On two mod p period maps: Ekedahl--Oort and fine Deligne--Lusztig stratifications

TL;DR

This paper compares two stratifications on the special fiber of a Shimura variety at an odd prime p, using perfectoid spaces and the Hodge-Tate period map, with applications to the non-emptiness of certain strata.

Contribution

It establishes a comparison between Ekedahl-Oort and fine Deligne-Lusztig stratifications via perfectoid methods and the Hodge-Tate period map.

Findings

Comparison of stratifications on perfectoid covers

Identification of conditions for non-emptiness of strata

Insights into the structure of Shimura varieties at p

Abstract

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime p>3. Consider its perfectoid cover S(p^\infty) and the Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the pull-back to S(p^\infty) of the Ekedahl-Oort stratification on the mod p special fiber of S and the pull back to S(p^\infty) of the fine Deligne-Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge-Tate period map. An application to the non-emptiness of Ekedhal-Oort strata is provided.