Differential operators on locally analytic Shimura varieties

TL;DR

This paper develops a geometric and analytic framework for infinite-level Shimura varieties, introducing differential operators and complexes to advance the understanding of their automorphic properties.

Contribution

It formulates a Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties and constructs new differential operators and complexes for automorphic analysis.

Findings

Reconstruction of Shimura varieties from perfectoid data and ${B}_{ ext{dR}}^{+}$-thickening.

Construction of generalized differential operators extending Pan's work.

Introduction of a BGG-Fontaine complex with conjectured automorphic properties.

Abstract

We investigate infinite-level Shimura varieties within the framework of analytic stacks of Clausen-Scholze, developing their smooth, completed, locally analytic, and de Rham realizations. We formulate a Grothendieck-Messing-Hodge-Tate period map, and establish a Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties. This theory, combined with a reformulation of Riemann-Hilbert correspondence, implies that the locally analytic infinite-level Shimura variety can be fully reconstructed purely from its perfectoid counterpart and its $\mathbb{B}_{\mathrm{dR}}^{+}$-thickening. Building upon this geometric structure, we systematically construct differential operators generalizing those of Pan, and we introduce a Bernstein-Gelfand-Gelfand-Fontaine complex based on dual BGG complexes, conjecturing its automorphic properties. These constructions will be used to establish a locally analytic Jacquet-Langlands correspondence in a companion paper ([Jia26a]).