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

Stéphane Bijakowski and Valentin Hernandez

Vol. 334 (2025), No. 1, 107–142

DOI: 10.2140/pjm.2025.334.107

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 study its special fiber by introducing a combinatorial stratification for which we can compute the closure relations. Even if there are “extra” components in the 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 $(1,n-1)$.

Keywords

Shimura varieties

Mathematical Subject Classification

Primary: 11G18, 14G35

Milestones

Received: 14 May 2024

Revised: 6 November 2024

Accepted: 6 December 2024

Published: 23 January 2025