Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T00:28:36.814Z Has data issue: false hasContentIssue false
  • English
  • Français

On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  05 April 2012

Jack Thorne
Jack Thorne
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor–Wiles method which allows one to weaken this hypothesis.

Keywords

Galois representationsGalois cohomologyrepresentation theory of p-adic groups

MSC classification

Secondary: 11F80: Galois representations 11R34: Galois cohomology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 4 , October 2012 , pp. 855 - 920
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Barnet-Lamb, T., Gee, T. and Geraghty, D., The Sato–Tate conjecture for Hilbert modular forms, J. Am. Math. Soc. 24 (2011), 411–469.CrossRefGoogle Scholar
2.Barnet-Lamb, T., Geraghty, D., Harris, M. and Taylor, R., A family of Calabi–Yau varieties and potential automorphy, II, Publ. RIMS Kyoto 47 (2011), 29–98.CrossRefGoogle Scholar
3.Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., Potential automorphy and change of weight, preprint.Google Scholar
4.Bernstein, I. N. and Zelevinsky, A. V., Induced representations of reductive p-adic groups, I, Annales Scient. Éc. Norm. Sup. 10(4) (1977), 441–472.CrossRefGoogle Scholar
5.Björner, A. and Brenti, F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Volume 231 (Springer, 2005).Google Scholar
6.Borel, A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer, 1991).CrossRefGoogle Scholar
7.Carter, R. W., Finite groups of Lie type: conjugacy classes and complex characters, Wiley Classics Library (Wiley, 1993; reprint of the 1985 original).Google Scholar
8.Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups, preprint.Google Scholar
9.Chenevier, G., Une application des variétés de Hecke des groupes unitaires, preprint.Google Scholar
10.Clozel, L., Harris, M. and Taylor, R., Automorphy for some l-adic lifts of automorphic mod l Galois representations (with Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by M.-F. Vignéras, Publ. Math. IHES 108 (2008), 1–181.CrossRefGoogle Scholar
11.Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Volume 11 (Interscience, 1962).Google Scholar
12.Gee, T., Automorphic lifts of prescribed types, Math. Annalen 350(1) (2011), 107–144.CrossRefGoogle Scholar
13.Gee, T. and Geraghty, D., Companion forms for unitary and symplectic groups, Duke Math. J., in press.Google Scholar
14.Geraghty, D., Modularity lifting theorems for ordinary Galois representations, preprint.Google Scholar
15.Gorenstein, D., Lyons, R. and Solomon, R., The classification of the finite simple groups, Number 3, Part I, Chapter A, Mathematical Surveys and Monographs, Volume 40 (American Mathematical Society, Providence, RI, 1998).Google Scholar
16.Gross, B. H., Algebraic modular forms, Israel J. Math. 113 (1999), 61–93.CrossRefGoogle Scholar
17.Guerberoff, L., Modularity lifting theorems for Galois representations of unitary type. Compositio Math. 147(4) (2011), 1022–1058.CrossRefGoogle Scholar
18.Guralnick, R. M., Small representations are completely reducible, J. Alg. 220(2) (1999), 531–541.CrossRefGoogle Scholar
19.Humphreys, J. E., Modular representations of finite groups of Lie type, London Mathematical Society Lecture Notes Series, Volume 326 (Cambridge University Press, 2006).Google Scholar
20.Hurley, J. F., A note on the centers of Lie algebras of classical type, in Lie Algebras and Related Topics, New Brunswick, NJ, 1981, Lecture Notes in Mathematics, Volume 933, pp. 111–116 (Springer, 1982).Google Scholar
21.Jantzen, J. C., Representations of algebraic groups, 2nd edn, Mathematical Surveys and Monographs, Volume 107 (American Mathematical Society, Providence, RI, 2003).Google Scholar
22.Kisin, M., Potentially semi-stable deformation rings, J. Am. Math. Soc. 21(2) (2008), 513–546.CrossRefGoogle Scholar
23.Kuperberg, G., Denseness and Zariski denseness of Jones braid representations, Geom. Topol. 15 (2011), 11–39.Google Scholar
24.Labesse, J.-P., Changement de base CM et sèries discrétes, preprint.Google Scholar
25.Lansky, J. M., Parahoric fixed spaces in unramified principal series representations, Pac. J. Math. 204(2) (2002), 433–443.CrossRefGoogle Scholar
26.Larsen, M. and Pink, R., Finite subgroups of algebraic groups, J. Am. Math. Soc. 24(4) (2011), 1105–1158.CrossRefGoogle Scholar
27.Magnus, W., On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7 (1954), 649–673.CrossRefGoogle Scholar
28.Mann, W. R., Local level raising for GLn, PhD thesis, Harvard University (2001).Google Scholar
29.Matsumura, H., Commutative ring theory (transl. from Japanese by M. Reid), 2nd edn, Cambridge Studies in Advanced Mathematics, Volume 8 (Cambridge University Press, 1989).Google Scholar
30.Milne, J. S., Algebraic groups, Lie groups, and their arithmetic subgroups (unpublished notes available at www.jmilne.org).Google Scholar
31.Prasad, D. and Raghuram, A., Representation theory of GL(n) over non-Archimedean local fields, in School on automorphic forms on GL(n), ICTP Lecture Notes, Volume 21, pp. 159–205 (Abdus Salam International Centre for Theoretical Physics, Trieste, 2008).Google Scholar
32.Serre, J.-P., Lie algebras and Lie groups, 2nd edn, Lecture Notes in Mathematics, Volume 1500 (Springer, 1992).CrossRefGoogle Scholar
33.Serre, J.-P., Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math. 116 (1994), 513–530. (1–3)CrossRefGoogle Scholar
34.Shalika, J. A., The multiplicity one theorem for GLn, Annals Math. 100 (1974), 171–193.CrossRefGoogle Scholar
35.Shin, S. W., Galois representations arising from some compact Shimura varieties, Annals Math. 173 (2011), 1645–1741.CrossRefGoogle Scholar
36.Springer, T. A., Twisted conjugacy in simply connected groups, Transform. Groups 11(3) (2006), 539–545.CrossRefGoogle Scholar
37.Springer, T. A., Linear algebraic groups, 2nd edn, Modern Birkhäuser Classics (Birkhäuser, 2009).Google Scholar
38.Steinberg, R., Automorphisms of classical Lie algebras, Pac. J. Math. 11 (1961), 1119–1129.CrossRefGoogle Scholar
39.Steinberg, R., Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, Number 80 (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar
40.Steinberg, R., Lectures on Chevalley groups(notes prepared by J. Faulkner and R. Wilson) (Yale University Press, New Haven, CT, 1968).Google Scholar
41.Taylor, R., Automorphy for some l-adic lifts of automorphic mod l Galois representations, II, Publ. Math. IHES 108 (2008), 183–239.CrossRefGoogle Scholar
42.Vignéras, M.-F., Représentations l-modulaires d'un groupe réductif p-adique avec l ≠ p, Progress in Mathematics, Volume 137 (Birkhäuser, 1996).Google Scholar
43.Vignéras, M.-F., Induced R-representations of p-adic reductive groups, Selecta Math. 4(4) (1998), 549–623.CrossRefGoogle Scholar