On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime $$p\ge 5$$ p ≥ 5 . Consider its perfectoid cover $$S^{\text {ad}}(p^\infty )$$ S ad ( p ∞ ) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to $$S^{\text {ad}}(p^\infty )$$ S ad ( p ∞ ) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to $$S^\text {ad}(p^\infty )$$ S ad ( p ∞ ) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.

On the geometric André–Oort conjecture for variations of Hodge structures

Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

Mixed Ax–Schanuel for the universal abelian varieties and some applications

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

The GL4 Rapoport–Zink Space

We give a description of the $\textrm{GL}_4$ Rapoport–Zink space, including the connected components, irreducible components, intersection behavior of the irreducible components, and Ekedahl–Oort stratification. As an application of this, we also give a description of the supersingular locus of the Shimura variety for the group $\textrm{GU}(2,2)$ over a prime split in the relevant imaginary quadratic field.

Vanishing theorems for the mod p cohomology of some simple Shimura varieties

We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

Effective estimates for the degrees of maximal special subvarieties

Let Z be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvariety of Z.

GALOIS CONJUGATES OF SPECIAL POINTS AND SPECIAL SUBVARIETIES IN SHIMURA VARIETIES

Let $S$ be a Shimura variety with reflex field $E$ . We prove that the action of $\text{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

Local Shimura Varieties: Minicourse Given by Peter Scholze

This chapter discusses Peter Scholze's minicourse on local Shimura varieties. The goal of these lectures is to describe a program to construct local Langlands correspondence. The construction is based on cohomology of so-called local Shimura varieties and generalizations thereof. It was predicted by Robert Kottwitz that for each local Shimura datum, there exists a so-called local Shimura variety, which is a pro-object in the category of rigid analytic spaces. Thus, local Shimura varieties are determined by a purely group-theoretic datum without any underlying deformation problem. This is now an unpublished theorem, by the work of Fargues, Kedlaya–Liu, and Caraiani–Scholze. The chapter then explains the approach to local Langlands correspondence via cohomology of Lubin–Tate spaces as well as Rapoport–Zink spaces. It also introduces a formal deformation problem and describes properties of the corresponding universal deformation formal scheme.

Norm-compatible systems of cohomology classes for GU(2,2)

We describe work of Faltings on the construction of étale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two-variable family of trace compatible classes in the cohomology of a unitary Shimura variety.