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The Frobenius action on the cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature (1,n-1) is semisimple, its cohomology is generally non-admissible, and for n=3 or 4, the cohomology of the supersingular locus can be completely described in terms of automorphic representations. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

December 18, 2025Journal of the Institute of Mathematics of Jussieu1 citationsOpen Access

COHOMOLOGY OF THE BASIC UNRAMIFIED PEL UNITARY RAPOPORT-ZINK SPACE OF SIGNATURE (1, n-1)

JMJoseph MüllerSorbonne Université

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Abstract

Abstract In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature (1, n-1) at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p -adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case n=3 or 4, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.

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