The Supersingular Locus of the Shimura Variety of

\text{GU}(2,n-2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{GU}(2,n-2)$$\end{document}

Abstract

We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to

\text{GU}(2,n-2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{GU}(2,n-2)$$\end{document}

. More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne–Lusztig varieties defined by explicit conditions after taking perfections. Moreover, we study the intersections of the irreducible components. Stratifications of Deligne–Lusztig varieties defined using powers of Frobenius action appear in the description of the intersections.

Keywords

Shimura variety / Supersingular locus / Affine Deligne–Lusztig variety / Demazure resolution / Rapoport–Zink space / Unitary group / Primary 11G18 / Secondary 14M15

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Funding

Division of Mathematical Sciences(2103150)

Japan Society for the Promotion of Science(22H00093)

The University of Tokyo

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