Potentially good reduction loci of Shimura varieties

TL;DR

This paper introduces the concept of the potentially good reduction locus in Shimura varieties, constructs a related partition of the adic space, and relates its cohomology to that of the entire Shimura variety, enhancing understanding of motive degenerations.

Contribution

It defines the potentially good reduction locus for Shimura varieties of preabelian type and constructs a partition of the associated adic space to analyze motive degenerations.

Findings

Existence of the potentially good reduction locus for preabelian type Shimura varieties.

Construction of a partition of the adic space describing motive degenerations.

Cohomology of the reduction locus matches that of the Shimura variety up to non-supercuspidal parts.

Abstract

In this paper, 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 non-supercuspidal parts.