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arXiv · 2502.17851

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

Joseph Muller
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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.

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Joseph Muller. 2025-02-25. Nearby cycles on the local model for the $\mathrm{GU}(n-1,1)$ PEL Shimura variety over a ramified prime. https://arxiv.org/abs/2502.17851

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