On the basic locus of GSpin Shimura varieties with vertex stabilizer level

TL;DR

This work analyzes the basic locus of GSpin Shimura varieties with vertex-lattice level at an odd prime $p$ by linking the global problem to a local Rapoport--Zink space $ cal{N}_L$. The authors describe the reduced locus $ cal{N}_{L, ext{red}}^{(ullet)}$ as a union of irreducible generalized Deligne--Lusztig varieties $S_oldsymbol{\Lambda}$ and $R_oldsymbol{\Lambda}$, through a duality with the lattices $L^lat$ and a careful analysis of special cycles. A key technical achievement is the construction of morphisms $ cal{Z}_{ ext{red}}^{(ullet)}(oldsymbol{\Lambda}) o S_oldsymbol{\Lambda}$ (and analogously for Y-cycles) that induce isomorphisms on tangent spaces, enabling the identification without perfection. The work also develops an explicit description of the Bruhat--Tits stratification of the basic locus, establishes the DL-structure of strata, and provides a framework that connects the basic locus to arithmetic applications such as Kudla–Rapoport-type intersections and Tate-type conjectures. Overall, the paper gives a concrete, non-perfected description of the basic locus in terms of classical DL varieties, extending prior results to the spinor-similitude setting with vertex-lattice level.

Abstract

We study the basic locus of Shimura varieties associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ with level structure given by the stabilizer of a vertex lattice. We give a description of the underlying reduced scheme of the associated Rapoport--Zink space, generalizing results of Howard--Pappas [HP17] and Oki [Oki20b], in the case of self-dual, and almost self dual level structure.

Theorems & Definitions (109)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 2.1
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4