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

Stéphane Bijakowski, Valentin Hernandez

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).

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

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).
Paper Structure (20 sections, 32 theorems, 106 equations)

This paper contains 20 sections, 32 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Case of a quadratic imaginary $F$
  3. Definition of the variety
  4. Geometry of the special fiber
  5. A remark on deformations of a $p$-divisible group
  6. Closure relations and geometry of strata
  7. Local rings
  8. Modular forms
  9. Sheaves
  10. Definition and vanishing modulo $p$
  11. Further strata for the case $(1,n)$
  12. Definition of the invariants
  13. The conjugate filtration
  14. Stratification when $n >1$
  15. Stratification when $n=1$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume $p \neq 2$. For all $h \leq \ell$, the stratum $X_{h,\ell}$ is non empty, smooth, and equidimensional of dimension $ab - \frac{(\ell - h)(\ell - h + 1)}{2}$. Moreover we have the closure relations, In particular $X$ is not smooth, and the smooth locus is the union of the $X_{h,h}$ for $0 \leq h \leq a$. Moreover $Y$ is flat over $O_F$, normal, and $X$ is Cohen-Macaulay.

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • ...and 71 more