Vol. 2, No. 2, 2020

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Potentially good reduction loci of Shimura varieties

Naoki Imai and Yoichi Mieda

Vol. 2 (2020), No. 2, 399–454
Abstract

We give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to nonsupercuspidal parts.

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Keywords
Shimura variety, good reduction, adic space, nearby cycle
Mathematical Subject Classification 2010
Primary: 14G35
Secondary: 11F70, 22E50
Milestones
Received: 14 February 2019
Accepted: 24 May 2019
Published: 2 August 2019
Authors
Naoki Imai
Graduate School of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba
Tokyo 153-8914
Japan
Yoichi Mieda
Graduate School of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba, Meguro-ku
Tokyo 153-8914
Japan