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 \(\ell\in [11,500,000]\).
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We adopt the convention that identity is a transvection so that the set of transvections for a given hyperplane H is a group.
References
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)
Arias-de-Reyna, S., Dieulefait, L, Wiese, G.: Classification of subgroups of symplectic groups over finite fields containing a transvection. Demonstratio Math. (2014, preprint)
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)
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)
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)
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)
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)
Dettweiler, M., Kühn, U., Reiter, S.: On Galois representations via Siegel modular forms of genus two. Math. Res. Lett. 8(4), 577–588 (2001)
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 theory
Dieulefait, L., Vila, N.: On the images of modular and geometric three-dimensional Galois representations. Am. J. Math. 126(2), 335–361 (2004)
Dieulefait, L., Vila, N.: Geometric families of 4-dimensional Galois representations with generically large images. Math. Z. 259(4), 879–893 (2008)
Dieulefait, L., Vila, N.: On the classification of geometric families of four-dimensional Galois representations. Math. Res. Lett. 18(4), 805–814 (2011)
Dieulefait, L., Wiese, G.: On modular forms and the inverse Galois problem. Trans. Am. Math. Soc. 363(9), 4569–4584 (2011)
Gramain, J.-B.: On defect groups for generalized blocks of the symmetric group. J. Lond. Math. Soc. (2) 78(1), 155–171 (2008)
Hall, C.: Big symplectic or orthogonal monodromy modulo l. Duke Math. J. 141(1), 179–203 (2008)
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 Kowalski
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. Robinson
Khare, C., Larsen, M., Savin, G.: Functoriality and the inverse Galois problem. Compos. Math. 144(3), 541–564 (2008)
Lang, S.: Abelian Varieties. Interscience Tracts in Pure and Applied Mathematics. No. 7. Interscience, New York/London (1959)
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)
Liu, Q.: Courbes stables de genre 2 et leur schéma de modules. Math. Ann. 295(2), 201–222 (1993)
Lockhart, P.: On the discriminant of a hyperelliptic curve. Trans. Am. Math. Soc. 342(2), 729–752 (1994)
Mumford, D.: A note of Shimura’s paper “Discontinuous groups and abelian varieties”. Math. Ann. 181, 345–351 (1969)
Ribet, K.A.: On \(\ell\)-adic representations attached to modular forms. Invent. Math. 28, 245–275 (1975)
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)
Serre, J.-P.: Œuvres. Collected papers IV. Springer, Berlin (2000). 1985–1998
Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972)
Stein, W.A., et al.: Sage Mathematics Software (Version 6.0). The Sage Development Team. http://www.sagemath.org (2014)
Stevenhagen, P., Lenstra, H. W.: Chebotarëv and his density theorem. Math. Intell. 18(2), 26–37 (1996)
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)
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)
Zarhin, Y.G.: Two-dimensional families of hyperelliptic jacobians with big monodromy (preprint, 2014) [arXiv:1310.6532]
Zywina, D.: The inverse Galois problem for \(\mathrm{PSL}_{2}(\mathbb{F}_{p})\) (preprint, 2013) [arXiv:1303.3646]
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.
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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. https://doi.org/10.1007/978-3-319-17987-2_8
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