Shimura Varieties Of Hodge Type Research Articles

Mod p points on shimura varieties of parahoric level

Abstract We study the $\overline {\mathbb {F}}_{p}$ -points of the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a conjecture of Kisin–Madapusi–Shin. We, furthermore, show that the mod p isogeny classes are of the form predicted by the Langlands–Rapoport conjecture (cf. Conjecture 9.2 of [Rap05]) if either the Shimura variety is proper or if the group at p is unramified. The main ingredient in our work is a global argument that allows us to reduce the conjecture to the case of very special parahoric level. This case is dealt with in the Appendix by Zhou. As a corollary to our arguments, we determine the connected components of Ekedahl–Oort strata.

Connected components of affine Deligne–Lusztig varieties for unramified groups

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

Vanishing theorems for Shimura varieties at unipotent level

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite \Gamma_1(p^\infty) -level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at p . This generalizes and strengthens the vanishing result proved in [A. Caraiani et al., Compos. Math. 156 (2020)]. As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari–Emerton for completed (Borel–Moore) homology.

Some Remarks on Ekedahl–Oort Stratifications

We study independence of symplectic embeddings of the theory of Ekedahl–Oort stratifications on Shimura varieties of Hodge type, by comparing two different embeddings with a third one. The main results are as follows. 1. The Ekedahl–Oort stratification is independent of the choices of symplectic embeddings. 2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl–Oort stratifications with respect to morphisms of Shimura varieties.

Mod p isogeny classes on Shimura varieties with parahoric level structure

We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. We also verify most of the He-Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are non-empty for these models.

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

AbstractWe prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h‐hyperspecial subgroups as pro‐étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.

Serre–Tate theory for Shimura varieties of Hodge type

We study the formal neighbourhood of a point in the \(\mu \)-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a “shifted cascade”. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms, analogous to the Serre–Tate canonical lift of an ordinary abelian variety.

The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type

We construct a generalisation of Mantovan’s almost product structure to Shimura varieties of Hodge type with hyperspecial level structure atppand deduce that the perfection of the Newton strata are proétale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani and Scholze’s generalisation of the almost product structure for Shimura varieties of Hodge type.

Automorphic Forms on the Stack of G-Zips

We define automorphic vector bundles on the stack of G-zips introduced by Moonen–Pink–Wedhorn–Ziegler and study their global sections. In particular, we give a combinatorial condition on the weight for the existence of nonzero mod p automorphic forms on Shimura varieties of Hodge-type. We attach to the highest weight of the representation $$V(\lambda )$$ a mod p automorphic form and we give a modular interpretation of this form in some cases.

L-Adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

Let $$(\mathsf {G},\mathsf {X})$$ be a Shimura datum of Hodge type. Let p be an odd prime such that $$\mathsf {G}_{\mathbb {Q}_p}$$ splits after a tamely ramified extension and $$p\not \mid |\pi _1(\mathsf {G}^\mathrm{der})|$$ . Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $$\mathsf {G}_{\mathbb {Q}_p}$$ is either unramified or residually split, we prove the generalisation of Mantovan’s formula for the l-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction modulo p of the Kisin–Pappas integral model.

Toroidal compactifications of integral models of Shimura varieties of Hodge type

We construct projective toroidal compacti cations for integral models of Shimura varieties of Hodge type that parameterize isogenies of abelian varieties with additional structure. We also construct integral models of the minimal (Satake-Baily-Borel) compacti cation. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compacti cations are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo center. Along the way, we demonstrate an interesting rationality property of Hodge cycles on abelian varieties with respect to p-adic analytic uniformizations.

On the Hodge–Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.

Nonemptiness of Newton strata of Shimura varieties of Hodge type

For a Shimura variety of Hodge type with hyperspecial level at a prime p, the Newton stratification on its special fiber at p is a stratification defined in terms of the isomorphism class of the rational Dieudonne module of parameterized abelian varieties endowed with a certain fixed set of Frobenius-invariant crystalline tensors (“Gℚp-isocrystal”). There has been a conjectural group-theoretic description of the F-isocrystals that are expected to show up in the special fiber. We confirm this conjecture. More precisely, for any Gℚp-isocrystal that is expected to appear (in a precise sense), we construct a special point whose reduction has associated F-isocrystal equal to the given one.

Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type

AbstractFor a Shimura variety of Hodge type with hyperspecial level structure at a primep, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models whenp> 2. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn, and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elementswin the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding towis smooth of dimensionl(w) (i.e., the length ofw) if it is non-empty. We also determine the closure of each stratum.

Generalized μ-ordinary Hasse invariants

We give a short proof for the existence of μ-ordinary Hasse invariants for the good reduction special fiber of Shimura varieties of Hodge-type using the stack of G-zips introduced by Moonen–Wedhorn and Pink–Wedhorn–Ziegler. We give an explicit formula for the power of the Hodge bundle that admits a Hasse invariant. When G is a Weil restriction of G1, we relate the Ekedahl–Oort strata of G and those of G1.

Rapoport–Zink spaces for spinor groups

After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.

The almost product structure of Newton strata in the Deformation space of a Barsotti–Tate group with crystalline Tate tensors

In this paper, we construct the almost product structure of the minimal Newton stratum in deformation spaces of Barsotti-Tate groups with crystalline Tate tensors, similar to Oort's and Mantovan's construction for Shimura varieties of PEL-type. It allows us to describe the geometry of the Newton stratum in terms of the geometry of two simpler objects, the central leaf and the isogeny leaf. This yields the dimension and the closure relations of the Newton strata in the deformation space. In particular, their nonemptiness shows that a generalisation of Grothendieck's conjecture of deformations of Barsotti-Tate groups with given Newton polygon holds. As an application, we determine analogous geometric properties of the Newton stratification of Shimura varieties of Hodge type and prove the equidimensionality of Rapoport-Zink spaces of Hodge type.

Canonical sections of the Hodge bundle over Ekedahl–Oort strata of Shimura varieties of Hodge type

We construct canonical non-vanishing global sections of powers of the Hodge bundle on each Ekedahl–Oort stratum of a Hodge type Shimura variety. In particular we recover the quasi-affineness of the Ekedahl–Oort strata. In the projective case, this gives a very short proof of non-emptiness of Ekedahl–Oort strata. It follows that the Newton strata are also nonempty, by a result of S. Nie. From the canonicity of our construction, we deduce the fact that the μ-ordinary locus is determined by the Ekedahl–Oort strata of its image under any embedding of Shimura varieties.

Level 𝑚 stratifications of versal deformations of 𝑝-divisible groups

Let k k be an algebraically closed field of characteristic p > 0 p>0 . Let c , d , m c,d,m be positive integers. Let D D be a p p -divisible group of codimension c c and dimension d d over k k . Let D \mathscr {D} be a versal deformation of D D over a smooth k k -scheme A \mathscr {A} which is equidimensional of dimension c d cd . We show that there exists a reduced, locally closed subscheme s D ( m ) \mathfrak {s}_D(m) of A \mathscr {A} that has the following property: a point y ∈ A ( k ) y\in \mathscr {A}(k) belongs to s D ( m ) ( k ) \mathfrak {s}_D(m)(k) if and only if y ∗ ( D ) [ p m ] y^*(\mathscr {D})[p^m] is isomorphic to D [ p m ] D[p^m] . We prove that s D ( m ) \mathfrak {s}_D(m) is regular and equidimensional of dimension c d − dim ⁡ ( A u t A u t ( D [ p m ] ) ) cd-\dim (\pmb {\mathrm {Aut}}(D[p^m])) . We give a proof of Traverso’s formula which for m ≫ 0 m\gg 0 computes the codimension of s D ( m ) \mathfrak {s}_D(m) in A \mathscr {A} (i.e., dim ⁡ ( A u t A u t ( D [ p m ] ) ) \dim (\pmb {\mathrm {Aut}}(D[p^m])) ) in terms of the Newton polygon of D D . We also provide a criterion of when s D ( m ) \mathfrak {s}_D(m) satisfies the purity property (i.e., it is an affine A \mathscr {A} -scheme). Similar results are proved for quasi Shimura p p -varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic ( 0 , p ) (0,p) .

Projective integral models of Shimura varieties of Hodge type with compact factors

Let $(G,X)$ be a Shimura pair of Hodge type such that $G$ is the Mumford--Tate group of some elements of $X$. We assume that for each simple factor $G_0$ of $G^{\ad}$ there exists a simple factor of $G_{0\dbR}$ which is compact. Let $N\Ge 3$. We show that for many compact open subgroups $K$ of $G(\dbA_f)$, the Shimura variety $\Sh(G,X)/K$ has a projective integral model $\scrN$ over $\dbZ[{1\over N}]$ which is a finite scheme over a certain Mumford moduli scheme $\scrA_{g,1,N}$. Equivalently, we show that if $A$ is an abelian variety over a number field and if the Mumford--Tate group of $A_{\dbC}$ is $G$, then $A$ has potentially good reduction everywhere. The last result represents significant progress towards the proof of a conjecture of Morita. If $\scrN$ is smooth over $\dbZ[{1\over N}]$, then it is a N\'eron model of its generic fibre. In this way one gets in arbitrary mixed characteristic, the very first examples of general nature of projective N\'eron models whose generic fibres are not finite schemes over abelian varieties.