Person:

Kisin, Mark

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.

Inside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-M´ezard predicts that part of the structure of this space can be described in terms of the local Langlands correspondence. For 2-dimensional representations the conjecture can be made precise. We explain some of the progress in this case, which reveals that the conjecture is intimately connected to the p-adic local Langlands correspondence, as well as to the Fontaine-Mazur conjecture.

We show that the mod p points on a Shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.