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Nearby cycles on the local model for the $\mathrm{GU}(n-1,1)$ PEL Shimura variety over a ramified prime

Joseph Muller

TL;DR

The paper determines the $\ell$-adic nearby cycles on the local model for the $\mathrm{GU}(n-1,1)$ PEL Shimura variety at a ramified prime, reducing the computation to a blow-up along the singular point. It proves a sharp dichotomy: when $n$ is odd, the nearby cycles are trivial; when $n$ is even, only $R^{n-1}\Psi_{\eta}\Lambda$ survives, supported at the singular point with Frobenius eigenvalue $\pm p^{\frac{n}{2}}$ and trivial inertia, with the sign determined by whether the underlying hermitian space is split or non-split. The method combines the semi-stable reduction of the blow-up, the monodromy filtration, and spectral sequences to express the nontrivial piece via the primitive cohomology of a smooth quadric, aligning with Krämer’s trace computations but providing a more precise, fully equivariant description. These results illuminate the local structure of ramified integral models and have implications for the cohomology of ramified PEL Shimura varieties.

Abstract

In this paper, we compute the cohomology sheaves of the $\ell$-adic nearby cycles on the local model of the PEL $\mathrm{GU}(n-1,1)$ Shimura variety over a ramified prime. The local model is known to have isolated singularities. If $n=2$ it has semi-stable reduction, and if $n\geq 3$ the blow-up at the singular point has semi-stable reduction. Thus, in principle one may compute the nearby cycles at least on the blow-up, then use proper base change to describe them on the original local model. As a result, we prove that the nearby cycles are trivial when $n$ is odd, and that only a single higher cohomology sheaf does not vanish when $n$ is even. In this case, we also describe the Galois action by computing the associated eigenvalue of the Frobenius.

Nearby cycles on the local model for the $\mathrm{GU}(n-1,1)$ PEL Shimura variety over a ramified prime

TL;DR

The paper determines the -adic nearby cycles on the local model for the PEL Shimura variety at a ramified prime, reducing the computation to a blow-up along the singular point. It proves a sharp dichotomy: when is odd, the nearby cycles are trivial; when is even, only survives, supported at the singular point with Frobenius eigenvalue and trivial inertia, with the sign determined by whether the underlying hermitian space is split or non-split. The method combines the semi-stable reduction of the blow-up, the monodromy filtration, and spectral sequences to express the nontrivial piece via the primitive cohomology of a smooth quadric, aligning with Krämer’s trace computations but providing a more precise, fully equivariant description. These results illuminate the local structure of ramified integral models and have implications for the cohomology of ramified PEL Shimura varieties.

Abstract

In this paper, we compute the cohomology sheaves of the -adic nearby cycles on the local model of the PEL Shimura variety over a ramified prime. The local model is known to have isolated singularities. If it has semi-stable reduction, and if the blow-up at the singular point has semi-stable reduction. Thus, in principle one may compute the nearby cycles at least on the blow-up, then use proper base change to describe them on the original local model. As a result, we prove that the nearby cycles are trivial when is odd, and that only a single higher cohomology sheaf does not vanish when is even. In this case, we also describe the Galois action by computing the associated eigenvalue of the Frobenius.

Paper Structure

This paper contains 4 sections, 6 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometry of the local model
  3. The nearby cycles on the local model
  4. Figures

Key Result

Theorem 1

We have where, for $n$ even, $i_{\overline{y^{\mathrm{sg}}}*}\Lambda[\epsilon p^{\frac{n}{2}}]$ is the skyscraper sheaf on $\overline{M^{\mathrm{loc}}}$ concentrated at the geometric singular point $\overline{y^{\mathrm{sg}}}$, with $\mathrm{Gal}(\overline E/E)$-action which is trivial on the inertia subgro

Figures (2)

  • Figure 1: The first page $(E^{K})_1^{a,b}$ of the monodromy spectral sequence for $M^{\psi,K}$.
  • Figure 2: The first page $(E^{Z_1})_1^{a,b}$ of the restriction of the monodromy spectral sequence to $Z_1$.

Theorems & Definitions (12)

  • Theorem
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 2 more