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

TL;DR

This paper establishes the existence of smooth integral models for Shimura varieties of Hodge type in unramified mixed characteristic, solving a conjecture of Langlands and advancing the theory of canonical models.

Contribution

It proves the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic, providing new proofs and progress on longstanding conjectures.

Findings

Existence of smooth integral models in unramified mixed characteristic

Solution to Langlands' conjecture for Hodge type Shimura varieties

Progress towards Milne and Reimann's conjectures on canonical models

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 provide a smooth solution (answer) to a conjecture (question) 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 with respect to h--hyperspecial subgroups as pro-\'etale covers of N\'eron models; 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.