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.Comment: 40 page

Geoengineering and Non-Ideal Theory

The strongest arguments for the permissibility of geoengineering (also known as climate engineering) rely implicitly on non-ideal theory—roughly, the theory of justice as applied to situations of partial compliance with principles of ideal justice. In an ideally just world, such arguments acknowledge, humanity should not deploy geoengineering; but in our imperfect world, society may need to complement mitigation and adaptation with geoengineering to reduce injustices associated with anthropogenic climate change. We interpret research proponents’ arguments as an application of a particular branch of non-ideal theory known as “clinical theory.” Clinical theory aims to identify politically feasible institutions or policies that would address existing (or impending) injustice without violating certain kinds of moral permissibility constraints. We argue for three implications of clinical theory: First, conditional on falling costs and feasibility, clinical theory provides strong support for some geoengineering techniques that aim to remove carbon dioxide from the atmosphere. Second, if some kinds of carbon dioxide removal technologies are supported by clinical theory, then clinical theory further supports using those technologies to enable “overshoot” scenarios in which developing countries exceed the cumulative emissions caps that would apply in ideal circumstances. Third, because of tensions between political feasibility and moral permissibility, clinical theory provides only weak support for geoengineering techniques that aim to manage incoming solar radiation

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. This completes the analysis begun in [BLGG11], which proved that all predicted Serre weights occur. Our methods are purely local, using the theory of (phi,Ghat)-modules to determine the possible reductions of certain two-dimensional crystalline representations.Comment: J. Amer. Math. Soc., to appear. Contains minor corrections from published versio

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

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), $F$ 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.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