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
Additional Information
- Sara Arias-de-Reyna
- Affiliation: Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg
- Address at time of publication: Departamento de Álgebra, Universidad de Sevilla, Avda. Reina Mercedes s/n. Apdo. 1160, CP 41080, Sevilla, Spain
- MR Author ID: 869958
- Email: sara_arias@us.es
- Luis Dieulefait
- Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
- MR Author ID: 671876
- Email: ldieulefait@ub.edu
- Gabor Wiese
- Affiliation: Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg
- Email: gabor.wiese@uni.lu
- Received by editor(s): September 23, 2013
- Received by editor(s) in revised form: January 30, 2015
- Published electronically: May 2, 2016
- Additional Notes: The first author worked on this article as a fellow of the Alexander-von-Humboldt foundation. She thanks the Université du Luxembourg for its hospitality during a long term visit. She was also partially supported by projects MTM2013-46231-P and MTM2015-66716-P of the Ministerio de Economía y Competitividad of Spain
The second author was supported by projects MTM2012-33830 and MTM2015-66716-P of the Ministerio de Economía y Competitividad of Spain and by an ICREA Academia Research Prize
The third author was partially supported by the DFG collaborative research centre TRR 45, the DFG priority program 1489 and the Fonds National de la Recherche Luxembourg (INTER/DFG/12/10/COMFGREP) - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 887-908
- MSC (2010): Primary 11F80, 20C25, 12F12
- DOI: https://doi.org/10.1090/tran/6708
- MathSciNet review: 3572258