Toric schemes and integral models for Shimura varieties with $\Gamma_1(p)$-type level

TL;DR

This paper develops a conjectural framework for $p$-integral models of Shimura varieties with specific level structures, utilizing root stacks and toric varieties, and proves the conjecture for PEL type cases with Iwahori level.

Contribution

It introduces a new conjectural approach to $p$-integral models using root stacks over local models, connecting local model theory with toric geometry, and confirms the conjecture in certain cases.

Findings

Proposes a conjectural theory for $p$-integral models with abelian quotient parahoric subgroups.

Constructs a root stack over local models based on the divisor theorem and toric varieties.

Proves the conjecture for PEL type Shimura varieties with Iwahori parahoric under additional conditions.

Abstract

We propose a conjectural theory of $p$-integral models of Shimura varieties with level structure at $p$ given by a class of normal subgroups of parahoric subgroups with abelian quotient group. The role of the theory of local models is played in this context by a certain root stack over the local model for parahoric level. The construction of this root stack is based on the "divisor theorem" (a foundational fact about local models) and on the theory of toric varieties in this context, both of which are of independent interest. We prove our conjecture in the case of Shimura varieties of PEL type when the parahoric is an Iwahori (under some additional conditions).