Towards a theory of local Shimura varieties

Abstract

This is a survey article that advertizes 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 $\ell$-adic cohomology of local Shimura varieties.

53 pages

References in corpus (8)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• A guide to the reduction modulo p of Shimura varieties
• On parahoric subgroups
• Basic loci in Shimura varieties of Coxeter type
• B(G) for all local and global fields
• $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties
• Lefschetz trace formula and l-adic cohomology of Rapoport-Zink tower for GSp(4)
• Lefschetz trace formula and l-adic cohomology of Lubin-Tate tower

Cited by in corpus (33)

• Rapoport-Zink spaces for spinor groups
• On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
• Coherent sheaves on the stack of Langlands parameters
• Fully Hodge-Newton decomposable Shimura varieties
• Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties
• The supersingular locus of unitary Shimura varieties with exotic good reduction
• On the Hodge-Newton filtration for p-divisible groups of Hodge type
• Théorie de la réduction pour les groupes p-divisibles
• Serre-Tate theory for Shimura varieties of Hodge type
• On the p-adic cohomology of the Lubin-Tate tower
• Rapoport-Zink Spaces For Local P-Shtukas and Their Local Models
• Good and semi-stable reductions of Shimura varieties
• On a decomposition of $p$-adic Coxeter orbits
• Central leaves on Shimura varieties with parahoric reduction
• On quaternionic unitary Rapoport-Zink spaces with parahoric level structures
• Eichler-Shimura Relations for Shimura Varieties of Hodge Type
• The Cohomology of Unramified Rapoport-Zink Spaces of EL-type and Harris's Conjecture
• Construction of a Rapoport-Zink space for $\mathrm{GU}(1,1)$ in the ramified $2$-adic case
• Notes on Rapoport--Zink spaces of Hodge type with parahoric level structure
• On integral local Shimura varieties
• On the Drinfeld moduli problem of p-divisible groups
• Deligne-Lusztig varieties and basic EKOR strata
• An averaging formula for the cohomology of PEL-type Rapoport--Zink spaces
• Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure
• On the generic part of the cohomology of local and global Shimura varieties
• On the supersingular locus of the Shimura variety for $\mathrm{GU}(2,2)$ over a ramified prime
• p-adic etale cohomology of period domains
• Convolution morphisms and Kottwitz conjecture
• On central leaves of Hodge-type Shimura varieties with parahoric level structure
• Stratifications in the reduction of Shimura varieties
• $l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients
• Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)
• Eichler-Shimura relations for local Shimura varieties