对称簇的整点与Satake相容性

具正特征的Kuga-Satake阿贝尔簇

Kuga-Satake Abelian Varieties in Positive Characteristic

Kuga和Satake将每个极化复K3曲面（X，L）与一个复阿贝尔簇相关联，称为（X，L）的Kuga-Satake阿贝尔簇。我们使用这个构造来定义具有某些能级结构的极化K3曲面的模空间与C上具有水平结构的极化阿贝尔簇的模空间之间的态射。本文研究了这些态射。我们首先证明它们定义在Q的有限扩张上。然后我们证明了它们的正特性。用这种方法我们给出了任意基上Kuga-Satake阿贝尔簇的一个间接构造。我们也给出了一些应用程序，这种建设的正则升降机的普通K3曲面。

Geometric Satake, categorical traces, and arithmetic of Shimura varieties

We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.

Cycles on Shimura varieties via geometric Satake

We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set…

任意秩的Drinfeld模簇和Drinfeld模形式的紧化

Compactification of Drinfeld modular varieties and Drinfeld Modular Forms of Arbitrary Rank

给出了一般Drinfeld模簇的Satake紧化的一个抽象刻画.我们证明了它的存在，是唯一的唯一同构，虽然我们没有给一个明确的分层Drinfeld模品种的秩较小，这也是预期的。我们构造了一个自然的充分可逆层，使得它的$k$次幂的整体瓣构成权$k$的（代数）Drinfeld模形式空间.我们展示了如何佐竹紧化和模形式的行为下的所有自然态射之间的德林费尔德模品种，特别是我们定义Hecke运营商。在某些特殊情况下，我们给出了明确的结果。

On the cohomology of certain non-compact Shimura varieties (with an appendix by Robert Kottwitz)

The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura varieties, to stabilize the result for Shimura varieties associated to unitary groups over $\mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace d'une correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne …

Inverse Satake transforms

Let H be a split reductive group over a local non-archimedean field, and let H^ denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of "Satake transform" and perform the analogous calculation for a large class of spherical varieties.