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

TL;DR

This paper computes the nearby cycles cohomology sheaves on the local model of a PEL Shimura variety with a ramified prime, revealing their triviality or non-vanishing depending on the parity of n.

Contribution

It provides explicit calculations of nearby cycles on the local model for the GU(n-1,1) Shimura variety, including the Galois action, especially in the ramified setting.

Findings

Nearby cycles are trivial for odd n.

For even n, only one higher cohomology sheaf is non-zero.

The Galois action is explicitly described for even n.

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, with level given by the stabilizer of a self-dual lattice. This 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. We compute the nearby cycles 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 Frobenius eigenvalue.