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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

  • Jack Thorne (a1)

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.

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1.Barnet-Lamb T., Gee T. and Geraghty D., The Sato–Tate conjecture for Hilbert modular forms, J. Am. Math. Soc. 24 (2011), 411469.
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), 2998.
3.Barnet-Lamb T., Gee T., Geraghty D. and Taylor R., Potential automorphy and change of weight, preprint.
4.Bernstein I. N. and Zelevinsky A. V., Induced representations of reductive p-adic groups, I, Annales Scient. Éc. Norm. Sup. 10(4) (1977), 441472.
5.Björner A. and Brenti F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Volume 231 (Springer, 2005).
6.Borel A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer, 1991).
7.Carter R. W., Finite groups of Lie type: conjugacy classes and complex characters, Wiley Classics Library (Wiley, 1993; reprint of the 1985 original).
8.Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, preprint.
9.Chenevier G., Une application des variétés de Hecke des groupes unitaires, preprint.
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Taylor R., Automorphy for some l-adic lifts of automorphic mod l Galois representations (with Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by M.-F. Vignéras, Publ.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), 1181.
11.Curtis C. W. and Reiner I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Volume 11 (Interscience, 1962).
12.Gee T., Automorphic lifts of prescribed types, Math. Annalen 350(1) (2011), 107144.
13.Gee T. and Geraghty D., Companion forms for unitary and symplectic groups, Duke Math. J., in press.
14.Geraghty D., Modularity lifting theorems for ordinary Galois representations, preprint.
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).
16.Gross B. H., Algebraic modular forms, Israel J. Math. 113 (1999), 6193.
17.Guerberoff L., Modularity lifting theorems for Galois representations of unitary type. Compositio Math. 147(4) (2011), 10221058.
18.Guralnick R. M., Small representations are completely reducible, J. Alg. 220(2) (1999), 531541.
19.Humphreys J. E., Modular representations of finite groups of Lie type, London Mathematical Society Lecture Notes Series, Volume 326 (Cambridge University Press, 2006).
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. 111116 (Springer, 1982).
21.Jantzen J. C., Representations of algebraic groups, 2nd edn, Mathematical Surveys and Monographs, Volume 107 (American Mathematical Society, Providence, RI, 2003).
22.Kisin M., Potentially semi-stable deformation rings, J. Am. Math. Soc. 21(2) (2008), 513546.
23.Kuperberg G., Denseness and Zariski denseness of Jones braid representations, Geom. Topol. 15 (2011), 1139.
24.Labesse J.-P., Changement de base CM et sèries discrétes, preprint.
25.Lansky J. M., Parahoric fixed spaces in unramified principal series representations, Pac. J. Math. 204(2) (2002), 433443.
26.Larsen M. and Pink R., Finite subgroups of algebraic groups, J. Am. Math. Soc. 24(4) (2011), 11051158.
27.Magnus W., On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7 (1954), 649673.
28.Mann W. R., Local level raising for GLn, PhD thesis, Harvard University (2001).
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).
30.Milne J. S., Algebraic groups, Lie groups, and their arithmetic subgroups (unpublished notes available at
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. 159205 (Abdus Salam International Centre for Theoretical Physics, Trieste, 2008).
32.Serre J.-P., Lie algebras and Lie groups, 2nd edn, Lecture Notes in Mathematics, Volume 1500 (Springer, 1992).
33.Serre J.-P., Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math. 116 (1994), 513530. (13)
34.Shalika J. A., The multiplicity one theorem for GLn, Annals Math. 100 (1974), 171193.
35.Shin S. W., Galois representations arising from some compact Shimura varieties, Annals Math. 173 (2011), 16451741.
36.Springer T. A., Twisted conjugacy in simply connected groups, Transform. Groups 11(3) (2006), 539545.
37.Springer T. A., Linear algebraic groups, 2nd edn, Modern Birkhäuser Classics (Birkhäuser, 2009).
38.Steinberg R., Automorphisms of classical Lie algebras, Pac. J. Math. 11 (1961), 11191129.
39.Steinberg R., Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, Number 80 (American Mathematical Society, Providence, RI, 1968).
40.Steinberg R., Lectures on Chevalley groups(notes prepared by J. Faulkner and R. Wilson) (Yale University Press, New Haven, CT, 1968).
41.Taylor R., Automorphy for some l-adic lifts of automorphic mod l Galois representations, II, Publ. Math. IHES 108 (2008), 183239.
42.Vignéras M.-F., Représentations l-modulaires d'un groupe réductif p-adique avec l ≠ p, Progress in Mathematics, Volume 137 (Birkhäuser, 1996).
43.Vignéras M.-F., Induced R-representations of p-adic reductive groups, Selecta Math. 4(4) (1998), 549623.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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