Abstract
We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.
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References
J. Arthur and L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Ann. Math. Stud., vol. 120, Princeton University Press, 1989.
C. Breuil and A. Mezard, Multiplicités modulaires et représentations de GL2(Z p ) et de \(\textup{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)\) en ℓ=p, Duke Math. J., 115 (2002), 205–310.
L. Clozel, M. Harris, and R. Taylor, Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations, this volume.
D. Eisenbud, Commutative Algebra with a View Towards Algebraic Geometry, Springer, 1994.
A. Grothendieck, Eléments de géométrie algébrique. IV. Etude locale des schémas et des morphismes de schémas. III., Publ. Math., Inst. Hautes Étud. Sci., 28 (1966).
M. Harris, N. Shepherd-Barron, and R. Taylor, Ihara’s lemma and potential automorphy, Ann. Math., to appear.
M. Harris and R. Taylor, The Geometry and Cohomology of some Simple Shimura Varieties, Ann. Math. Stud., vol. 151, Princeton University Press, 2001.
M. Kisin, Moduli of finite flat groups schemes and modularity, Ann. Math., to appear.
H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.
C. Skinner and A. Wiles, Base change and a problem of Serre, Duke Math. J., 107 (2001), 15–25.
R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math., 141 (1995), 553–572.
R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., 20 (2007), 467–493.
A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math., 141 (1995), 443–551.
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Taylor, R. Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ.math.IHES 108, 183–239 (2008). https://doi.org/10.1007/s10240-008-0015-2
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DOI: https://doi.org/10.1007/s10240-008-0015-2