arXiv subjects
Publications and source records attributed to Toby Gee.
We prove a natural analogue of the Sato-Tate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place.
We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2<k<p, which is totally ordinary at p and has tamely ramified Galois representation at all primes dividing p, then there is a "companion form" of parallel weight k':=p+1-k. This work generalises results of Gross and Coleman-Voloch for modular forms over Q.
We prove a companion forms theorem for mod l Hilbert modular forms. This work generalises results of Gross and Coleman--Voloch for modular forms over Q, and gives a new proof of their results in many cases. The methods used are completely different to previous work in this area, and rely on modularity lifting theorems and the general theory of deformations of Galois representations.
We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod $p$ Hilbert modular forms, by making use of modularity lifting theorems and computations in $p$-adic Hodge theory.
We study the possible weights of an irreducible two-dimensional mod p representation of the absolute Galois group of F which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.
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.
We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.
We prove a modularity lifting theorem for potentially Barostti-Tate representations over totally real fields, generalising recent results of Kisin. Unfortunately, there was an error in the original version of this paper, meaning that we can only obtain a slightly weaker result in the case where the representations are potentially ordinary; an erratum has been added explaining this error.