STACK

. This paper is about the arithmetic geometry of the reduction modulo p of Shimura varieties with parahoric level structure. We realize the EKOR-stratiﬁcation on the special ﬁber of the Siegel modular stack at parahoric level as the ﬁbers of a smooth morphism into an algebraic stack parametrizing homogeneously polarized chains of certain truncated displays.


Introduction
This paper is about the arithmetic geometry of the reduction modulo p of Shimura varieties with parahoric level structure. We realize the EKOR-stratification on the special fiber of the Siegel modular stack at parahoric level as the fibers of a smooth morphism into an algebraic stack parametrizing homogeneously polarized chains of certain truncated displays.
Fix a prime p, an integer g ∈ Z ≥1 and a subset J ⊆ Z with J + 2gZ ⊆ J and −J ⊆ J. We consider the moduli stack A g,J /Z p of polarized chains of Abelian varieties of type (g, J) introduced by de Jong in [Jon93] and in full generality by Rapoport and Zink in [RZ96] and denote by A g,J /F p its special fiber. It is naturally an integral model for the Siegel modular stack at parahoric level K = K · GSp 2g ( Z (p) ), where K ⊆ GSp 2g (Q p ) is the stabilizer of the standard lattice chain of type J. We let G/Z p be the parahoric model of GSp 2g corresponding to K (i.e. such that we have K = G(Z p )) and also setK = G(Z p ).
There exists a natural map, called the central leaves map, whereK σ denotes the σ-conjugation action g.x = gxσ(g) −1 (and σ denotes the Frobenius). It is given by sending a polarized chain of Abelian varieties to the Frobenius of the associated rational Dieudonné module (see the work of Oort [Oor02] and He and Rapoport [HR15]). The image of this map is given byK σ \X 2g,J where X 2g,J = w∈Admg,JK wK is a certain union of double cosets in GSp 2g (Q p ). He and Rapoport also consider the composition υ K of Υ K with the projection K σ \X 2g,J →K σ \(K 1 \X 2g,J ) =K σ (K 1 ×K 1 )\X 2g,J , whereK 1 ⊆K is the pro-unipotent radical, i.e. the kernel of the homomorphism G(Z p ) → (G Fp ) rdt (F p ), and call the (finitely many) fibers of υ K Ekedahl-Kottwitz-Oort-Rapoport (EKOR) strata. One can project even further ontoK\X 2g,J /K = Adm g,J . The resulting fibers are called Kottwitz-Rapoport (KR) strata. The EKOR stratification is thus a refinement of the KR stratification. In fact both agree in the Iwahori case J = Z (see [HR15,Corollary 6.2]).
The KR strata are smooth by work of Rapoport and Zink. In fact they show even more: There exists a certain normal projective scheme M loc over F p , called the (special fiber of the) local model, defined essentially by linear algebra, that comes with a G-action (see [RZ96,Definition 3.27]). Its F p -points identify withK\X 2g,J , withK acting via g.x = xσ(g) −1 . The slightly odd normalization of the action is in order to be compatible with the σ-conjugation action ofK on X 2g,J . There then is a natural smooth morphism A g,J → [G\M loc ] parametrizing the Hodge filtration (the proof of smoothness is via Grothendieck-Messing theory, see the argument in [RZ96,Proposition 3.33]). The fibers of this morphism are precisely the KR strata.
The EKOR strata are also smooth (see [HR15,Remark 6.22]). The argument He and Rapoport give is via comparing the EKOR stratification to the KR stratification at Iwahori level (whose strata are smooth), using a result from [GH12].
In the hyperspecial case J = 2gZ (or J = g + 2gZ) the EKOR stratification is also just called the Ekedahl-Oort (EO) stratification and was first considered by Oort in [Oor01]. Two points of A g = A g,2gZ given by two (principally polarized) Abelian varieties A, A ′ over some algebraically closed field k/F p lie in the same EO stratum if and only A[p] ∼ = A ′ [p]. Viehmann and Wedhorn ( [VW10]) and Zhang ([Zha18]) realize the EO stratification as the fibers of a smooth morphism into a certain stack of zips (in the sense of Moonen, Pink, Wedhorn and Ziegler, see [MW04] and [PWZ15]).
Thus it is natural to ask the following question: Question 0.1. Is it possible to canonically realize the map υ K (or maybe even Υ K ) as a smooth morphism from A g,J into some algebraic stack that is defined in terms of linear algebraic/group theoretic data?
The existence of such a smooth morphism would in particular give a new proof of the smoothness of the EKOR strata and the closure relations between them. More importantly, it also provides a tool that may be applied to further study the geometry of A g,J and cycles on it.
Let us give an overview of results that have been obtained so far (and that we are aware of), also considering more general Shimura varieties than the Siegel modular variety: • Moonen and Wedhorn ( [MW04]) introduce the notion of an F -zip that is a characteristic p analog of the notion of a Hodge structure. Given an Abelian variety A over some F p -algebra R the de Rham cohomology H 1 dR (A/R) naturally has the structure of an F -zip. If R is perfect then the datum of H 1 dR (A/R) with its F -zip structure is equivalent to the datum of the Dieudonné module of the p-torsion A[p].
Pink, Wedhorn and Ziegler ( [PWZ15]) define a group theoretic version of the notion of an F -zip (that is called G-zip).
• Viehmann and Wedhorn ( [VW10]) define a moduli space of F -zips with polarization and endomorphism structure (that the call D-zips) in a PEL-type situation with hyperspecial level structure (that in particular includes the hyperspecial Siegel case). They construct a morphism from the special fiber of the associated Shimura variety to this stack of D-zips that parametrizes the EO stratification. Then they show that this morphism is flat and use this to deduce that the EO strata are non-empty and quasi-affine and to compute their dimension (smoothness of the EO strata was already shown by Vasiu in [Vas06]).
Zhang ([Zha18]) constructs a morphism from the special fiber of the Kisin integral model (see [Kis10]) of a Hodge type Shimura variety to the group-theoretic stack of G-zips. They then show that this morphism is smooth and thus gives an EO stratification with the desired properties. Shen and Zhang ([SZ17]) later generalize this to Shimura varieties of Abelian type.
Hesse ([Hes20]) considers an explicit moduli space of chains of F -zips with polarization and constructs a morphism from the Siegel modular stack A g,J into this stack. However it appears that such a morphism is not well-behaved in a non-hyperspecial situation (see also Subsection 3.7). • Xiao and Zhu ( [XZ17]) construct a perfect stack of local shtukas Sht loc µ as well as truncated versions Sht loc(n,m) µ in a hyperspecial situation. For a Shimura variety of Hodge type (still at hyperspecial level) they construct a morphism from the perfection of its special fiber into Sht loc µ that gives a central leaves map Υ K . They show that the induced morphisms to Sht loc(n,m) µ are perfectly smooth (but there is a detail in the proof of this result that I don't understand). They also construct a natural perfectly smooth forgetful morphism from Sht loc(2,1) µ into the perfection of the stack of G-zips and in particular recover the smoothness result of Zhang after perfection. • Shen, Yu and Zhang ([SYZ21]) generalize the previous work of Shen, Xiao, Zhang and Zhu to Shimura varieties of Abelian type at parahoric level (where the construction of integral models is due to Kisin and Pappas, see [KP18]). They give two constructions.
-They construct a morphism from each KR stratum into a certain stack of G-zips, parametrizing the EKOR strata contained in this KR stratum. They also show that this morphism is smooth, thus establishing the smoothness of the EKOR strata. -They also construct perfect stacks of local shtukas Sht loc K,µ and truncated versions Sht are perfectly smooth (the proof is by the same argument as in [XZ17] and there is the same detail that I don't understand). • Zink ([Zin02]) defines a stack of displays over Spf(Z p ) and Bültel and Pappas ([BP18]) define a group theoretic version of this notion in a hyperspecial situation (that is called (G, µ)-display). In characteristic p this gives a deperfection of the notion of local shtuka from [XZ17].
In the parahoric Hodge type situation there is a definition of display due to Pappas ([Pap22]) but this definition is only over p-torsionfree p-adic rings. Pappas also constructs a parahoric display on the p-completion of the integral model of the Shimura variety. • Scholze and Weinstein ([SW20]) define a v-stack of G-shtukas (of type µ) over Spd(Z p ), the special fiber of which agrees with the stack Sht loc K,µ of [SYZ21]. It is however not so clear what the relation between shtukas and displays should be.
Pappas and Rapoport ([PR21]) construct, in a Hodge type situation with parahoric level structure, a morphism from the (v-sheaf associated to the) integral model of the Shimura variety to this stack of G-shtukas.
Content. Let us explain the content of the present work.
We start by recalling the definition of a (3n-)display in the sense of [Zin02], phrased in our own way (that is very much in line with [BP18, Definition 3.2.1]): Definition 0.2 (Definition 2.11). Let R ∈ Alg(F p ) and 0 ≤ d ≤ h. Then a display (of type (h, d) over R) is a triple (M, Fil(M ), F div ) that is given as follows: • M is a finite projective W (R)-module of rank h.
Here W (R) denotes the ring of Witt vectors of R with augmentation ideal I R = ker(W (R) → R). The object σ-Fil(M ) is a certain W (R)-module attached to (M, Fil(M )) (see Definition 2.8). It agrees with the Frobenius-twist Fil(M ) σ if R is perfect but is better behaved in the sense that it is always finite projective (see Remark 2.10).
The category of displays of type (h, d) carries a natural action of the symmetric monoidal category of tuples (L, ι) consisting of an invertible W (R)-module L and an isomorphism ι : L σ → L. We denote this action by ⊗.
Moreover, if h = 2d then there is also a natural duality on the category of displays of type (h, d) that we denote by (•) ∨ . is not algebraic). Here the word "truncated" refers to the use of truncated Witt vectors and m (resp. n) roughly measures how truncated the module M (resp. the divided Frobenius F div ) is. For two tuples (m, n) and (m ′ , n ′ ) with m ≥ m ′ and n ≥ n ′ we have a natural morphism of stacks Disp (m,n) → Disp (m ′ ,n ′ ) forgetting some structure.
One should note that this notion of a truncated display is different from the one defined by Lau in [Lau10, Definition 3.4] (see also Subsection 2.4) but instead should be compared to the restricted local shtukas from [XZ17, Definition 5.3.1].
We have the following theorem of Lau: Theorem 0.3 ([Lau10, Theorem 5.1], Theorem 2.14). There is a natural functor where BT h,d (R) denotes the category of p-divisible groups over R of height h and dimension h − d. It induces an equivalence of categories between formal p-divisible groups and nilpotent displays and is compatible with the natural dualities on both sides.
As we are interested in studying the moduli space A g,J it makes sense to make the following definition. Note that in the main text this definition is phrased in a different way, using the notation introduced in Subsection 1.2 and Subsection 1.3.
Definition 0.4 (Definition 3.11 and Definition 3.13). A homogeneously polarized chain of displays (of type Here we call an R-module homomorphism N → N ′ between finite projective R-modules of constant rank k if its image is a direct summand of N ′ that is finite projective of rank k. ) are isomorphisms such that we have the compatibilities θ j • ρ i,j = ρ i+2g,j+2g • θ i and ρ i,i+2g = pθ i . • (L, ι) is as in Definition 0.2.
Homogeneously polarized chains of displays of type (2g, J) again form a stack HPolChDisp as before. Here we allow n to take the additional value 1-rdt formally lying between 0 and 1 (the "rdt" stands for "reductive quotient"). It roughly corresponds to only having a divided Frobenius on the graded pieces of the 1-truncated chain (compare with Definition 3.3).
We compare this definition with the ones in [VW10] and [Hes20] in Subsection 3.7. We also show that HPolChDisp (m,n) 2g,J admits the following description (actually in the main text we choose to present our results in a way so that this will be a tautology): where we use the following notation.
• G/Z p is the parahoric model of GSp 2g associated to K.
• For k ∈ Z >0 ∪ {∞}, L (k) G denotes the Greenberg realization of G Z/p k Z and for k = 1-rdt we set L (1-rdt) G := (G Fp ) rdt to be the reductive quotient of the special fiber of G. • M loc,(n) → M loc is a certain L (n) G-torsor where M loc denotes the special fiber of the local model associated to the group G.
The proof of this lemma has two main ingredients: • Any two homogeneously polarized chains ( is again a homogeneously polarized chain (see Proposition 3.9 and Definition 3.10).

Thus our definition of HPolChDisp
(m,n) 2g,J gives (up to a slightly different normalization) a deperfection of the stack of parahoric restricted local shtukas from [SYZ21, Section 4]. We discuss this in a little more detail in Subsection 3.8.
On F p -points the projection M loc,(∞) → M loc identifies with the map X 2g,J →K\X 2g,J where X 2g,J is the subset of GSp 2g (Q p ) introduced above. The action we take the quotient by in Proposition 0.5 corresponds to the σ-conjugation action ofK on X 2g,J so that we obtain an equivalence of groupoids Consequently we also obtain HPolChDisp Thus the natural morphism A g,J → HPolChDisp gives a way to view the map Υ K as a map between geometric objects over F p and its composition with the projection to HPolChDisp (∞,1-rdt) 2g,J (or even HPolChDisp (m,1-rdt) 2g,J for 1 < n < ∞) gives a similar way of viewing the map υ K parametrizing the EKORstratification.
Our main result is now the following, giving a non-perfect version of [SYZ21,Theorem 4.4.3] in the Siegel case: Theorem 0.6 (Theorem 5.12). For n < ∞ the natural morphism A g,J → HPolChDisp (m,n) g,J is smooth.
Let us outline the strategy of the proof: • By the Serre-Tate theorem (see for example the book [Mes72, Theorem V.2.3] by Messing) the smoothness of the morphism at a point corresponding to a polarized chain of Abelian varieties only depends on the associated polarized chain of p-divisible groups (Lemma 5.6 (1)). • Using Theorem 0.3 we then show that the morphism is smooth along the formal locus |A fml g,J | ⊆ |A g,J |, i.e. the subset of all points corresponding to chains of Abelian varieties with formal p-divisible groups (Lemma 5.6 (3)).
• To obtain the smoothness in general we show that for every polarized chain of p-divisible groups there exists a polarized chain of Abelian varieties inducing it such that the corresponding point in |A g,J | specializes to one in the formal locus (Corollary 5.11). The openness of the smooth locus then implies that the morphism A g,J → HPolChDisp (m,n) g,J is smooth everywhere as desired.
Although the definition of our stacks HPolChDisp As a natural next step we intend to generalize our results to the case of a more general Shimura variety at parahoric level instead of the Siegel modular variety and study if it is possible to also extend the results to the p-complete world. It could be expected that there exists a stack of (G, µ)-displays over Spf(O E ) (where E/Q p is the reflex field) as well as truncated versions for parahoric G recovering the definition from [BP18] in the hyperspecial case and giving our definition when passing to the special fiber. In a situation where (G, µ) comes from a Shimura datum there then should be a natural (formally) smooth morphism from the p-completion of the corresponding Shimura variety into the stack of truncated (G, µ)-displays. As already mentioned above, partial results in this direction have been achieved by Pappas in [Pap22].
Acknowledgements. I heartily thank Ulrich Görtz for introducing me to the subject of Shimura varieties and their special fibers and for all the support I received from him throughout this project. Furthermore I thank Jochen Heinloth, Ludvig Modin, Herman Rohrbach and Torsten Wedhorn for helpful conversations. This work was partially funded by the DFG Graduiertenkolleg 2553. • We denote by Cat ⊕ the category of preadditive categories.
• Let C, D be preadditive categories with duality. A functor between preadditive categories with duality (from C to D) is a tuple (F, α) consisting of an additive functor F : C → D and an isomorphism α : is commutative for all x ∈ C. • Let F, G : C → D be two functors between preadditive categories with duality. An isomorphism (from F to G) is an isomorphism between functors ω : F → G such that the diagram is commutative for all x ∈ C. • We obtain the category of preadditive categories with duality that we denote by Cat ∨ .
Note that functors in Cat ∨ preserve (anti-)symmetric morphisms and that symmetric morphisms in C are the same thing as antisymmetric morphisms in C − and vice versa.

Actions of Symmetric Monoidal Categories on Preadditive Categories (with Duality).
• Symmetric monoidal categories are always assumed to be endowed with a preadditive structure and rigid. We denote by Cat ⊗ the category of symmetric monoidal categories. • Every M ∈ Cat ⊗ can be naturally considered as a commutative monoid object in Cat ∨ by using the duality defined by a ∨ := Hom(a, 1) for a ∈ M. • Let M ∈ Cat ⊗ and C ∈ Cat ⊕ . Then an action of M on C is a biadditive functor ⊗ : These isomorphisms are required to satisfy some coherence conditions. One can also give the following more precise definition (that we won't use in this paper): The category of preadditive categories with an action of a symmetric monoidal category is given as the category of algebras over the operad introduced by Liu and Zheng in [LZ12, Definition 1.5.6] in the category Cat ⊕ (that we equip with the cartesian symmetric monoidal structure). For a definition of operads and algebras over them we refer to the book [Lur17, Section 2.1] by Lurie.
• Let M ∈ Cat ⊗ and C ∈ Cat ∨ . Then an action of M on C is an action of M on C in the above sense that is further equipped with a natural isomorphism ((a ⊗ x) ∨ → a ∨ ⊗ x ∨ ) a∈M,x∈C that is required to satisfy some further coherence conditions. Again we can make this precise by saying that the category of predditive categories with duality and an action of a symmetric monoidal category is given as the category of algebras over the same operad as before, but now in the category Cat ∨ (that we also equip with the cartesian symmetric monoidal structure).
• Suppose we are given an action of M ∈ Cat ⊗ on C ∈ Cat ∨ as above. Then a morphism f : x → a⊗x ∨ with x ∈ C and a ∈ M invertible is called (anti-)symmetric if it agrees with the (negative of the) composition In the case where a = 1 this recovers the previous definition of (anti-)symmetric.

(Homogeneously) Polarized Objects.
• Let C ∈ Cat ∨ . Then we define the groupoid Pol(C) ∈ Grpd of polarized objects in C as the groupoid of tuples (x, λ) with x ∈ C and λ : x → x ∨ an antisymmetric isomorphism. • Now suppose we are given an action of M ∈ Cat ⊗ on C ∈ Cat ∨ . Then we define the groupoid HPol(C) = HPol M (C) ∈ Grpd of (M-)homogeneously polarized objects in C as the groupoid of tuples (x, a, λ) with x ∈ C, a ∈ M and λ : x → a ⊗ x ∨ an antisymmetric isomorphism. In this situation we have an action of the group G := Aut M (1) on Pol(C) by g.(x, λ) := (x, gλ) and the functor is G-equivariant for the trivial G-action on HPol(C). Moreover we have a fiber sequence (where the second functor is given by (x, a, λ) → a) that induces the just defined G-action.

Stacks.
• The stacks and sheaves we consider will mostly live on the category Alg(F p ) op that we endow with the flat topology. We denote by Stack the category of stacks (with values in categories). Likewise we denote by Stack ≃ , Stack ⊕ , Stack ∨ and Stack ⊗ the categories of stacks with values in Grpd, Cat ⊕ , Cat ∨ and Cat ⊗ . • Given X ∈ Stack we denote by X ≃ ∈ Stack ≃ its groupoid core.
• Given a partially ordered set I we denote by Stack I the functor category Fun(I, Stack) and use similar notation when Stack is replaced by its decorated variants.
As an example, an object X (•) ∈ Stack ⊕,N consists of the following data: -For every n ∈ N a stack X (n) with values in preadditive categories.
-Isomorphisms ρ n,n ∼ = id X (n) for n ∈ N and ρ n ′ ,n ′′ • ρ n,n ′ ∼ = ρ n,n ′′ for n ≥ n ′ ≥ n ′′ in N. These data have to satisfy the following coherence conditions: The compositions are required to be the identities on the respective morphisms (for n ≥ n ′ ≥ n ′′ ≥ n ′′′ ).

Witt vectors.
• p is a fixed prime. R always denotes an F p -algebra.
• N denotes the set Z >0 ∪ {∞} considered as a partially ordered set with respect to ≥. Similarly we also set N 0 := Z ≥0 ∪ {∞}. • For n ∈ N we denote by W n (R) the ring of n-truncated Witt vectors over R. We also set I n,R := ker(W n (R) → R). Note that I n+1,R is killed by ker(W n+1 (R) → W n (R)) so that it is a W n (R)-module.
• We denote the Frobenius on W n (R) by σ. Given a W n (R)-module M we denote its base change along the Frobenius (i.e. its Frobenius twist) by The inverse of the Verschiebung gives a W n (R)-module homomorphism σ div : I σ n+1,R → W n (R) that we call divided Frobenius.
• Given a functor G : Alg(Z p ) → Grp from the category of Z p -algebras to the category of groups and n ∈ N we define (this is usually called the Greenberg realization of G). Varying n, this gives an h . We also extend Vect (•) to an object in Stack ⊗,N0 by setting Vect (0) to be the trivial stack.

Pairs and Displays
The notion of a (3n-)display was introduced by Zink in [Zin02] and gives a generalization of the theory of Dieudonné modules to not necessarily perfect base rings. Lau defines a truncated version of the notion of a display in [Lau10].
In this section we recall the definitions of (truncated) pairs and displays. However we should warn the reader that our definitions do not agree with [Lau10, Definitions 3.2 and 3.4] but are better suited for our purposes.

Pairs.
Definition 2.1. We define the stack Pair (•) ∈ Stack ∨,N of (truncated) pairs as follows: • Pair The above remark gives a description of the stack Pair (1) . We now give a similar description of Pair (n) for general n ∈ N: Definition 2.3. Define the stack Pair (•) ∈ Stack ∨,N as follows: • The dual of an object (L, T ) ∈ Pair • The dual of a morphism a b c d in Pair by fixing the rank of L ⊕ T as h and Pair h,d . Proof. Fully faithfulness of the morphism is immediate so that we need to check essential surjectivity. Thus, let (M, Fil(M )) ∈ Pair (m) (R). We clearly can choose a complement T ⊆ M/I m,R M of Fil(M )/I m,R M . Now observe that the W m (R)-scheme X given by Notation 2.6. We write N 2 > for the set of pairs (m, n) with m, n ∈ N 0 such that m ≥ n + 1 (where we formally set ∞ + 1 := ∞). We define a partial order on this set by declaring (m, n) ≥ (m ′ , n ′ ) if and only if m ≥ m ′ and n ≥ n ′ . For fixed (m, n) ∈ N 2 > we will use the following notation: Remark 2.7. If R is perfect then I n+1,R = pW n+1 (R) and In the following definition we implicitly choose a normal decomposition for every (truncated) pair. This amounts to choosing a (quasi-)inverse of the equivalence from Lemma 2.4. • We also need to define a duality coherence datum for the morphism σ-Fil, i.e. a natural isomorphism We define this to be the isomorphism We would like to see that this is equal to This follows from the following identities, some of which were already given in Notation 2.6: that can be checked to have the following two properties: • It is natural in the pair M and compatible with base change and changing (m, n).
• If R is perfect then it is actually an isomorphism.
Definition 2.11. We define the stack Disp (•,•) ∈ Stack ∨,N 2 > by declaring Disp (m,n) (R) to be the category of tuples M = (M, Fil(M ), F div M ) that are given as follows: Definition 2.13. Let M ∈ Disp (m,n) (R) and suppose that n = 0.
is trivial. We denote by Disp (m,n),nilp ∈ Stack ⊕ the substack of Disp (m,n) given by the nilpotent displays.
2.4. Comparison with [Lau10]. In this subsection we compare our definitions to the ones in [Lau10]. Let us write Pair      Lau is the stack of (3n-)displays in the sense of [Zin02].
2.5. The display of a p-divisible group. We have the following result due to Lau, relating p-divisible groups and displays: Theorem 2.14 ( [Lau10]). There is a natural morphism in Stack ∨ with the following properties: Proof. Everything except for possibly the last part follows from [Lau10], see in particular Proposition 4.1 and Theorem 5.1. For the last statement, we can assume that R is an algebraically closed field (see [Stacks, Tag 0FWG]). In this situation the functor D coincides with classical (covariant) Dieudonné theory and the claim is a standard result.

(Polarized) Chains of Pairs and Displays
Notation 3.1. Throughout the section fix h ∈ Z >0 and ∅ = J ⊆ Z with J + hZ ⊆ J. We then also define E = E(J) to be the set of subsets {i, j} ⊆ J with i, j being consecutive in J. For e = {i, j} ∈ E we set |e| := |i − j|. At some points we will specialize to the situation where h = 2g is even and −J ⊆ J. In this case we call (h, J) symmetric. Also set N 1-rdt := N ∪ {1-rdt} and N 0,1-rdt := N 0 ∪ {1-rdt} where 1-rdt is an element with 0 < 1-rdt < 1 (where rdt refers to "reductive quotient"). We will later also use a variant N 2 1-rdt,> of N 2 > that is given by the set of pairs (m, n) with m, n ∈ N 0,1-rdt such that m ≥ n + 1 (where we formally set 1-rdt + 1 := 2).

Chains.
Definition 3.2. We define the stack ChVect  h,J (R) to be the category of tuples M = (M, ρ, θ) = ((M i ) i∈J , (ρ i,j ) i,j∈J,i≤j , (θ i ) i∈J ) that are given as follows: • θ i : M i → M i+h are isomorphisms such that we have the compatibility θ j • ρ i,j = ρ i+h,j+h • θ i and such that ρ i,i+h = pθ i . h,J carries a duality given by

The objects of ChVect
Definition 3.3. We extend ChVect d,J to an object of Stack ⊕,N 1-rdt (resp. of Stack ∨,N 1-rdt if (h, J) is symmetric) as follows: • Set ChVect           -The isomorphisms θ −j and θ −i together induce an isomorphism ker(ρ −j,−i ) ∼ = ker(ρ −j+h,−i+h ). Altogether this gives the desired datum Remark 3.4. We further extend ChVect h,J to an object of Stack ⊕,N 0,1-rdt (resp. Stack ∨,N 0,1-rdt ) by setting ChVect and the forgetful morphism ChPair h,J naturally is a morphism in Stack ∨,N . We still have an action of Vect    • Lemma 3.8. Let 0 ≤ ℓ ≤ h (and 0 ≤ d ≤ h as usual) and consider the moduli problem . Then X is a reduced scheme. Proof. This is a special case of the result [HO08, Theorem 4.2] proved by Helm and Osserman.   h,d (R) satisfying the same conditions as in Lemma 3.7 (for some 0 ≤ ℓ ≤ h). Then also the R-module homomorphism is of constant rank ℓ.
Proof. We may work locally on R so that we may assume M = M ′ = W m (R) d and ρ = 1 ℓ 0 0 p·1 d−ℓ and ρ ′ = p·1 ℓ 0 0 1 h−ℓ by Lemma 3.7. In this situation we can easily reduce to the case m = ∞ and using Lemma 3.8 we may also assume that R is reduced by passing to the universal case. But then showing that R ⊗ Wn(R) σ-Fil(ρ) is of constant rank ℓ can be done on geometric fibers (see the Stacks Project [Stacks, Tag 0FWG]) so that we may assume that R = k is an algebraically closed field.
The morphism ρ now gives rise to a commutative diagram of finite free W (k)-modules of rank h d,J,h (R) whenever n ∈ N and as the composition ChPair h,J for n = 0, 1-rdt. Proposition 3.9 ensures that this is well-defined.
If (h, J, d) is symmetric then this even gives a morphism in Stack ∨,N 2 1-rdt,> . Moreover the morphism σ-Fil is equivariant with respect to the morphism (Vect  Definition 3.11. We define the stack ChDisp (whose definition, see Definition 2.11, we extend in the evident way to N 2 1-rdt,> ).
The group Aut M (1) acting on PolChVect Remark 3.14. We have Vect , the standard chain of type (h, Z) as follows: i+h is the multiplication by p −1 . In the case that h = 2g is even we also define λ std for j = 1, . . . , g, −e ∨ 2g+1−j for j = g + 1, . . . , 2g.
By restriction this also defines a standard (polarized) chain of type (h, J) for every (h, J) (that is symmetric). By abuse of notation we denote this again by V std .
Note that V std gives rise to points in ChVect 2g,J (F p ). We define G h,J , G Pol 2g,J , G HPol 2g,J : Alg(Z p ) → Grp to be the automorphism groups of the various V std (where G HPol 2g,J is the group of automorphisms of (V std , ρ std , θ std ) that respect λ std up to a scalar). These are affine Z p -algebraic groups with generic fibers GL h , Sp 2g , GSp 2g respectively. Finally we also set L (1-rdt) G h,J to be the reductive quotient of L (1) G h,J and similarly for G Pol 2g,J and G HPol 2g,J . [f ] · (a 0 , . . . , a n−1 ) = (f a 0 , f p a 1 , . . . , f p n−1 a n−1 ) it is immediate that this ring homomorphism is in fact an isomorphism.
Corollary 3.21. The natural morphisms in Stack ≃ are equivalences for all n ∈ N 1-rdt .
Proof. We prove the claim for the first of the three morphisms and only for n > 1-rdt (the case n = 1-rdt is easy). The argument for the remaining two morphisms is the same. First suppose that n < ∞. We claim that ChVect   h,J (W n (•)) and the only difference between the two is that on the left hand side there is a condition on the rank of the reduction of the ρ i,j modulo I n,R while on the right hand side there is a condition on the rank of the reduction of the ρ i,j modulo pW n (R). But by Lemma 3.7 these two conditions are actually the same (where we use Lemma 3.20 to compare the Zariski topologies on R and W n (R)). From this claim it readily follows that every object of ChVect • The first action is of the group L (n) G h,J . A group element g ∈ L (n) G h,J (R) acts as (Fil, F div ) → (Fil, gF div ).
Here we regard g as a morphism (V std , Fil) → (V std , g(Fil)) in ChPair where the quotient is for the σ-conjugation action.
In the polarized situations we have similar actions with G h,J replaced by G Pol 2g,J and G HPol 2g,J respectively that are defined by the exact same formulas. We also have quotient stack descriptions  are the (special fibers of the) usual local models for the general linear and the symplectic group (see [RZ96,Definition 3.27] and also the work of Görtz [Gör01] and [Gör03]). In particular these objects are reduced projective F p -schemes.
For n ∈ N 1-rdt the projection M loc,(n) h,J,d → M loc h,J,d is a L (n) G h,J -torsor (for the left multiplication action) and the same is true in the polarized cases.
Proof. This is immediate from the definitions.
The same statements are also true in the polarized cases.
Proof. This follows from Proposition 3.23 as ChDisp 3.7. Comparison with [VW10] and [Hes20]. Let us denote by Zip the stack of F -zips in the sense of [MW04, Definition 1.5] (with filtration-type supported in {0, 1}) and write Zip h,d ⊆ Zip for the substack given by those F -zips that have rank h and whose Hodge filtration has rank d. This is equipped with a duality and carries an action of Vect σ=id 1 (defined by setting Vect σ=id 1 (R) to be the symmetric monoidal category of invertible R-modules L together with an isomorphism L σ ∼ = L).
Applying Subsection 1.3 we can define the stacks PolZip 2g and HPolZip 2g of (homogeneously) polarized zips of type (2g, g). HPolZip 2g then is precisely the stack of D-zips from [ 2g,2gZ → HPolZip 2g . These morphisms are smooth (after perfection this is shown in [XZ17,Lemma 5.3.6] and the argument carries over to the non-perfect world).
In the same way as what we did one can also put zips in chains and obtains stacks ChZip h,J,d , PolChZip g,J , HPolChZip g,J . The last one of these is then precisely the stack of G K -zips from [Hes20]. However, the natural forgetful morphisms ChDisp 3.8. Comparison with [SYZ21]. In this subsection we compare our definitions (after perfection) with the more conceptual group-theoretic ones in [SYZ21, Section 4] (see also [XZ17, Section 5]).
Notation 3.25. Fix an algebraic closure Q p of Q p with residue field F p (an algebraic closure of F p ). Writȇ Z p := W ∞ (F p ) andQ p :=Z p [p −1 ]. Finally write Alg(F p ) pf ⊆ Alg(F p ) for the category of perfect F p -algebras R. We equip Alg(F p ) pf,op with the flat topology.
Let G/Q p be a connected reductive group and let G/Z p be a parahoric model for G. Moreover let µ be a minuscule conjugacy class of cocharacters of G Q p with reflex field E ⊆ Q p and denote by k E ⊆ F p the residue field of E.
Definition 3.26. We recall the following definitions that are mostly taken from [SYZ21,Section 4]. Note that we use slightly different normalizations, see Remark 3.28.
• The loop group of G is the sheaf of groups • The (Witt vector) affine flag variety of G is the sheaf Gr G := L (∞) G\LG on Alg(F p ) pf,op . We will consider Gr G equipped with the left action of L (∞) G that is given by g.x := xσ(g) −1 .   Remark 3.28. Here is an attempt to explain the difference between the normalization in [SYZ21] and our normalization.
Suppose we are given a p-divisible group X of height h and dimension h − d over R. Then we have the associated (covariant) Dieudonné module (M, F ). We can now do the following two things.
• In general we know that the assertion is true after perfection and from this it follows that it is also true after restricting to the generic L (1) G-orbits in M loc,can by the explicit description of the canonical deperfection (see [Ans+22, Proof of Proposition 3.3]). However, the complete statement does not seem to follow from the general machinery of canonical deperfections.
Remark 3.36. In the case where G is reductive, Sht loc(∞,∞),can G,µ agrees with the special fiber of the stack of (G, µ)-displays from [BP18, Definition 3.2.1]. This can be seen essentially by the same argument as in the proof of Proposition 3.34.
We also recall from [XZ17, Lemma 5.3.6] that for G reductive there is a natural perfectly smooth forgetful morphism from Sht loc(2,1) G,µ to the perfection of the stack of G-zips of type µ. As in Subsection 3.7 this morphism deperfects to a smooth map from Sht loc(2,1),can G,µ to the stack of G-zips of type µ.

Some Generalities on Algebraic Stacks
Definition 4.1. Let X ∈ Stack ≃ .
• The reduction of X is the largest reduced substack X red ⊆ X .
Definition 4.2. Let f : X → Y be a morphism in Stack ≃ , let k/F p be an algebraically closed field and let ξ ∈ X (k). Then we say that f has the Artinian lifting property (ALP) at ξ if every lifting problem with B ′ → B a surjective F p -algebra homomorphism between Artinian local F p -algebras with residue field k has a solution.
Example 4.3. Formally smooth morphisms of stacks have the ALP at all of their geometric points.
Let k/F p be an algebraically closed field, ξ ∈ X (k) and suppose that f has the ALP at ξ. Then g • f has the ALP at ξ if and only if g has the ALP at υ := f (ξ) ∈ Y(k).
Proof. The "if"-direction is straightforward to check. For the "only if"-direction assume that g • f has the ALP at ξ and suppose we are given a lifting problem Lemma 4.5. Let f : X → Y be a morphism of algebraic stacks. Suppose that Y is locally Noetherian and that f is locally of finite type. Let k/F p be an algebraically closed field and let ξ ∈ X (k). Then f is smooth at ξ ∈ |X | if and only if f has the ALP at ξ.

Main Result
Fix (g, J) as before.
5.1. The Siegel Modular Stack A g,J . The following definition of the Siegel modular stack A g,J with parahoric level structure of type J at p is due to Rapoport and Zink (see [RZ96, Definition 6.9]).
Definition 5.1. We define the stack ChAbVar g,J ∈ Stack ∨ of chains of Abelian varieties (of type (g, J)) by setting AbVar g,J (R) to be the category of diagrams (A, ρ) = ((A i ) i , (ρ i,j ) i,j ) of shape J in AbVar g (R) such that the ρ i,j : A i → A j are isogenies of degree p j−i and such that ker(ρ i,i+2g ) = A i [p]. The duality is given by . Given (A, ρ) ∈ ChAbVar g,J (R) we formally set ρ i,j := ρ −1 j,i for i > j (this is a quasi-isogeny). A principal polarization of (A, ρ) ∈ ChAbVar g,J (R) is a symmetric isomorphism λ = (λ i ) i : Finally we define the Siegel modular stack A g,J ∈ Stack ≃ by setting A g,J (R) to be the groupoid of tuples (A, ρ, λ) where (A, ρ) ∈ ChAbVar g,J (R) and λ is a principal polarization on (A, ρ). Definition 5.3. Completely analogously to Definition 5.1 we define the stack ChBT 2g,J ∈ Stack ∨ of chains of p-divisible groups (X, ρ) and the stack PolChBT 2g,J ∈ Stack ≃ of chains of p-divisible groups (X, ρ) equipped with an antisymmetric isomorphism λ : (X, ρ) → (X, ρ) ∨ (again called a polarization).
Note that given (X, ρ) ∈ ChBT 2g,J there exist unique isomorphisms θ i : X i → X i+2g satisfying pθ i = ρ i,i+2g . Also note that there is a natural "forgetful morphism" A g,J → PolChBT 2g,J that even factors over PolChBT red 2g,J . We denote by PolChBT fml 2g,J ⊆ PolChBT 2g,J the substack consisting of those (X, ρ, λ) such that each X i is a formal p-divisible group and by A fml g,J ⊆ A g,J its preimage (this is a formal neighborhood of a closed subset of |A g,J |).
Construction 5.4. Given (X, ρ, λ) ∈ PolChBT 2g,J (R) we can apply D (from Subsection 2.5) to the data X i , ρ i,j , θ i , λ i . If R is reduced then using Theorem 2.14 we see that we obtain an object in PolChDisp Lemma 5.6. We have the following properties: (1) A g,J → PolChBT 2g,J is surjective (in fact A g,J (k) → PolChBT 2g,J (k) is essentially surjective for all k/F p algebraically closed) and formallyétale.
(2) Let B be an Artinian local F p -algebra B with algebraically closed residue field. Then we have PolChBT (2): This follows formally from (1) together with the reducedness of A g,J .

Spec(B)
PolChBT red 2g,J Spec(B ′ ) PolChDisp (∞,∞) g,J with B ′ → B a surjection of Artinian local F p -algebras with algebraically closed residue fields such that the given point in PolChBT red 2g,J (B) classifies a (polarized) chain of formal p-divisible groups (i.e. is contained in PolChBT fml 2g,J (B)). Then applying Theorem 2.14 we obtain a deformation in PolChBT 2g,J (B ′ ) such that applying D gives an object of PolChDisp (∞,∞) g,J (B ′ ) isomorphic to the given one. By the second part of this lemma, this deformation is actually contained in PolChBT red 2g,J (B ′ ) and thus gives a solution to the lifting problem.
We have a natural map of sets |PolChBT 2g,J | → B g sending (X, ρ, λ) ∈ PolChBT 2g,J (k) to the Newton polygon of any of the X i . We denote the fibers of this map by S BT ν and their preimages in |A g,J |, i.e. the Newton strata, by S ν .
Proposition 5.8. We have the following properties: • The S ν ⊆ |A g,J | are locally closed subsets.
Proof. This is all contained in [HR15]. See in particular Axiom 3.5, Theorem 5.6 and Section 7.
Lemma 5.9. Let ν, ν ′ ∈ B g with ν ≤ ν ′ and let x ∈ S BT ν ′ . Then there exists a preimage y ∈ S ν ′ of x that specializes to a point in S ν .
Proof. Choose a field k/F p and X = (X, ρ, λ) ∈ PolChBT 2g,J (k) representing x. By Proposition 5.8 there exists a point in S ν ′ specializing to a point in S ν . This specialization now can be realized by a point A ′ = (A ′ , ρ ′ , λ ′ ) ∈ A g,J (R) for some discrete valuation ring R over k. Write X ′ ∈ PolChBT 2g,J (R) for the image of A ′ , write K := Frac(R) and choose an algebraic closure K of K. Now X ′ K and X K both lie in S BT g,J,ν ′ , hence there exists an isogeny f : X ′ K → X K in ChBT g,J of height gN for some N ∈ Z ≥0 such that we have f * λ = p N · λ ′ . After replacing K by a finite field extension (inside K) we may assume that f is defined over K.
By construction A ′′ still realizes a specialization from S ν ′ to S ν and moreover f gives rise to an isomorphism X ′′ K → X K in PolChBT 2g,J K (where X ′′ ∈ PolChBT 2g,J (R) denotes the image of A ′′ ). Thus setting y ∈ S ν ′ ⊆ |A g,J | to be the point corresponding to A ′′ K finishes the proof.
Remark 5.10. Lemma 5.9 can be rephrased as follows: Given a central leaf C ⊆ |A g,J | (see [Oor02]) contained in a Newton stratum S ν ′ and some ν ≤ ν ′ we have C ∩ S ν = ∅.
Corollary 5.11. Every point x ∈ |PolChBT 2g,J | has a preimage y ∈ |A g,J | specializing to a point in |A fml g,J |.  has the ALP at all points in PolChBT fml 2g,J (k) for k/F p algebraically closed. Applying Lemma 4.4 and Lemma 4.5 we deduce that |A fml g,J | is contained in the smooth locus U ⊆ |A g,J | of υ (m,n) and that U = π −1 (V ) for some V ⊆ |PolChBT 2g,J |. Now Corollary 5.11 together with the openness of U ⊆ |A g,J | imply that V = |PolChBT 2g,J | and consequently that U = |A g,J | as desired.
Now C x → Spec(F p ) is a gerbe, hence smooth, so that we can deduce the smoothness of EKOR x over F p .