Person:

Kisin, Mark

We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’s conjecture. The main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of ℚ2.

The aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].

Using Lubin-Tate groups, we develop a variant of Fontaine's theory of $(\varphi, \Gamma)$-modules, and we use it to give a description of the Galois stable lattices inside certain crystalline representations.

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 ℚ3. 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.

Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck–Katz p-curvature conjecture for many locally symmetric varieties, including Hilbert–Blumenthal modular varieties and the moduli space of abelian varieties Graphic when g > 1.

We prove new cases of the Fontaine-Mazur conjecture, that a 2 -dimensional p -adic representation rho of G_{{Q}, S} which is potentially semi-stable at p with distinct Hodge-Tate weights arises from a twist of a modular eigenform of weight k>= 2 . Our approach is via the Breuil-Mezard conjecture, which we prove (many cases of) by combining a global argument with recent results of Colmez and Berger-Breuil on the p -adic local Langlands correspondence.

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

We investigate an analogue of the Grothendieck p-curvature conjecture, where the vanishing of the p-curvature is replaced by the stronger condition, that the module with connection mod p underlies a XDX -module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a ℤZ -variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod p Tannakian fundamental groups, which in particular resolves in the projective case a conjecture of Matzat–van der Put.

We show that Colmez's functor from GL2(Qp) representations to GQp representation produces essentially all two dimensional representations of GQp. The method compares the deformation theory for the two kinds of representations: An group calculation of Colmez implies that the deformation space for GL2(Qp) representations is closed in that for GQp-representations. A local version of the Gouvêa-Mazur "infinite fern'' argument shows that this closed subspace is also dense.