We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular p-adic integral models for these varieties over odd primes p at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.
In this note, we discuss several forms of the Eichler-Shimura relation for the compactly supported cohomology of local Shimura varieties, using the work of Fargues-Scholze.
Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative approach to vanishing results of Caraiani-Scholze for the cohomology of unitary Shimura varieties.
We give a simple proof that Kottwitz's PEL type integral models of Shimura varieties admit closed embeddings into Siegel integral models. We also show that Rapoport's and Kottwitz's integral models agree with Kisin's integral models for relevant Shimura data.
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oort's method for Siegel modular varieties.
In this paper we study the geometry of reduction modulo $p$ of the Kisin-Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR strata on these Shimura varieties, using the theories of $G$-zips and mixed characteristic local $\mathcal{G}$-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.
For infinitely many Hurwitz spaces parametrizing cyclic covers of the projective line, we provide a method to determine the integral PEL datum of the Shimura variety that contains the image of the Hurwitz space under the Torelli morphism.
We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.
We show that the integral models of Shimura varieties of Rapoport, Smithling and Zhang in relation to variants of the arithmetic Gan-Gross-Prasad conjecture, the arithmetic fundamental lemma conjecture and the arithmetic transfer conjecture are canonical in the sense prescribed by Pappas. In particular, we prove that they are isomorphic to the models constructed by Kisin and Pappas.
We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.
The Kodaira dimension of Shimura varieties has been studied by many people. Kondo and Gritsenko-Hulek-Sankaran studied the singularities of orthogonal Shimura varieties related to the moduli spaces of polarized K3 surfaces. They proved that they have canonical singularities and are of general type if the polarization degree is sufficiently large. In this paper, we work on similar problems for unitary Shimura varieties. We show that they have canonical singularities if $n > 4$. As an application, we show that certain unitary Shimura varieties associated with Hermitian forms over the rings of in…
We use the method of Bruinier--Raum to show that symmetric formal Fourier--Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension for unitary Shimura varieties defined in these cases.
We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Thélène--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$.
We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.
This paper is a continuation of [G-dS1]. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call "tautological foliations", are defined on Hilbert modular varieties, and lift to characteristic $0$. The second, the "$V$-foliations", are defined on unitary Shimura varieties in characteristic $p$ only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are $p$-closed, and the locus where they are smooth. Where not smooth, we construct a "successive blow up" …
Let $S$ be the special fibre of a Shimura variety of Hodge type, with good reduction at a place above $p$. We give an alternative construction of the zip period map for $S$, which is used to define the Ekedahl-Oort strata of $S$. The method employed is local, $p$-adic, and group-theoretic in nature.
We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.
For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a general conjecture formulated in previous papers by the authors. Furthermore, we prove in these cases that any mod $p$ automorphic form whose weight lies in a specific region of the weight space is divisible by a partial Hasse invariant. This generalizes to other Shimura varieties previous results of Diamond--Kassaei on Hilbert modular forms.
