Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An irreducibility criterion for group representations, with arithmetic applications


Authors: Matteo Longo and Stefano Vigni
Journal: Proc. Amer. Math. Soc. 138 (2010), 3437-3447
MSC (2010): Primary 20C12, 11F80
DOI: https://doi.org/10.1090/S0002-9939-2010-10485-9
Published electronically: May 17, 2010
MathSciNet review: 2661544
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a criterion for the irreducibility of an integral group representation $ \rho$ over the fraction field of a Noetherian domain $ R$ in terms of suitably defined reductions of $ \rho$ at prime ideals of $ R$. As applications, we give irreducibility results for universal deformations of residual representations, with special attention to universal deformations of residual Galois representations associated with modular forms of weight at least $ 2$.


References [Enhancements On Off] (What's this?)

  • 1. M. F. Atiyah, I. G. MacDonald, Introduction to commutative algebra, Addison-Wesley Publishing Company, London, 1969. MR 0242802 (39:4129)
  • 2. N. Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley, Reading, MA, 1972. MR 0360549 (50:12997)
  • 3. I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), no. 1, 54-106. MR 0016094 (7:509h)
  • 4. C. W . Curtis, I. Reiner, Methods of representation theory. Volume I, John Wiley & Sons, Inc., New York, 1981. MR 632548 (82i:20001)
  • 5. F. Q. Gouvêa, Deformations of Galois representations, in Arithmetic algebraic geometry, B. Conrad and K. Rubin (eds.), IAS/Park City Math. Ser. 9, American Mathematical Society, Providence, RI, 2001, 233-406. MR 1860043 (2003a:11061)
  • 6. B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. 3 (1990), no. 4, 929-960. MR 1071117 (92a:11077)
  • 7. H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989. MR 1011461 (90i:13001)
  • 8. B. Mazur, Deforming Galois representations, in Galois groups over $ \mathbb{Q}$, Y. Ihara, K. Ribet and J.-P. Serre (eds.), MSRI Publications 16, Springer-Verlag, New York, 1989, 385-437. MR 1012172 (90k:11057)
  • 9. B. Mazur, An introduction to the deformation theory of Galois representations, in Modular forms and Fermat's last theorem, G. Cornell, J. H. Silverman and G. Stevens (eds.), Springer-Verlag, New York, 1997, 243-311. MR 1638481
  • 10. J.-P. Serre, Three letters to Walter Feit on group representations and quaternions, J. Algebra 319 (2008), no. 2, 549-557. MR 2381795 (2008m:20017)
  • 11. T. Weston, Unobstructed modular deformation problems, Amer. J. Math. 126 (2004), no. 6, 1237-1252. MR 2102394 (2006c:11061)
  • 12. T. Weston, Explicit unobstructed primes for modular deformation problems of squarefree level, J. Number Theory 110 (2005), no. 1, 199-218. MR 2114681 (2006g:11105)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20C12, 11F80

Retrieve articles in all journals with MSC (2010): 20C12, 11F80


Additional Information

Matteo Longo
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: mlongo@math.unipd.it

Stefano Vigni
Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain
Email: stefano.vigni@upc.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10485-9
Keywords: Group representations, Noetherian domains, reductions modulo primes
Received by editor(s): December 23, 2009
Published electronically: May 17, 2010
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society