Preprints

Functoriality and the Inverse Galois problem (joint with Michael Larsen and Gordan Savin)

pdf file We prove that given an even integer n, a prime p, and an integer t, there is a positive integer k divisible by t, such that PSp_n(Fpk) appears as a Galois group over Q. This generalises a result of Gabor Wiese which inspired our work. (to appear in Compositio Mathematica)

Modularity of Galois representations and motives with good reduction properties

pdf file These are some musings on motives, automorphic forms and Galois representations suggested by the work below on Serre's conjecture. The material is close to the slides from various talks I gave in Boston and Paris, April--June 2006 on Serre's conjecture which are available at http://www.math.harvard.edu/ev/documents.html. Errata: On page 86, l. 5 and l. 6, delete the phrase up to semisimiplification''. (appeared in Journal of Ramanujan Math. Soc.)

Serre's modularity conjecture (II) (joint with Wintenberger)

pdf file This provides proofs of the technical results, Theorems 4.1 and 5.1, of the paper below.

Serre's modularity conjecture (I) (joint with Wintenberger)

pdf file This is the first part of a work that proves Serre's conjecture for odd conductors.

Serre's modularity conjecture: a survey of the level one case

dvi file, ps file This is a survey paper: to appear in Proceedings of the LMS Durham conference.

Serre's modularity conjecture: the level one case

dvi file, ps file This paper proves the level one case of Serre's conjecture: to appear in Duke Math J.

On Serre's conjecture for 2-dimensional mod p representations of the absolute Galois group of the rationals (joint with Jean-Pierre Wintenberger)

pdf file to appear in Annals of Math.

Converging sequences of p-adic Galois representations and density theorems (joint with J. Bellaiche, G. Chenevier and Michael Larsen)

to appear in IMRN

Transcendental \ell-adic Galois representations (joint with Michael Larsen and Ravi Ramakrishna)

to appear in Mathematical Research Letters

Constructing semisimple p-adic Galois representations with prescribed properties (joint with Michael Larsen and Ravi Ramakrishna)

(to appear in American J of Mathematics).

Mod \ell representations of arithmetic fundamental groups: I and II(joint with Gebhard Boeckle)

Part I (to appear in Duke Math J): dvi file ps file

Part II (to appear in Compositio): dvi file ps file

Finiteness results for mod $\ell$ Galois representations over function fields (joint with Gebhard Boeckle)

Reciprocity law for compatible systems of abelian mod p Galois representations

(to appear in Canadian Journal of Mathematics).

(to appear in Journal de Theorie des Nombres Bordeaux). Errata: In Theorems 1 and 2 we must assume that the minimal field of definition of the adjoint of the residual representation is k (this is needed for Lemma 1).

Belyi parametrisations of elliptic curves and congruence defects

to appear in proceedings of CNTA

Reduction of homomorphisms mod p and algebraicity (joint with Dipendra Prasad)

(to appear in Journal of Number Theory).

(to appear in Journal of Number Theory).

Compatible systems of mod p Galois representations and Hecke characters

(to appear in Mathematical Research Letters). Errata: In Defintion 1, in the second bulleted item the phrase for each place r of K not in S'' should occur at the beginning of the item

Hasse invariant and group cohomology (joint with Bas Edixhoven)

(to appear in Documenta Mathematica).

On isomorphisms between deformation rings and Hecke rings (with appendix by G. Boeckle)

(to appear in Inventiones mathematicae).

Finiteness of Selmer groups and deformation rings (joint with Ravi Ramakrishna)

(to appear in Inventiones mathematicae).

Limits of residually irreducible p-adic Galois representations

(to appear in Proceedings of the American Math. Soc.).

F-split Galois representations are potentially abelian

(to appear in Proceedings of the American Math. Soc.).