Person:

Taylor, Richard L.

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GL n over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight.

If ρ: Gal(Qac/Q) → GL2(C) is a continuous odd irreducible representation with nonsolvable image, then under certain local hypotheses we prove that is the representation associated to a weight 1 modular form and hence that the L-function of
has an analytic continuation to the entire complex plane.

We prove that the L-function of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

We prove the compatibility of local and global Langlands correspondences for (GL_n), which was proved up to semisimplification in M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. More precisely, for the (n)-dimensional (l-)adic representation (R_l(\Pi)) of the Galois group of an imaginary CM-field (L) attached to a conjugate self-dual regular algebraic cuspidal automorphic representation (\Pi) of (GL_n(\mathbb{A}_l)), which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of (R_l(\Pi)) to the decomposition group of a place (v) of (L) not dividing (l) corresponds to (\Pi_v) by the local Langlands correspondence. If (\Pi_v) is square integrable for some finite place (v \not\vert l ) we deduce that (R_l(\Pi)) is irreducible. We also obtain conditional results in the case of (v\vert l).

To the memory of Olga Taussky-Todd

We prove that some new infinite families of odd two dimensional icosahedral representations of the absolute Galois group of Q are modular and hence satsify the Artin conjecture. We also give an account of work of Ramakrishna on lifting mod l Galois representations to characteristic zero.

We extend the methods of Wiles and of Taylor and Wiles from (GL_2) to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, (l)-adic lifts of certain automorphic mod (l) Galois representations of any dimension. We also make a conjecture about the structure of mod (l) automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for (GL_2) . Following Wiles’ method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.