• Corpus ID: 25848547

# Towards a theory of local Shimura varieties

@article{Rapoport2014TowardsAT,
title={Towards a theory of local Shimura varieties},
author={Michael Rapoport and Eva Viehmann},
journal={arXiv: Algebraic Geometry},
year={2014}
}
• Published 13 January 2014
• Mathematics
• arXiv: Algebraic Geometry
This is a survey article that advertises the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also review their theory in the light of this idea. We also discuss conjectures on the l-adic cohomology of local Shimura varieties.
The supersingular locus of unitary Shimura varieties with exotic good reduction
In this paper, we use a group-theoretic approach to give a concrete description of the geometric structure of the supersingular locus of unitary Shimura varieties with exotic good reduction. This
On the generic part of the cohomology of local and global Shimura varieties
Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative
Serre–Tate theory for Shimura varieties of Hodge type
• Mathematics
• 2016
We study the formal neighbourhood of a point in $\mu$-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a shifted cascade.
ON THE SUPERCUSPIDAL COHOMOLOGY OF BASIC LOCAL SHIMURA VARIETIES
We prove that the supercuspidal cohomology of basic local Shimura varieties is concentrated in the middle degree, under a mild (and probably necessary) condition related to the local Langlands
Zelevinsky Duality on Basic Local Shimura Varieties
. We give a simple proof of a general result describing the action of the Zelevinsky involution on the cohomology of certain basic local Shimura varieties, using the machinery of Fargues-Scholze
Good and semi-stable reductions of Shimura varieties
• Mathematics
Journal de l’École polytechnique — Mathématiques
• 2020
We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp.
Eichler-Shimura relations for local Shimura varieties
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.
Rapoport–Zink spaces for spinor groups
• Mathematics
Compositio Mathematica
• 2017
After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge
Honda–Tate theory for Shimura varieties
• Mathematics
Duke Mathematical Journal
• 2022
A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are non-empty provided the
On the cohomology of Rapoport-Zink spaces of Hodge type
We prove the conjecture of Harris on the cohomology of Rapoport-Zink spaces for unramified reductive groups, under the "Hodge-Newton decomposibility" assumption. Our proof is based upon Mantovan's

## References

SHOWING 1-10 OF 105 REFERENCES
Local models of Shimura varieties, I. Geometry and combinatorics
• Mathematics
• 2010
We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and
On the flatness of models of certain Shimura varieties of PEL-type
Abstract. Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of $\Q_p$. Rapoport and Zink have defined a model of the Shimura variety over the ring
On irreducible components of Rapoport-Zink spaces
Under a mild condition, we prove that the action of the group of self-quasi-isogenies on the set of irreducible components of a Rapoport-Zink space has finite orbits. Our method allows both ramified
Local models in the ramified case. III Unitary groups
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2009
Abstract We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p-adic
Basic loci in Shimura varieties of Coxeter type
• Mathematics
• 2013
This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and
A guide to the reduction modulo p of Shimura varieties
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
STRATIFICATIONS OF HILBERT MODULAR VARIETIES
We consider g-dimensional abelian varieties in characteristic p, with a given action of OL the ring of integers of a totally real field L of degree g. A stratification of the associated moduli spaces
Local models of Shimura varieties and a conjecture of Kottwitz
• Mathematics
• 2011
We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to
Étale Cohomology of Rigid Analytic Varieties and Adic Spaces
Summary of the results on the etale cohomology of rigid analytic varieties - Adic spaces - The etale site of a rigid analytic variety and an adic space - Comparison theorems - Base change theorems -