Related papers: The test function conjecture for parahoric local m…
Kottwitz's conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze's more…
This is a report on results and methods in the reduction modulo p of Shimura varieties with parahoric level structure. In the first part, the local theory, we explain the concepts of parahoric subgroups, of the mu-admissible and…
We determine the Galois representations inside the $l$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to…
In this paper, we consider the geometric special fibers of local models of Shimura varieties and of moduli of $\bG$-Shtukas with parahoric level structure. We investigate two problems with respect to the irreducibility of local models.…
In this article, we prove that a version of Tate conjectures for certain Deligne-Lusztig varieties implies the Kudla-Rapoport conjecture for unitary Shimura varieties with maximal parahoric level at unramified primes. Furthermore, we prove…
We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then prove that associated to each torsion…
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of…
We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the…
We study localized versions of the spectral action of Fargues--Scholze, using methods from higher algebra. As our main motivation and application, we deduce a formula for the cohomology of moduli spaces of local shtukas under certain…
We establish basic results on p-adic shtukas and apply them to the theory of local and global Shimura varieties, and on their interrelation. We construct canonical integral models for (local, and global) Shimura varieties of Hodge type with…
We show how to characterize integral models of Shimura varieties over places of the reflex field where the level subgroup is parahoric by formulating a definition of a "canonical" integral model. We then prove that in Hodge type cases and…
We prove the Pappas-Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show,…
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…
A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to…
We give new proofs of two functional relations for the alternating analogues of Tornheim's double zeta function. Using the functional relations, we give new proofs of some evaluation formulas found by H. Tsumura for these alternating…
We study the local factor at p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at p by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the…
We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity. We show that these conditions are directly related to important conjectures in number theory coming from…
In this note, we discuss several forms of the Eichler-Shimura relation for the compactly supported cohomology of local Shimura varieties, using the work of Fargues-Scholze.
We study the conditions imposed by conjectures of Arthur and Kottwitz on the Galois representations occurring in the cohomology of Shimura varieties.
We define etale cohomology of the moduli spaces of mixed characteristic local shtukas so that it gives smooth representations including the case where the relevant elements of the Kottwitz set are both non-basic. Then we relate the etale…