Cohomology of the Bruhat-Tits strata in the supersingular locus of the GU(1, n − 1) Shimura variety at a ramified prime

The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme $\mathcal{N}$ . In the case when p is an inert prime, we define special cycles ${\mathcal{Z}}({\bold x})$ in $\mathcal{N}$ , associated to collections ${\bold x}$ of m ‘special homomorphisms’ with fundamental matrix T∈Herm m (O k ). When m=n and T is nonsingular, we show that the cycle ${\mathcal{Z}}({\bold x})$ is either empty or is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient condition, in terms of T, for ${\mathcal{Z}}({\bold x})$ to be irreducible. When ${\mathcal{Z}}({\bold x})$ is zero dimensional—in which case it reduces to a single point—we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.

In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink space whose special fibre is described by the Bruhat-Tits stratification. The description for this supersingular locus in the case where the prime is inert in imaginary quadratic field is already known to Howard and Pappas by exploiting the exceptional isomorphism. Our method is more direct without using the exceptional isomorphism.

We describe the supersingular locus of a GU(2,2) Shimura variety at a prime inert in the corresponding quadratic imaginary field.

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.

Corrigendum to the article On the supersingular locus of the GU(2,2) Shimura variety

We complete the study of the supersingular locus \(\mathcal{M}^{\mathrm{ss}}\) in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert. This was started by the first author in Can. J. Math. 62, 668–720 (2010) where complete results were obtained for n=2,3. The supersingular locus \(\mathcal{M}^{\mathrm{ss}}\) is uniformized by a formal scheme \(\mathcal{N}\) which is a moduli space of so-called unitary p-divisible groups. It depends on the choice of a unitary isocrystal N. We define a stratification of \(\mathcal{N}\) indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of N. We show that the combinatorial behavior of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the irreducible components of \(\mathcal{N}_{\mathrm{red}}\)) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that \(\mathcal{M}^{\mathrm{ss}}\) is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of \(\mathcal{M}^{\mathrm{ss}}\).

In this paper we study the supersingular locus of the reduction modulopof the Shimura variety for GU(1,s) in the case of an inert primep. Using Dieudonné theory we define a stratification of the corresponding moduli space ofp-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.In the case of GU(1, 2), we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to $$\textrm{GU}(2,n-2)$$ GU ( 2 , n - 2 ) . 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.

We study the structure of the supersingular locus of the Rapoport–Zink integral model of the Shimura variety for [Formula: see text] over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structure of the basic loci. More precisely, the former one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the projective line.

We give a description of the $\textrm{GL}_4$ Rapoport–Zink space, including the connected components, irreducible components, intersection behavior of the irreducible components, and Ekedahl–Oort stratification. As an application of this, we also give a description of the supersingular locus of the Shimura variety for the group $\textrm{GU}(2,2)$ over a prime split in the relevant imaginary quadratic field.

The supersingular locus of the Shimura variety for $$\hbox {GU}(1, n-1)$$ GU ( 1 , n - 1 ) over a ramified prime

In this article we study the special fiber of the Rapoport–Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat–Tits stratification and is intimately related to the Bruhat–Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.

We enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport–Zink space for the quaternionic unitary similitude group of degree 2, with a special parahoric level. We prove that such a scheme is purely 2-dimensional, and every irreducible component is isomorphic to the Fermat surface. We also determine its connected components, irreducible components and their intersection behaviors by means of the Bruhat–Tits building of \({{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)\). In addition, we compute the intersection multiplicity of the GGP cycles associated to an embedding of the considering Rapoport–Zink space into the Rapoport–Zink space for the unramified \({{\,\mathrm{GU}\,}}_{2,2}\) with hyperspecial level for the minuscule case.

In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of Gross and Keating (Invent Math 112(225–245):2051, 1993) and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over $$\mathbb {F}_p$$ and the sum of the Fourier coefficients of the Siegel-Eisenstein series for $$\mathrm {Sp}_4/\mathbb {Q}$$ of weight 2, which is independent of $$p \left( > 2\right) $$ . In addition, we will explain a description of the local intersection multiplicities of the special cycles over $$\mathbb {F}_p$$ on the supersingular locus of the ‘special fiber’ of the Shimura varieties for $$\mathrm {GSpin}(n,2), n\le 3$$ in terms of the Siegel series directly.