In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial $\ell$-adic étale cohomology classes appearing in the the generic fiber of Lubin-Tate and Rapoprt-Zink towers. We also study the behavior of the vector bundle functor on the fundamental curve in $p$-adic Hodge theory, defined by Fargues-Fontaine, under multilinear morphisms.
In this article, we describe the underlying reduced scheme of a quaternionic unitary Rapoport-Zink space with Iwahori level structure. In a previous work, we have studied the quaternionic unitary Rapoport-Zink space with a special maximal parahoric level structure. We will describe the morphism between these two Rapoport-Zink spaces and the fiber of this morphism. As an application of this result, we describe the the quaternionic unitary Rapoport-Zink space with any parahoric level structure.
In this article, we treat two questions on Rapoport--Zink spaces of Hodge type constructed by Hamacher and Kim. One of which is their singularities, and the other is $p$-adic uniformization of Shimura varieties. More precisely, we prove that the singularity of a Rapoport--Zink space is controlled by its asssociated local model, and the basic locus of a Kisin--Pappas integral model of a Shimura variety is uniformized by the corresponding Rapoport--Zink space. These results extend the known facts by Rapoport and Zink in the case of PEL type.
When $p>2$, we construct a Hodge-type analogue of Rapoport-Zink spaces under the unramifiedness assumption, as formal schemes parametrising "deformations" (up to quasi-isogeny) of $p$-divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of "local Shimura varieties" conjectured by Rapoport and Viehmann.
Let (G, μ) be a pair of a reductive group G over the p-adic integers and a minuscule cocharacter μ of G defined over an unramified extension. We introduce and study "(G, μ)-displays" which generalize Zink's Witt vector displays. We use these to define certain Rapoport-Zink formal schemes purely group theoretically, i.e. without p-divisible groups.
Under a mild condition, we prove that the action of the group of self-quasi-isogenies on the set of irreducible components of a Rapoport-Zink space has finite orbits. Our method allows both ramified and non-basic cases. As a consequence, we obtain some finiteness results on the representation obtained from the l-adic cohomology of a Rapoport-Zink tower.
We develop a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of the canonical integral models of Shimura varieties of Hodge type at primes of good reduction. We then apply the general theory to the special case of Shimura varieties associated with groups of spinor similitudes, and, in the basic case, determine explicitly the reduced scheme underlying the Rapoport-Zink formal scheme.
We give a description of the $\mathrm{GL}_4$ 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 $\mathrm{GU}(2,2)$ over a prime split in the relevant imaginary quadratic field.
We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport-Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan-Gross-Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic-geometric side of the arithmetic fundamental lemma proved by Rapoport-Terstiege-Zhang in the minuscule case.
Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the $l$-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport-Zink spaces which are Hodge type and Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink space into an appropriate space of EL type, for which th…
We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of…
In this paper, we investigate how the Zelevinsky involution appears in the l-adic cohomology of the Rapoport-Zink tower. We generalize the result of Fargues on the Drinfeld tower to the Rapoport-Zink towers for symplectic similitude groups.
We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport-Zink functors associated to triples $(G,\{μ\},[b])$, where $G$ is a flat affine group scheme over $\mathbb{Z}_p$ and $μ$ is a cocharacter of $G$ defined over a finite unramified extension of $\mathbb{Z}_p$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport-Zink spaces given by Bültel and Papp…
In this article, we give a concrete description of the underlying reduced subscheme of the Rapoport--Zink spaces for spinor similitude groups with special maximal parahoric (and non-hyperspecial) level structure. Moreover, we give two applications of the above result. One of which is describing the structure of the basic loci of mod $p$ reductions of Kisin--Pappas integral models of Shimura varieties for spinor similitude groups with special maximal parahoric level structure at $p$. The other is constructing a variant of the result of He, Li and Zhu, which gives a formula on the intersection m…
Rapoport-Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let $\mathscr{M}_{\infty}$ be an infinite-level Rapoport-Zink space of EL type, and let $\mathscr{M}_{\infty}^\circ$ be one geometrically connected component of it. We show that $\mathscr{M}_{\infty}^{\circ}$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the in…
We generalize the work of M. Harris and R. Taylor on the local Langlands correspondence for the linear group over $\mathbb{Q}_p$. We prove some cases of the Kottwitz conjectures for the supercuspidal part of the compactly supported $\ell$-adic cohomology of Rapoport-Zink rigid-analytic spaces. This means that we decompose this part in terms of local Langlands correspondences. In particular, we prove the first case of a non-abelian reciprocity law constructed geometrically and associated to groups other than inner forms of the linear group. More precisely, we do this for the unramified unitary …
In this paper, we study the l-adic cohomology of the Rapoport-Zink tower for GSp(4). We prove that the smooth representation of GSp_4(Q_p) obtained as the i-th compactly supported l-adic cohomology of the Rapoport-Zink tower has no quasi-cuspidal subquotient unless i=2,3,4. Our proof is purely local and does not require global automorphic methods.
We investigate the alternating sum of the l-adic cohomology of the Rapoport-Zink tower for GSp(4) by the Lefschetz trace formula. Under some assumptions on L-packets of GSp(4) and its inner form, we observe that the local Jacquet-Langlands correspondence appears in the cohomology.
We enlarge the class of Rapoport-Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. These such-obtained Rapoport-Zink spaces are called 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 formal 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 t…
Local shtukas are the function field analogs for $p$-divisible groups. Similar to the $p$-adic theory, one defines Rapoport-Zink functors and Rapoport-Zink spaces for these local shtukas. The associated Hodge-Pink structures are described uniquely by a morphism, called the period morphism of the moduli problem. We will prove the ind-representability of the Rapoport-Zink functor in a particular case and compute the corresponding Rapoport-Zink space as well as the corresponding period morphism. In this case, the period morphism is given by the Carlitz logarithm.
In this paper we study the cohomology of PEL-type Rapoport-Zink spaces associated to unramified unitary similitude groups over $\Q_p$ in an odd number of variables. We extend the results of Kaletha-Minguez-Shin-White to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapport-Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.
In this article we first survey the analogy between Shimura varieties (resp. Rapoport-Zink spaces) and moduli stacks for global G-shtukas (resp. Rapooprt Zink spaces for local P-shtukas). This part is intended to enrich the dictionary between the arithmetic of number fields and function fields a bit further. Furthermore, to complete this picture, we also study some local properties of Rapoport-Zink spaces for local P-shtukas by constructing local models for them. This provides a "local" complementary to a previous work of the author, which was devoted to the study of the local models for the m…
We prove under certain assumptions a formula for the cohomology of PEL-type Rapoport--Zink spaces that "averages" over the Kottwitz set. Our formula generalizes that of Shin's beyond the EL-type case and is proven by combining Mantovan's formula with descriptions of the cohomology of Shimura and Igusa varieties. We then use this averaging formula to derive a conjectural description of the cohomology of Rapoport--Zink spaces, generalizing our earlier work for EL-type spaces. Along the way, we give a description of the cohomology of Igusa varieties in terms of the Langlands correspondence, gener…
We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, μ}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, μ}(ρ)$ for $ρ$ a s…
As an example of relative p-adic Hodge theory, we sketch the construction of the universal admissible filtration of an isocrystal (ϕ$-module) over the completion of the maximal unramified extension of Q_p, together with the associated universal crystalline local system.
In their book Rapoport and Zink constructed rigid analytic period spaces $F^{wa}$ for Fontaine's filtered isocrystals, and period morphisms from PEL moduli spaces of $p$-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism $F^a \to F^{wa}$ of rigid analytic spaces and of a universal local system of $Q_p$-vector spaces on $F^a$. For Hodge-Tate weights $n-1$ and $n$ we construct in this article an intrinsic Berkovich open subspace $F^0$ of $F^{wa}$ and the universal local system on $F^0$. We conjecture that the rigid-analytic space associ…
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.
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p-adic Hodge theory. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of p-divisible groups over the ring of integers of a complete algebraically closed fi…
In this paper, we study the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ by using the Bruhat-Tits stratification on its special fiber. As such, we apply the same method that we developped for the unramified case in two previous papers. More precisely, we first investigate the cohomology of a given closed Bruhat-Tits stratum. It is isomorphic to a generalized Deligne-Lusztig variety which is in general not smooth, and is associated to a finite group of symplectic similitudes. We determine the weights of the Frobenius and most of the unipotent representations…
In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.
We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $μ$ ; this condition consists in a filtration on the sheaf of differentials $ω_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying …
The Kottwitz conjecture describes the cohomology of basic Rapoport-Zink spaces using local Langlands correspondences. In this paper, via geometrical studies of some Kottwitz-type Shimura varieties, we prove this conjecture for basic simple unramified unitary PEL type Rapoport-Zink spaces of signature (1, n-1).
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 has 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 q…
W. Zhang's arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport-Zink space. In the minuscule case, Rapoport-Terstiege-Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.
Let $G$ be a reductive group scheme over the $p$-adic integers, and let $μ$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,μ)$-displays over $p$-nilpotent rings $R$ to formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors. When $R/pR$ has a $p$-basis étale locally, we show that this defines an equivalence between the two categories. The definition of the functor relies on the construction of a $G$-crystal associated with any adjoint nilpotent $(G,μ)$-display, which extends the construction of the Dieudonné crystal ass…
Let $K$ be a $p$-adic field. We continue to develop the theory of rigid analytic $p$-divisible groups over $K$. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic $p$-divisible groups "in finite level" without perfectoid spaces. We then establish some results about families of rigid analytic $p$-divisible groups. This allows us to prove a "minimality" result in the sense of birationnal geometry for integral models of unramified Rapoport-Zink spaces.
Starting from our work on Harder-Narasimhan filtrations of finite flat group schemes over a $p$-adic field, we developp a theory of Harder-Narasimhan filtrations for $p$-divisible groups. We apply this to the study of the geometry of period morphisms for Rapoport-Zink spaces and to the $p$-adic geometry of Shimura varieties. We define and study in particular some fundamental domains for the action of Hecke correspondences.
We show that the Zink equivalence between p-divisible groups and Dieudonne displays over a complete local ring with perfect residue field of characteristic p is compatible with duality. The proof relies on a new explicit formula for the p-divisible group associated to a Dieudonne display.
In this note 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 give a simple and natural proof for the compatibility of the Hasse invariant with duality. We then study a $p$-divisible group with an action of the ring of integers of a finite ramified extension of $\mathbb{Q}_p$. We suppose that it satisfies the Pappas-Rapoport condition ; in that case the Hasse invariant is a product of partial Hasse invariants, each of which can be expressed in terms of primitive Hasse invariants. We then show that the dual of the $p$-divisible group naturally satisfies a Pappas-Rapoport condition, and prove the compatibility with the duality for the partial and primit…
This survey article explains the construction of Rapoport-Zink local models and their use in understanding various questions relating to the singularities in the reduction modulo p of certain Shimura varieties with parahoric level structure at p.
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latte…
For arbitrary reductive groups $G$ defined over a finite field, we decompose Newton strata in the special fiber of moduli spaces of global $G$-shtukas into a product of Rapoport-Zink spaces and Igusa varieties. This allows us to compare the $\ell$-adic cohomology of these spaces together with the various actions on them.
