On the geometry of the Pappas-Rapoport models in the (AR) case

TL;DR

This paper investigates the geometric and singularity properties of Pappas-Rapoport models of certain ramified unitary Shimura varieties, revealing their normality, Cohen-Macaulayness, and stratification structure, with implications for modular forms.

Contribution

It provides a detailed analysis of the geometry of Pappas-Rapoport models in the ramified case, including their singularities, stratification, and relation to modular forms, which was previously not well-understood.

Findings

The models are normal with Cohen-Macaulay special fibers.

A combinatorial stratification of the special fiber is introduced and analyzed.

Extra components in the special fiber do not affect mod p modular forms in regular degree.

Abstract

We study some integral model of P.E.L. Shimura varieties of type A for ramified primes. Precisely, we look at the Pappas-Rapoport model (or splitting model) of some unitary Shimura varieties for which there is ramification in the degree 2 CM extension. We show that the model isn't smooth, but that it is normal with Cohen-Macaulay special fiber. We moreover study its special fiber by introducing a combinatorial stratification for which we can compute the closure relations. Even if there are "extra" components in special fiber, we prove that those do not contribute to mod p modular forms in regular degree. We also study the interaction of the stratification with the natural stratification given by the vanishing of some partial Hasse invariants, in the case of signature (1,n-1).