# Automorphy for some l-adic lifts of automorphic mod l Galois representations. II

@article{Taylor2008AutomorphyFS, title={Automorphy for some l-adic lifts of automorphic mod l Galois representations. II}, author={Richard Taylor}, journal={Publications math{\'e}matiques}, year={2008}, volume={108}, pages={183-239} }

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

SHOWING 1-10 OF 83 REFERENCES

### Automorphy for some l-adic lifts of automorphic mod l Galois representations

- Mathematics
- 2008

We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate…

### Compatibility of Local and Global Langlands Correspondences

- Mathematics
- 2004

We prove the compatibility of local and global Langlands correspondences for GLn, which was proved up to semisimplification in [HT]. More precisely, for the ndimensional l-adic representation Rl(Π)…

### Types and Hecke algebras for principal series representations of split reductive p-adic groups

- Mathematics
- 1998

### Simple algebras, base change, and the advanced theory of the trace formula

- Mathematics
- 1989

A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups.…

### The Taylor-Wiles construction and multiplicity one

- Mathematics
- 1997

Wiles’ proof [17] of the modularity of semistable elliptic curves over Q relies on a construction of Taylor and Wiles [16] showing that certain Hecke algebras are complete intersections. These Hecke…

### Base change and a problem of Serre

- Mathematics
- 2001

We establish a version of “level-lowering” for mod p Galois representations arising from the reductions of representations associated to Hilbert modular forms. In particular, we show that…

### Induced R-representations of p-adic reductive groups

- Mathematics
- 1998

This article concerns the relation between the non cuspidal irreducible smooth representations of the F -points of a reductive connected algebraic group over F over an R-vector space, and the simple…

### Deformations of Galois representations and Hecke algebras

- Mathematics
- 1996

In Guise of Introduction - GL1 and Class-Field Theory Deformation Theory Deformations of Galois Representations The Universal Ring Functorialities Obstructions Estimates on Dimension Nearly Ordinary…

### Schur algebras of reductive p-adic groups, I

- Mathematics
- 2003

We give a link—through the affine Schur algebra—between the representations of the p-affine Schur algebra of GL(n) over R and the smooth R-representations of the p-adic group GL(n,Qp) over any…