Peter Scholze

Projectivity of the Witt vector affine Grassmannian

We prove that the Witt vector affine Grassmannian, which parametrizes W(k)-lattices in

The Langlands-Kottwitz method and deformation spaces of

W(k)[\frac{1}{p}]^n

-divisible groups

W(k)[1p]n for a perfect field k of characteristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.

Topological Realisations of Absolute Galois Groups

In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in [38], following the stabilization given by Kottwitz in [25]. This introduces endoscopic transfers of the functions φτ,h introduced in [38]. We state a general conjecture relating these endoscopic transfers with Langlands parameters. We verify this conjecture in all cases of EL type, and deduce new results about the endoscopic part of the cohomology of Shimura varieties. This allows us to simplify the construction of Galois representations attached to conjugate self-dual regular algebraic cuspidal automorphic representations of GLn, as previously constructed by one of us, [41].

On torsion in the cohomology of locally symmetric varieties

We extend the results of Kottwitz, [17], on points of Shimura varieties over finite fields to cases of bad reduction. The ”test function” whose twisted orbital integrals appear in the final expression is defined geometrically using deformation spaces of p-divisible groups.

Moduli of

Let

p

F

-divisible groups

be a field of characteristic