21 research outputs found
The Sato-Tate conjecture for Hilbert modular forms
We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely,
we prove the natural generalisation of the Sato-Tate conjecture for regular
algebraic cuspidal automorphic representations of \GL_2(\A_F), a totally
real field, which are not of CM type. The argument is based on the potential
automorphy techniques developed by Taylor et. al., but makes use of automorphy
lifting theorems over ramified fields, together with a 'topological' argument
with local deformation rings. In particular, we give a new proof of the
conjecture for modular forms, which does not make use of potential automorphy
theorems for non-ordinary -dimensional Galois representations.Comment: 59 pages. Essentially final version, to appear in Journal of the AMS.
This version does not incorporate any minor changes (e.g. typographical
changes) made in proo
Congruences between Hilbert modular forms: constructing ordinary lifts
Under mild hypotheses, we prove that if F is a totally real field, k is the
algebraic closure of the finite field with l elements and r : G_F --> GL_2(k)
is irreducible and modular, then there is a finite solvable totally real
extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each
place dividing l. We deduce a similar result for r itself, under the assumption
that at places v|l the representation r|_{G_F_v} is reducible. This allows us
to deduce improvements to results in the literature on modularity lifting
theorems for potentially Barsotti-Tate representations and the
Buzzard-Diamond-Jarvis conjecture. The proof makes use of a novel lifting
technique, going via rank 4 unitary groups.Comment: 48 page
Local-global compatibility for l=p, II
We prove the compatibility at places dividing l of the local and global
Langlands correspondences for the l-adic Galois representations associated to
regular algebraic essentially (conjugate) self-dual cuspidal automorphic
representations of GL_n over an imaginary CM or totally real field. We prove
this compatibility up to semisimplification in all cases, and up to Frobenius
semisimplification in the case of Shin-regular weight.Comment: 13 page
Serre weights for rank two unitary groups
We study the weight part of (a generalisation of) Serre's conjecture for mod
l Galois representations associated to automorphic representations on rank two
unitary groups for odd primes l. We propose a conjectural set of Serre weights,
agreeing with all conjectures in the literature, and under a mild assumption on
the image of the mod l Galois representation we are able to show that any
modular representation is modular of each conjectured weight. We make no
assumptions on the ramification or inertial degrees of l. Our main innovation
is to make use of the lifting techniques introduced in our recent papers.Comment: 43 page
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Local-global compatibility for l=p, I
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.Mathematic