Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-09T07:24:20.986Z Has data issue: false hasContentIssue false
  • English
  • Français

LIFTING N-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 June 2018

JAYANTA MANOHARMAYUM
JAYANTA MANOHARMAYUM*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, United Kingdom e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let F be a number field, let N ≥ 3 be an integer, and let k be a finite field of characteristic ℓ. We show that if ρ:GF → GLN(k) is a continuous representation with image of ρ containing SLN(k) then, under moderate conditions at primes dividing ℓ∞, there is a continuous representation ρ:GF → GLN(W(k)) unramified outside finitely many primes with ρ ~ρ mod ℓ. Stronger results are presented for ρ:Gℚ → GL3(k).

MSC classification

Primary: 11F80: Galois representations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 1 , January 2019 , pp. 115 - 150
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Böckle, G., A local-to-global principle for deformations of Galois representations, J. Reine Angew. Math. 509 (1999), 199–236.Google Scholar
2. Böckle, G., Presentations of universal deformation rings, in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 24–58.Google Scholar
3. Cline, E., Parshall, B. and Scott, L., Cohomology of finite groups of Lie type. I, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 169–191.Google Scholar
4. Clozel, L., Harris, M. and Taylor, R., Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.Google Scholar
5. Curtis, C. W. and Reiner, I., Methods of representation theory, Vol. I (John Wiley & Sons Inc., New York, 1981).Google Scholar
6. Ellenberg, J. S., Serre's conjecture over , Ann. Math. (2) 161 (3) (2005), 1111–1142.Google Scholar
7. Hamblen, S., Lifting n-dimensional Galois representations, Can. J. Math. 60 (5) (2008), 1028–1049.Google Scholar
8. Khare, C. and Wintenberger, J.-P., On Serre's conjecture for 2-dimensional mod p representations of Gal(ℚ/ℚ), Ann. of Math. (2) 169 (1) (2009), 229–253.Google Scholar
9. Manoharmayum, J., On the modularity of certain GL2() Galois representations, Math. Res. Lett. 8 (5–6) (2001), 703–712.Google Scholar
10. Mazur, B., An introduction to the deformation theory of Galois representations, in Modular forms and Fermat's last theorem (Boston, MA, 1995) (Springer, New York, 1997), 243–311.Google Scholar
11. Mazur, B., Deforming Galois representations, in Galois groups over Q (Berkeley, CA, 1987), Mathematical Sciences Research Institute Publications, vol. 16 (Springer, New York, 1989), 385–437.Google Scholar
12. Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of number fields, second ed., Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 323 (Springer-Verlag, Berlin, 2008).Google Scholar
13. Ramakrishna, R., Lifting Galois representations, Invent. Math. 138 (3) (1999), 537–562.Google Scholar
14. Schlessinger, M., Functors of Artin rings, Trans. Am. Math. Soc. 130 (1968), 208–222.Google Scholar
15. Taylor, R., On icosahedral Artin representations. II, Am. J. Math. 125 (3) (2003), 549–566.Google Scholar
16. Upton, M. G., Galois representations attached to Picard curves, J. Algebra 322 (4) (2009), 1038–1059.Google Scholar