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

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

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