We construct smooth integral models of Shimura varieties of abelian type at primes where the level structure is hyperspecial.

We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of ‫ޑ‬ 3 . The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations.

We prove the Breuil-Mézard conjecture for 2-dimensional potentially Barsotti-Tate representations of the absolute Galois group G K , K a finite extension of Qp, for any p > 2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Qp is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre's conjecture, proving a variety of results including the Buzzard-Diamond-Jarvis conjecture.

No abstract