[Paper Review] Towards a theory of local Shimura varieties

Michael Rapoport, Eva Viehmann|arXiv (Cornell University)|Jan 13, 2014

Advanced Algebra and Geometry86 references82 citations

TL;DR

This paper advocates for a theory of $p$-adic local Shimura varieties—towers of rigid-analytic spaces associated to local Shimura data $(G,[b],"){\mu\})$—extending the classical theory of global Shimura varieties. It proposes that such spaces, modeled on Rapoport-Zink spaces, carry compatible actions of $G(\mathbb{Q}_p)$, $J(\mathbb{Q}_p)$, and the Weil group $W_E$, with cohomology realizing local Langlands correspondences, and formulates conjectures on $\ell$-adic cohomology, including the Kottwitz and Harris conjectures.

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.

Motivation & Objective

• To establish a conceptual framework for $p$-adic local analogues of Shimura varieties, analogous to the classical global theory.
• To unify the theory of Rapoport-Zink spaces under a broader geometric and group-theoretic principle.
• To formulate conjectures on the $\ell$-adic cohomology of such local Shimura varieties, particularly the Kottwitz and Harris conjectures.
• To explore the role of the $\sigma$-centralizer group $J(\mathbb{Q}_p)$ and Weil descent in the cohomological realization of automorphic representations.
• To reframe the theory of RZ-spaces in a more intrinsic, group-theoretic way, stripping away their moduli-theoretic origins.

Proposed method

• Propose a definition of a local Shimura datum as a triple $(G,[b],\{\mu\})$ with axioms on reductive group $G$ over $\mathbb{Q}_p$, $\sigma$-conjugacy class $[b]$, and cocharacter class $\{\mu\}$.
• Define a local Shimura variety as a tower of rigid-analytic spaces $\{\mathbb{M}^K\}$ over $\breve{E}$, with compatible actions of $G(\mathbb{Q}_p)$, $J(\mathbb{Q}_p)$, and Weil group $W_E$.
• Construct the tower as a generic fiber of a formal scheme solving a moduli problem of $p$-divisible groups with level structures.
• Use the period domain $\breve{\mathcal{F}}(G,b,\{\mu\})$ as a target for the tower maps, with equivariance under all group actions.
• Introduce the cohomological conjectures: the Kottwitz conjecture for basic $[b]$, and the Harris conjecture for non-basic $[b]$.
• Apply the theory of perfectoid spaces and Fargues-Fontaine curve to potentially realize the tower directly in the generic fiber, bypassing formal schemes.

Experimental results

Research questions

• RQ1 Can a theory of $p$-adic local Shimura varieties be constructed directly as a tower of rigid-analytic spaces, without first building a formal scheme?
• RQ2 How do the $\ell$-adic cohomology groups of such local Shimura varieties realize local Langlands correspondences via simultaneous actions of $G(\mathbb{Q}_p)$, $J(\mathbb{Q}_p)$, and $W_E$?
• RQ3 What is the precise relationship between the cohomology of local Shimura varieties and the representation theory of $p$-adic groups, especially for supercuspidal representations?
• RQ4 Under what conditions does the cohomology of a local Shimura variety decompose according to the Harris conjecture, particularly when $[b]$ is not basic?
• RQ5 Can the theory of RZ-spaces be reinterpreted in a purely group-theoretic way, independent of their moduli interpretation?

Key findings

• The Kottwitz conjecture is proposed as a description of the discrete part of the $\ell$-adic cohomology when $[b]$ is basic, linking cohomology to discrete series representations.
• The Harris conjecture provides an inductive formula for the cohomology in the non-basic case, expressed as a sum over $\{\mu'\}_L \in I_{b,\{\mu\},L}$, with cohomology of smaller Levi subgroups.
• Theorem 8.11 by Mantovan confirms the Harris conjecture for unramified simple integral RZ-data, showing that the cohomology of the $G$-tower is induced from cohomology of $L$-towers.
• Theorem 8.8 establishes that if $\{\mu'\} \in I^{G}_{b,\{\mu\},L}$, then the $L$-dominant representative of $\{\mu'\}_L$ equals the $L$-dominant representative of $\{\mu\}_L$, under conditions on the fundamental group and $\Gamma$-invariants.
• The existence of a slope filtration for $p$-divisible groups over RZ spaces is shown to be sufficient for the Harris conjecture, and this is now known to hold in full generality via Shen's work.
• The paper corrects an error in earlier literature by showing that $|I^{G}_{b,\{\mu\},L}| = 1$ does not hold in general, and that the correct condition is the existence of $\{\mu'\}_L \in I_{b,\{\mu\},L}$ with $\mu'_{L-\text{dom}} = \mu_{\text{dom}}$.