 1993). Contemporary Mathematics, vol. 186, pp. 51–63. American Mathematical Society, Providence (1995)Google Scholar
  26. Serre, J.-P.: Œuvres. Collected papers IV. Springer, Berlin (2000). 1985–1998MATHGoogle Scholar
  27. Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972)MathSciNetCrossRefMATHGoogle Scholar
  28. Stein, W.A., et al.: Sage Mathematics Software (Version 6.0). The Sage Development Team. http://www.sagemath.org (2014)
  29. Stevenhagen, P., Lenstra, H. W.: Chebotarëv and his density theorem. Math. Intell. 18(2), 26–37 (1996)MathSciNetCrossRefMATHGoogle Scholar
  30. Taylor, J.: Families of irreducible representations of \(S_{2} \wr S_{3}\). https://documents.epfl.ch/users/j/jt/jtaylor/www/PDF/representations_of_S2wrS3.pdf (2012)
  31. Wiese, G.: On projective linear groups over finite fields as Galois groups over the rational numbers. In: Modular Forms on Schiermonnikoog, pp. 343–350. Cambridge University Press, Cambridge (2008)Google Scholar
  32. Zarhin, Y.G.: Two-dimensional families of hyperelliptic jacobians with big monodromy (preprint, 2014) [arXiv:1310.6532]Google Scholar
  33. Zywina, D.: The inverse Galois problem for \(\mathrm{PSL}_{2}(\mathbb{F}_{p})\) (preprint, 2013) [arXiv:1303.3646]Google Scholar

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

Personalised recommendations