Toby Gee

Automorphic Forms and Galois Representations: The conjectural connections between automorphic representations and Galois representations

We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.

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 GL2(AF ), 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 n-dimensional Galois representations.

The Buzzard-Diamond-Jarvis conjecture for unitary groups

Let p > 2 be prime. We prove the weight part of Serre’s conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard–Diamond–Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. Our methods are purely local, using the theory of (φ, Ĝ)-modules to determine the possible reductions of certain two-dimensional crystalline representations.

Companion forms for unitary and symplectic 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.

Serre weights for mod p Hilbert modular forms: the totally ramified case

Abstract We study the possible weights of an irreducible 2-dimensional modular mod p representation of Gal(/F), where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. 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.

Slopes of Modular Forms

We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in Buzzard (Asterisque 298:1–15, 2005), discuss strategies for making further progress, and examine other related questions.

Explicit reduction modulo p of certain 2-dimensional crystalline representations, II

We complete the calculations begun in [BG09], using the p-adic local Langlands correspondence for GL2(Q_p) to give a complete description of the reduction modulo p of the 2-dimensional crystalline representations of G_{Q_p} of slope less than 1, when p > 2.

Companion forms over totally real fields, II

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.

p-adic Hodge-theoretic properties of étale cohomology with mod p coefficients, and the cohomology of Shimura varieties

We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation with Hodge-Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serres conjecture.