Abstract
We study compatible families of four-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. We apply our result to an example constructed by Jasper Scholten (A non-selfdual 4-dimensional Galois representation, www.math.uiuc.edu/Algebraic-Number-Theory/0183, 1999), obtaining a family of linear groups and one of unitary groups as Galois groups over \({\mathbb{Q}}\) .
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Kostrikin A.I., Shafarevich I.R. (eds) (1989). Algebra IV, Encyclopedia of Mathematical Sciences, vol. 37. Springer, Berlin
Ash, A., Gunnells, P., McConnell, M.: Cohomology of congruence subgroups of SL(4, Z) II (2007) (Preprint)
Berthelot, P.: Altérations de variétés algébriques [d’après A. J. de Jong], in “Séminaire Bourbaki”, vol. 1995/96, exp. 815, pp. 273–311; Astérisque (1997)
Dieulefait L. (2002). On the images of the Galois representations attached to genus 2 Siegel modular forms. J. Reine Angew. Math. 553: 183–200
Dieulefait L. (2002). Explicit determination of the images of the Galois representations attached to abelian surfaces with \({\rm End}(A) = {\mathbb{Z}}\). Exp. Math. 11: 503–512
Dieulefait L. and Vila N. (2004). On the images of modular and geometric three-dimensional Galois representations. Am. J. Math. 126: 335–361
Dieulefait, L., Vila, N.: The classification of geometric families of four-dimensional Galois representations, in preparation
Fontaine, J.M., Laffaille, G.: Construction de représentations p-adiques. Ann. Sci. Éc. Norm. Sup., 4e série, t. 15, 547–608 (1982)
Fontaine J.M and Messing W. (1987). p-adic periods and p-adic etale cohomology, in “Currents Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985)”. Contemp. Math. 67: 179–207
Gunnells, P.E.: Modular symbols and Hecke operators, Algorithmic number theory (Leiden, 2000), Lecture Notes in Computer Science, pp. 347–358, 1838. Springer, Berlin (2000)
Kleidman, P.B.: The subgroup structure of some finite simple groups. Ph.D. thesis, Cambridge (1986)
Kleidman P. and Liebeck M. (1990). The subgroup structure of the finite classical groups. London Math. Soc. LNS 129. Combridge University Press, Combridge
Malle G. and Matzat B.H. (1999). Inverse Galois theory. Springer, Berlin
Neumann P. M. and Praeger C. E. (1992). A recognition algorithm for special linear groups. Proc. Lond. Math. Soc. 65(3): 555–603
Scholten, J.: A non-selfdual 4-dimensional Galois representation, www.math.uiuc.edu/Algebraic-Number-Theory/0183 (1999)
Serre J-P. (1968). Abelian ℓ-adic representations and elliptic curves. Benjamin, Reading
Serre J-P. (1972). Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15: 259–331
Serre J-P. (2000). Oeuvres, vol. 4. Springer, Berlin, 1–55
Völklein H. (1993). Braid group action via GL n (q) and U n (q) and Galois realizations. Israel J. Math. 82: 405–427
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by MEC grant MTM2006-04895.
Rights and permissions
About this article
Cite this article
Dieulefait, L., Vila, N. Geometric families of 4-dimensional Galois representations with generically large images. Math. Z. 259, 879–893 (2008). https://doi.org/10.1007/s00209-007-0253-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-007-0253-x