On Local Models with Special Parahoric Level Structure

TL;DR

This paper studies local models of PEL-type Shimura varieties with special parahoric level structures, proving geometric properties like irreducibility, normality, and Frobenius splitting of their special fibers.

Contribution

It establishes the irreducibility and geometric properties of local models with special parahoric level structures in the ramified unitary case.

Findings

Special fiber is irreducible and generically reduced.

Special fiber is normal, Frobenius split, and has rational singularities.

Contains an open subset isomorphic to affine space.

Abstract

We consider the local model of a Shimura variety of PEL type, with the unitary similitudes corresponding to a ramified quadratic extension of $\mathbb{Q}_p$ as defining group. We examine the cases where the level structure at $p$ is given by a parahoric that is the stabilizer of a selfdual periodic lattice chain and that is special in the sense of Bruhat--Tits theory. We prove that in these cases the special fiber of the local model is irreducible and generically reduced; consequently, the special fiber is reduced and is normal, Frobenius split, and with only rational singularities. In addition, we show that in these cases the local model contains an open subset that is isomorphic to affine space.