Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations

Arias-de-Reyna, Sara; Dieulefait, Luis; Wiese, Gabor

doi:10.1090/tran/6708

Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations
HTML articles powered by AMS MathViewer

by Sara Arias-de-Reyna, Luis Dieulefait and Gabor Wiese PDF

Trans. Amer. Math. Soc. 369 (2017), 887-908 Request permission

Abstract:

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem.

In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.

References

Sara Arias-de-Reyna, Luis V. Dieulefait, Sug Woo Shin, and Gabor Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties, Math. Ann. 361 (2015), no. 3-4, 909–925. MR 3319552, DOI 10.1007/s00208-014-1091-x
Sara Arias-de-Reyna, Luis Dieulefait, and Gabor Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem, II: Transvections and huge image, Pacific J. Math. 281 (2016), no. 1, 1–16. MR 3459964, DOI 10.2140/pjm.2016.281.1
E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
Henri Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 213–237 (French). MR 1279611, DOI 10.1090/conm/165/01601
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Luis Dieulefait and Gabor Wiese, On modular forms and the inverse Galois problem, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4569–4584. MR 2806684, DOI 10.1090/S0002-9947-2011-05477-2
Loo-Keng Hua, On the automorphisms of the symplectic group over any field, Ann. of Math. (2) 49 (1948), 739–759. MR 27764, DOI 10.2307/1969397
Chandrashekhar Khare, Michael Larsen, and Gordan Savin, Functoriality and the inverse Galois problem, Compos. Math. 144 (2008), no. 3, 541–564. MR 2422339, DOI 10.1112/S0010437X07003284
Barry Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 243–311. MR 1638481
Richard Pink, Compact subgroups of linear algebraic groups, J. Algebra 206 (1998), no. 2, 438–504. MR 1637068, DOI 10.1006/jabr.1998.7439
Kenneth A. Ribet, On $l$-adic representations attached to modular forms. II, Glasgow Math. J. 27 (1985), 185–194. MR 819838, DOI 10.1017/S0017089500006170
Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, Research Notes in Mathematics, vol. 7, A K Peters Ltd., Wellesley, MA, 1998, With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original.
Christopher Skinner, A note on the $p$-adic Galois representations attached to Hilbert modular forms, Doc. Math. 14 (2009), 241–258. MR 2538615
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2