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Let p > 2 be prime. We complete the proof of the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the totally ramified case, by proving that any weight which occurs is a predicted weight. Our methods are a mixture of local and global techniques, and in the course of the proof we establish some purely local results(More)
We prove the main conjectures of [Bre12] (including a generali-sation from the principal series to the cuspidal case) and [Dem], subject to a mild global hypothesis that we make in order to apply certain R = T theorems. More precisely, we prove a multiplicity one result for the mod p cohomology of a Shimura curve at Iwahori level, and we show that certain(More)
We study the possible weights of an irreducible 2-dimensional mod p representation of Gal(F /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(More)
Motivated by Gauss's first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that(More)
We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Ga-lois representations. To achieve this, we extend results of Breuil and Mézard (classifying Galois lattices in semistable representations in terms of " strongly divisible modules ") to the potentially crystalline case in(More)