Stratification des vari\'et\'es de Hilbert en pr\'esence de ramification

TL;DR

This paper investigates the geometric stratification of Hilbert Shimura varieties' special fibers, demonstrating the stratification's goodness and flatness properties, contrasting with the Hilbert-Siegel case.

Contribution

It proves the Hodge polygon stratification is good for Hilbert Pappas-Rapoport models and describes the dimension of strata using affine Grassmannian convolution results.

Findings

Hodge polygon stratification is good in the Hilbert case.

The dimension of strata is explicitly described.

The forgetful morphism is flat on Hodge strata.

Abstract

In this paper we study the geometry of the special fiber of Pappas-Rapoport models of Shimura varieties in the Hilbert case. More precisely we prove that the stratification induced by the Hodge polygon is a good stratification, which is false in the Hilbert-Siegel case for $g\geq 2$. Then we use known results on the convolution product of affine Grassmannians to describe the dimension of the strata and obtain that the forgetting morphism to the Kottwitz PEL model is flat in restriction to Hodge strata.