Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

Abstract

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.

A Geometric Jacquet-Langlands Transfer for automorphic forms of higher weights

“…Choose ν to be a place of E lying over p. We write O E,(ν) for the localization of O E at ν. Results of Kisin [6] and Vasiu [14] state that for any Hodge type Shimura datum (G, X), there is a smooth integral canonical model S K (G, X) of Sh K (G, X), which is defined over O E,(ν) . Let k ν denote the residue field of O E,ν and fix an algebraic closure kν of k ν .…”

“…Write k ν for the residue field of E 1 E 2 at ν. Results of Kisin [6] and Vasiu [14] state that there exists a canonical smooth integral model of Sh K (G i , X i ) over O E,(ν) . Let d i = dim Sh K (G i , X i ) and Sh µi denote the mod p fiber of this canonical integral model, base changed to kν .…”

Perfectoid Shimura Varieties of Abelian Type

“…[11]) and Vasiu (cf. [32,33], as well as the more recent [34,35]) have proved that integral canonical models for Shimura varieties of abelian type (in the case of hyperspecial levels at p) exist. These works and the results of Scholze in [27] are the motivation of this paper.…”