1 Introduction

This paper deals with the mod p geometry and arithmetic of some Shimura varieties at ramified places. More precisely, we study the reduction modulo p of the splitting models for PEL-type Shimura varieties constructed by Pappas-Rapoport in [Reference Pappas and Rapoport49]. For smooth splitting models, we explore the mod p Hodge structures on their special fibers by constructing universal F-zips with additional structure of fixed type (determined by the Hodge cocharacter). Then we derive some consequences for the geometry and coherent cohomology. In particular, we reprove the main results of Bijakowski-Hernandez [Reference Bijakowski and Hernandez6] by different methods. We also clarify and generalize the work of Reduzzi-Xiao [Reference Reduzzi and Xiao56] in the ramified Hilbert case.

PEL moduli spaces are central objects to study in arithmetic geometry and the Langlands program. If the associated reductive group is unramified at p, Kottwitz constructed smooth integral PEL moduli spaces in [Reference Kottwitz30]. In the general setting, Rapoport and Zink in [Reference Rapoport55] introduced similar and generalized integral PEL moduli spaces ${\mathscr {A}}^{\operatorname {\mathrm {naive}}}$ . These are called naive integral models, as shown by Pappas in [Reference Pappas45] that in some ramified unitary case, the moduli scheme ${\mathscr {A}}^{\operatorname {\mathrm {naive}}}$ fails to be flat over ${\mathcal {O}}_E$ , the ring of integers of the local reflex field E. To study integral models of PEL-type Shimura varieties in the ramified case, in [Reference Pappas and Rapoport49] Pappas and Rapoport introduced the so-called splitting integral models ${\mathscr {A}}^{\mathrm {spl}}$ . Roughly speaking, in the setting of [Reference Pappas and Rapoport49], the associated p-adic reductive group G has the form of a Weil restriction and the ramification comes mainly from restriction of scalars. After fixing the data, the model ${\mathscr {A}}^{\mathrm {spl}}$ is proposed as a relative moduli space over the naive integral model ${\mathscr {A}}^{\operatorname {\mathrm {naive}}}$ , the latter constructed in [Reference Rapoport55]. For any scheme S over ${\mathcal {O}}_E$ , ${\mathscr {A}}^{\operatorname {\mathrm {naive}}}(S)$ classifies abelian schemes with PEL structure $\underline {A}=(A,\lambda , \iota ,\alpha )$ over S. Fix a sufficiently large field extension $F|E$ . For a scheme S over ${\mathcal {O}}_F$ , ${\mathscr {A}}^{\mathrm {spl}}(S)$ classifies $(\underline {A}, \underline {\mathscr {F}_{\bullet }})$ , where $\underline {A}\in {\mathscr {A}}^{\operatorname {\mathrm {naive}}}(S)$ and $\underline {\mathscr {F}_{\bullet }}$ is a filtration on the cotangent bundle $\omega _{A/S}$ , satisfying certain conditions related to the ramification data. Pappas and Rapoport proved that the model ${\mathscr {A}}^{\mathrm {spl}}$ over ${\mathcal {O}}_F$ admits nice properties. In particular, there is a scheme $\mathbb {M}^{\mathrm {spl}}$ (called the splitting local model) over ${\mathcal {O}}_F$ , so that ${\mathscr {A}}^{\mathrm {spl}}$ and $\mathbb {M}^{\mathrm {spl}}$ sit in a local model diagram. Moreover, by construction $\mathbb {M}^{\mathrm {spl}}$ can be realized as a twisted product of unramified local models. There is a natural morphism ${\mathscr {A}}^{\mathrm {spl}}\rightarrow {\mathscr {A}}^{\operatorname {\mathrm {naive}}}$ , which is the composition of the forgetful morphism ${\mathscr {A}}^{\mathrm {spl}}\rightarrow {\mathscr {A}}^{\operatorname {\mathrm {naive}}}_{{\mathcal {O}}_F}$ over ${\mathcal {O}}_F$ and the base change projection map ${\mathscr {A}}^{\operatorname {\mathrm {naive}}}_{{\mathcal {O}}_F}\rightarrow {\mathscr {A}}^{\operatorname {\mathrm {naive}}}$ . Pappas and Rapoport defined the canonical model ${\mathscr {A}}$ as the scheme-theoretic image of this morphism. Then ${\mathscr {A}}$ is a flat integral model over ${\mathcal {O}}_E$ . In case that the group G is tamely ramified and $p\nmid |\pi _1(G_{der})|$ , then up to the Hasse principle the scheme ${\mathscr {A}}$ should coincide with the Kisin-Pappas integral models [Reference Kisin and Pappas28]. We refer to [Reference Pappas and Rapoport49] for more information and details on the general theory of splitting models (so far only available in the PEL-type case) and to [Reference Pappas46, Reference Pappas47, Reference Pappas and Rapoport51] for some recent progress on the canonical models.

The most well-known splitting models come from the example of Hilbert modular varieties. In this case, ${\mathscr {A}}={\mathscr {A}}^{\operatorname {\mathrm {naive}}}$ and the special fiber of ${\mathscr {A}}$ was previously studied by Deligne and Pappas in [Reference Deligne and Pappas13]. Let L be a totally real field of degree $g>1$ and p a prime number. Let $\kappa |\mathbb {F}_p$ be a large enough finite field and $\mathcal {M}^{\textrm {DP}}={\mathscr {A}}\otimes \kappa $ the Deligne-Pappas moduli space over $\kappa $ , which parameterizes abelian schemes with real multiplication given by ${\mathcal {O}}_L$ together with polarization and level structure. If p is unramified in L, this is a smooth scheme (a special case of the Kottwitz models [Reference Kottwitz30]). Here we are mainly concerned with the case where p ramifies in L. Then $\mathcal {M}^{\textrm {DP}}$ is only a normal scheme that is singular (cf. [Reference Deligne and Pappas13]). By contrast, the special fiber of ${\mathscr {A}}^{\mathrm {spl}}$ over $\kappa $ , in this case denoted by $\mathcal {M}^{\textrm {PR}}$ , is smooth and the natural morphism

$$\begin{align*}\mathcal{M}^{\textrm{PR}}\rightarrow \mathcal{M}^{\textrm{DP}}\end{align*}$$

is a resolution of singularities. In [Reference Reduzzi and Xiao56] Reduzzi and Xiao constructed g partial Hasse invariants on $\mathcal {M}^{\textrm {PR}}$ by carefully exploring the structure of Pappas-Rapoport filtrations. On the other hand, in the ramified case the number of partial Hasse invariants on $\mathcal {M}^{\textrm {DP}}$ is strictly less than g (see the introduction of [Reference Reduzzi and Xiao56] and the references therein). Reduzzi and Xiao applied these g partial Hasse invariants to $\mathcal {M}^{\textrm {PR}}$ to construct Galois pseudorepresentations attached to torsion Hecke eigenclasses in the coherent cohomology. This shows a big advantage to work with splitting models. The space $\mathcal {M}^{\textrm {PR}}$ and the partial Hasse invariants on it have been serving as a basic tool in the study of geometry and arithmetic of Hilbert modular varieties, for example, see the recent works of Sasaki [Reference Sasaki58] and Diamond-Kassaei [Reference Diamond and Kassaei14].

In the more general PEL setting, recently Bijakowski and Hernandez in [Reference Bijakowski and Hernandez6] studied some aspects of the mod p geometry of splitting models. More precisely, they proved that ${\mathscr {A}}^{\mathrm {spl}}$ with maximal level at p is smooth under some conditions on the PEL datum. Roughly, these conditions are to ensure that at a p-adic place $v_i$ , the group has the form $\operatorname {\mathrm {Res}}_{F_i|{\mathbb {Q}}_p}H_i$ , where $H_i$ is unramified over $F_i$ , and the level at p is hyperspecial for these $H_i$ . In particular, this excludes the ramified unitary groups (labeled as type (AR) in [Reference Bijakowski and Hernandez6]) as local factors. Bijakowski and Hernandez also proved that the $\mu $ -ordinary locus (defined as the maximal Newton stratum) is open and dense in the special fiber ${\mathscr {A}}^{\mathrm {spl}}_0$ of ${\mathscr {A}}^{\mathrm {spl}}$ . For this, they introduced a so-called Hodge stratification and proved that the maximal Hodge stratum (which contains the $\mu $ -ordinary locus) is open and dense.

In this paper, we essentially work in the same setting as that in [Reference Bijakowski and Hernandez6]. Our first key observation is that one can modify the local model diagram between ${\mathscr {A}}^{\mathrm {spl}}$ and $\mathbb {M}^{\mathrm {spl}}$ constructed in [Reference Pappas and Rapoport49] to a local model diagram between ${\mathscr {A}}^{\mathrm {spl}}$ and $\prod _{i,j.l}\mathbb {M}^{\mathrm {loc}}(\mathcal {G}_{i,j}^l,\mu _{i,j}^l)$ , a product of unramified local models related to the PEL datum, cf. Proposition A.8. This is not quite surprising, as we already mentioned above that the splitting local model $\mathbb {M}^{\mathrm {spl}}=\mathbb {M}^{\mathrm {spl}}(\mathcal {G},\mu )$ was constructed in [Reference Pappas and Rapoport49] as a twisted product of unramified local models $\mathbb {M}^{\mathrm {loc}}(\mathcal {G}_{i,j}^l,\mu _{i,j}^l)$ . Let G be the reductive group defined by the PEL datum, which we assume to be connected (thus we exclude the type D case), and $\{\mu \}$ the attached geometric conjugacy class of Hodge cocharacters of G. In the following we often fix a suitable choice of $\mu $ in this conjugacy class. Recall that $E|\mathbb {Q}_p$ is the field of definition of $\{\mu \}$ and $F|E$ is a large enough extension (so that G splits over F). The group scheme related to the modified local model diagram is $\mathcal {G}^{\mathrm {spl}}$ , a reductive group over ${\mathcal {O}}_F$ , which is the reductive model of the split group $G_F$ . On the other hand, let $\mathcal {G}$ be the parahoric model of G over $\mathbb {Z}_p$ associated to the integral PEL datum. The local model diagram of [Reference Pappas and Rapoport49] corresponds to a morphism of algebraic stacks

$$\begin{align*}{\mathscr{A}}^{\mathrm{spl}}\rightarrow [\mathbb{M}^{\mathrm{spl}}(\mathcal{G},\mu)/\mathcal{G}_{{\mathcal{O}}_F}],\end{align*}$$

which is more suited to study the canonical model ${\mathscr {A}}$ and its related canonical local model. On the other hand, the local model diagram here corresponds to a morphism

$$\begin{align*}{\mathscr{A}}^{\mathrm{spl}}\rightarrow [\mathbb{M}^{\mathrm{loc}}(\mathcal{G}^{\mathrm{spl}},\mu)/\mathcal{G}^{\mathrm{spl}}],\end{align*}$$

where $ \mathbb {M}^{\mathrm {loc}}(\mathcal {G}^{\mathrm {spl}},\mu )=\prod _{i,j,l}\mathbb {M}^{\mathrm {loc}}(\mathcal {G}_{i,j}^l,\mu _{i,j}^l).$ If there is an i such that the finite extension $F_i|\mathbb {Q}_p$ is ramified, then in general

$$\begin{align*}\mathcal{G}_{{\mathcal{O}}_F}\neq \mathcal{G}^{\mathrm{spl}}.\end{align*}$$

From the modified diagram here, we can immediately deduce the smoothness of ${\mathscr {A}}^{\mathrm {spl}}$ if the local factors $\mathcal {G}_i$ satisfy the same condition as that in [Reference Bijakowski and Hernandez6] (compare [Reference Bijakowski and Hernandez6] Theorem 2.30 and Remark 2.31). Under this condition, the parahoric subgroup $K_p=\mathcal {G}(\mathbb {Z}_p)$ is in fact very special in the sense of [Reference Zhu71].

In fact, the above observation on the modified local model diagram leads us to go much further. Recall that the theory of F-zips was introduced and studied by Moonen-Wedhorn in [Reference Moonen and Wedhorn43] as a candidate for a mod p Hodge structure. This notion has been promoted and enlarged by Fontaine-Jannsen [Reference Fontaine and Jannsen17], Drinfeld [Reference Drinfeld15], and Bhatt-Lurie [Reference Bhatt9]. Here we work with the generalization of F-zips in another direction: the notion of G-zips for a reductive group G over a finite field, cf. [Reference Pink, Wedhorn and Ziegler53, Reference Pink, Wedhorn and Ziegler54]. Back to splitting models of PEL-type Shimura varieties, we will work with smooth splitting models ${\mathscr {A}}^{\mathrm {spl}}$ from now on (in particular, this excludes the ramified unitary group case). Let $\kappa $ be the residue field of ${\mathcal {O}}_F$ and ${\mathscr {A}}^{\mathrm {spl}}_0$ the special fiber of ${\mathscr {A}}^{\mathrm {spl}}$ over $\kappa $ .

Let us first comment on the related reductive groups appearing here. For ${\mathscr {A}}^{\mathrm {spl}}_0$ , the group $\mathcal {G}^{\mathrm {spl}}_0$ replaces $\mathcal {G}_0^{\textrm {rdt}}$ , the maximal reductive quotient of the special fiber $\mathcal {G}_0$ of $\mathcal {G}$ . The group $\mathcal {G}_0^{\textrm {rdt}}$ is mainly related to the geometry of ${\mathscr {A}}_0$ , the special fiber of the canonical model ${\mathscr {A}}$ . Recall that there is a natural morphism ${\mathscr {A}}^{\mathrm {spl}}_0\rightarrow {\mathscr {A}}_0$ . The difference between $\mathcal {G}^{\mathrm {spl}}_0$ and $\mathcal {G}_0^{\textrm {rdt}}$ reflects the ramification data. Indeed, at a p-adic place $v_i$ , the local factors of the two groups admit the following description:

$$\begin{align*}\mathcal{G}^{\mathrm{spl}}_{0,i}=\operatorname{\mathrm{Res}}_{\kappa_i|{\mathbb{F}}_p}H_i^{e_i},\quad \mathcal{G}_{0,i}^{\textrm{rdt}}= \operatorname{\mathrm{Res}}_{\kappa_i|{\mathbb{F}}_p}H_i,\end{align*}$$

where $\kappa _i$ is the residue field of $F_i$ , $e_i$ is the ramification index of the extension $F_i|\mathbb {Q}_p$ , and $H_i$ is a reductive group over $\kappa _i$ . Note that the Hodge cocharacter $\mu $ of G over F naturally admits a reduction to a cocharacter of $\mathcal {G}^{\mathrm {spl}}_0$ over $\kappa $ .

Now we briefly explain the construction of the universal $\mathcal {G}^{\mathrm {spl}}_0$ -zip of type $\mu $ . In the unramified PEL case, we have ${\mathscr {A}}_0={\mathscr {A}}^{\operatorname {\mathrm {naive}}}_0={\mathscr {A}}^{\mathrm {spl}}_0$ and the construction is direct, cf. [Reference Moonen and Wedhorn43, Reference Viehmann and Wedhorn65] and [Reference Zhang70]. However, in the ramified case, the construction becomes rather indirect and complicated. Since a general PEL datum is involved, we illustrate the ideas by working with the example of Hilbert-Siegel case. For the general case, see sections 2 and 3. Then the PEL datum $(B, \ast , V, \left \langle \cdot , \cdot \right \rangle , {\mathcal {O}}_B,\Lambda )$ is of type C and $B=L$ is a totally real field, which we assume $[L:\mathbb {Q}]>1$ . We further assume that there is only one finite place v of L over p and the local field $F_1:=L_v$ is ramified over $\mathbb {Q}_p$ , i.e., the ramification index $e=[F_1:F_1^{\textrm {ur}}]>1$ , where $F_1^{\textrm {ur}}$ is the maximal unramified subextension of $F_1$ over $\mathbb {Q}_p$ . Let $f=[F_1^{\textrm {ur}}:\mathbb {Q}_p]=[\kappa _1:\mathbb {F}_p]$ , where $\kappa _1$ is the residue field of $F_1$ , so that $[F_1:\mathbb {Q}_p]=ef$ . Let $K^p\subset G(\mathbb {A}_f^p)$ be a fixed sufficiently small open compact subgroup. Recall $F|F_1$ is a fixed large enough extension with residue field $\kappa $ . For any $\kappa $ -scheme S, let $\underline {A}=(A, \lambda , \iota , \alpha )$ be an S-point of $\mathscr {A}^{\operatorname {\mathrm {naive}}}=\mathscr {A}^{\operatorname {\mathrm {naive}}}_{K^p}$ . Then we have an exact sequence of locally free sheaves of ${\mathcal {O}}_S$ -modules

$$\begin{align*}0 \to \omega_{A/S} \to H_{\operatorname{\mathrm{dR}}}^1(A/S) \to \operatorname{\mathrm{Lie}}_{A^\vee/S} \to 0.\end{align*}$$

In the unramified case, this exact sequence, together with the Frobenius and Verschiebung morphisms, defines the F-zip with additional structure of type $\mu $ over ${\mathscr {A}}^{\mathrm {spl}}_0$ . However, in our ramified case here, it turns out a posteriori that the type of the Hodge filtrations constructed in this way will vary, which reflects the singularities of ${\mathscr {A}}_0$ in some sense.

Indeed, we have the Kottwitz-Rapoport stratification on ${\mathscr {A}}_0$ (see later), and on each stratum by [Reference Shen, Yu and Zhang59] we have a $\mathcal {G}_0^{\textrm {rdt}}$ -zip of a certain type (dependent on this stratum). Recall the natural map ${\mathscr {A}}_0^{\mathrm {spl}}\rightarrow {\mathscr {A}}_0$ . We can pullback these $\mathcal {G}_0^{\textrm {rdt}}$ -zips to the corresponding preimages of KR strata in ${\mathscr {A}}_0^{\mathrm {spl}}$ . But these do not give the correct object that we want. Here, to construct the $\mathcal {G}^{\mathrm {spl}}_0$ -zip of fixed type $\mu $ over ${\mathscr {A}}^{\mathrm {spl}}_0$ , we have to take into account the splitting structures. It is here that our modified local model diagram (over $\kappa $ )

$$\begin{align*}{\mathscr{A}}^{\mathrm{spl}}_0\rightarrow [\mathbb{M}^{\mathrm{loc}}(\mathcal{G}^{\mathrm{spl}},\mu)_0/\mathcal{G}^{\mathrm{spl}}_{\kappa}]\end{align*}$$

plays the key role. Under our smoothness assumption, the underlying topological space of $[\mathbb {M}^{\mathrm {loc}}(\mathcal {G}^{\mathrm {spl}},\mu )_0/\mathcal {G}^{\mathrm {spl}}_{\kappa }]$ has only one point. Then the construction is guided by the constructions in [Reference Shen, Yu and Zhang59] subsections 3.3 and 3.4. On the other hand, we remark that it is very hard to work with the original local model diagram ${\mathscr {A}}^{\mathrm {spl}}_0\rightarrow [\mathbb {M}^{\mathrm {spl}}(\mathcal {G},\mu )_0/\mathcal {G}_{\kappa }]$ of [Reference Pappas and Rapoport49]. The quotient stack $[\mathbb {M}^{\mathrm {spl}}(\mathcal {G},\mu )_0/\mathcal {G}_{\kappa }]$ is quite complicated, and in fact it is not known whether the set of ${\mathcal {G}}_\kappa $ -orbits on $\mathbb {M}^{\mathrm {spl}}(\mathcal {G},\mu )_0$ is finite or not in general (cf. [Reference Berger3] section 8 and [Reference Berger4]).

Let us discuss in more detail the construction of the $\mathcal {G}^{\mathrm {spl}}_0$ -zip of fixed type $\mu $ over ${\mathscr {A}}^{\mathrm {spl}}_0$ . Let $(\underline {A}, \underline {\mathscr {F}_{\bullet }})$ be an S-valued point of ${\mathscr {A}_0}^{\mathrm {spl}}$ . Then by definition $\underline {A}=(A, \lambda , \iota ,\alpha )\in \mathscr {A}^{\operatorname {\mathrm {naive}}}(S)$ , $\underline {\mathscr {F}_{\bullet }}=\{\mathscr {F}_{j}^l\}$ is a Pappas-Rapoport filtration of the ${\mathcal {O}}_{F_1} \otimes _{\mathbb {Z}_p} {\mathcal {O}}_S$ -module $\omega _{A/S}$ . To explain this term, we write $\mathcal {H} =H_{\operatorname {\mathrm {dR}}}^1(A/S)$ , which is an ${\mathcal {O}}_{F_1^{\textrm {ur}}} \otimes _{\mathbb {Z}_p} {\mathcal {O}}_S$ -module, hence it has a decomposition

$$\begin{align*}\mathcal{H} = \bigoplus_{j} \mathcal{H}_{j},\end{align*}$$

where $\mathcal {H}_{j}$ is the locally free sub ${\mathcal {O}}_S$ -module of $\mathcal {H}$ , where ${\mathcal {O}}_{F_1^{\textrm {ur}}}$ acts by the fixed embedding $\sigma _j: {\mathcal {O}}_{F_1^{\textrm {ur}}}\rightarrow {\mathcal {O}}_F$ . For each j, there is a pairing on $\mathcal {H}_j$ induced from the pairing on $\mathcal {H}$ coming from the polarization $\lambda $ . Similarly, we have a decomposition $\omega _{A/S}=\bigoplus _j\omega _j$ with $\omega _j\subset \mathcal {H}_j$ for each $1\leq j\leq f$ . The Pappas-Rapoport filtration is by definition a filtration of locally direct ${\mathcal {O}}_S$ -factors of each $\omega _{j}$ .

$$\begin{align*}0 = \mathscr{F}_{j}^0 \subset \mathscr{F}_{j}^1 \subset \cdots \subset \mathscr{F}_{j}^{e} = {\omega_{j}} \subset \mathcal{H}_{j}\end{align*}$$

with ${\mathcal {O}}_{F_1}/ (p) \simeq \kappa _1[T]/(T^{e}) = \kappa _1[\varepsilon _1]$ action, such that $\kappa _1$ acts on $\mathscr {F}_{j}^l$ by $\sigma _{j}: \mathcal {O}_{F_1^{\mathrm {ur}}} \to \kappa $ ; for all $1 \leq l \leq e$ , $\mathscr {F}_{j}^l / \mathscr {F}_{j}^{l-1}$ is locally free of rank $d_{j}^l$ and $\varepsilon _1 \mathscr {F}_{j}^l \subset \mathscr {F}_{j}^{l-1}$ .

The ${\mathcal {O}}_{F_1^{\textrm {ur}}}$ -action on $\Lambda $ induces a decomposition $\Lambda =\bigoplus _{j}\Lambda _j$ . For each embedding $\sigma ^l_j: {\mathcal {O}}_{F_1}\hookrightarrow {\mathcal {O}}_F$ which extends $\sigma _j: {\mathcal {O}}_{F_1^{\textrm {ur}}}\hookrightarrow {\mathcal {O}}_F$ , we set $\Lambda _j^l=\Lambda _j\otimes _{{\mathcal {O}}_{F_1},\sigma ^l_j}{\mathcal {O}}_F$ (note that the lattice $\Lambda _{j}^l$ is denoted by $\Xi _{j}^l$ in [Reference Pappas and Rapoport49, Proposition 5.2]) and $\Lambda ^l=\bigoplus _j\Lambda _j^l$ . Set $ \Lambda _{j,0}^l=\Lambda _j^l\otimes _{{\mathcal {O}}_F}\kappa $ and $\Lambda ^{\mathrm {spl}}_0=\bigoplus _l\bigoplus _j\Lambda _{j,0}^l$ . One can show that there is a standard F-zip structure on $\Lambda ^{\mathrm {spl}}_0$ , whose Hodge filtration is given by the cocharacter $\mu $ over $\kappa $ .

For $S={\mathscr {A}_0}^{\mathrm {spl}}$ and $(\underline {{A}}, \underline {\mathscr {F}_{\bullet }}) \in {\mathscr {A}_0}^{\mathrm {spl}}(S)$ the universal object, one can associate a module

$$\begin{align*}\mathcal{M}:= \bigoplus_{ l=1}^e \bigoplus_{ j=1}^f \mathcal{M}_{j}^l, \quad\text{ with }\quad\mathcal{M}_{j}^l := \varepsilon_1^{-1}\mathscr{F}_{j}^{l-1} / \mathscr{F}_{j}^{l-1}.\end{align*}$$

Then each $\mathcal {M}_{j}^l$ is a locally free ${\mathcal {O}}_S$ -module, and locally isomorphic to $\Lambda _{j}^l \otimes _{{\mathcal {O}}_F} {\mathcal {O}}_S$ , cf. [Reference Pappas and Rapoport49]. Now the idea is to transfer the Hodge and conjugate filtrations on $\mathcal {H}$ to filtrations on each $\mathcal {M}^l:=\bigoplus _j\mathcal {M}_{j}^l$ , and to show that the F-zip structure on $\mathcal {H}$ induces an F-zip structure on $\mathcal {M}$ . To this end, one can first complete the filtration of $\omega $ to a full filtration of $\mathcal {H}$ , then apply the Frobenius and Verschiebung morphisms successively to the full filtration, and make use of the usual relation $\textrm {Ker}\, F=\textrm {Im}\, V, \textrm {Ker}\, V=\textrm {Im}\, F$ to succeed. In fact, we proceed by an alternative and equivalent approach. For any $1\leq j \leq f$ , inspired by [Reference Reduzzi and Xiao56] we construct explicit linear maps $F_{j}^l: \mathcal {M}_{j}^{l-1} \to \mathcal {M}_{j}^{l},\; V_{j}^l: \mathcal {M}_{j}^l \to \mathcal {M}_{j}^{l-1}$ for each $2 \leq l \leq e$ (which are easy to construct), and $\sigma ^{-1}$ -linear ( $\sigma $ -linear) maps $F_{j}^1: \mathcal {M}_{j-1}^{e} \to \mathcal {M}_{j}^1,\; V_{j}^1:\mathcal {M}_{j}^1 \to \mathcal {M}_{j-1}^{e}$ (which are induced by the Frobenius and Verschiebung, and the definition of $V_{j}^1$ is a little subtle). Now for each $1\leq l\leq e$ , $\mathcal {M}^l = \bigoplus _{j=1}^f \mathcal {M}_{j}^l$ is a locally free ${\mathcal {O}}_S[\varepsilon _1]$ -module with $\kappa _1$ action, and locally ${\mathcal {M}}^l \simeq \Lambda ^l \otimes _{{\mathcal {O}}_F} {\mathcal {O}}_S.$ For each l, set $F^l=\bigoplus _jF_{j}^l$ and $V^l=\bigoplus _jV_{j}^l$ . We have the following linear morphisms:

and semilinear morphisms ( $F^{1}$ is $\sigma $ -linear and $V^{1}$ is $\sigma ^{-1}$ -linear)

Then (see Lemmas 3.7 and 3.6 respectively)

1. (1) For $2\leq l\leq e$ , we have $\operatorname {\mathrm {Im}}\,F^l = \operatorname {\mathrm {Ker}}\,V^l, \, \operatorname {\mathrm {Ker}}\,F^l = \operatorname {\mathrm {Im}}\,V^l.$

2. (2) There is a canonical isomorphism $g: {\mathcal {M}}^1 \to {\mathcal {M}}^1$ sending $\operatorname {\mathrm {Ker}}\,V^1$ to $\operatorname {\mathrm {Im}}\,F^1$ .

Therefore we get an F-zip

$$\begin{align*}({\mathcal{M}} = \bigoplus_l {\mathcal{M}}^l, C = \bigoplus_l \operatorname{\mathrm{Ker}}\,F^l, D = \bigoplus_l \operatorname{\mathrm{Ker}}\,V^l, \varphi_{\bullet})\end{align*}$$

over ${\mathscr {A}}_0^{\mathrm {spl}}$ . Moreover, there are natural compatible symplectic forms and ${\mathcal {O}}_L$ -action on $\mathcal {M}$ . Comparing with the standard F-zip structure on $\Lambda ^{\mathrm {spl}}_0$ , we get the $\mathcal {G}^{\mathrm {spl}}_0$ -zip of type $\mu $ over ${\mathscr {A}}_0^{\mathrm {spl}}$ . Roughly, the process of transforming $\mathcal {H}$ to ${\mathcal {M}}$ is a way of “semisimplification”. During this transformation the different types of $\mathcal {G}^{\operatorname {\mathrm {rdt}}}_0$ -zips become uniform as $\mathcal {G}^{\mathrm {spl}}_0$ -zips, which reflects some key features of the resolution of singularities ${\mathscr {A}}_0^{\mathrm {spl}}\rightarrow {\mathscr {A}}_0$ . Note that even in the Hilbert case, the above construction reveals more complete geometric information than that in [Reference Reduzzi and Xiao56]. For more details, see Proposition 3.8 and subsection 3.2.

Let $k=\overline {\kappa }$ be an algebraic closure of $\kappa $ . By Theorem 1.1, we get an induced stratification

$$\begin{align*}{\mathscr{A}}^{\mathrm{spl}}_k=\coprod_{w\in {}^JW}{\mathscr{A}}^{\mathrm{spl},w}_k\end{align*}$$

which we call the Ekedahl-Oort (EO) stratification. Here the index set ${}^JW$ is the subset of the (absolute) Weyl group of $\mathcal {G}^{\mathrm {spl}}_0$ defined by the cocharacter $\mu $ in the usual way, which is equipped with the partial order $\preceq $ as that in [Reference Pink, Wedhorn and Ziegler53, Reference Pink, Wedhorn and Ziegler54]. By Theorem 1.1, each stratum ${\mathscr {A}}^{\mathrm {spl},w}_k$ is non-empty, smooth, and we have the usual closure relation for the EO strata given by $\preceq $ . In fact, the nonemptiness is more subtle, requiring additional efforts. We prove the nonemptiness by comparing the minimal EO stratum of ${\mathscr {A}}^{\mathrm {spl}}_k$ with the minimal EKOR stratum of ${\mathscr {A}}_k$ (see below) in Proposition 3.11, and show the minimal EKOR stratum is nonempty in the Appendix (Proposition A.15) by adapting the method of He-Zhou [Reference He and Zhou26].

We can compare the EO strata with some other naturally raised strata. Recall the natural morphism

$$\begin{align*}\pi:{\mathscr{A}}^{\mathrm{spl}}_k\rightarrow {\mathscr{A}}_k,\end{align*}$$

which is a resolution of singularities: as in the Hilbert case ([Reference Deligne and Pappas13]) the scheme ${\mathscr {A}}_k$ is usually singular in the ramified case. On ${\mathscr {A}}_k$ , we have the following stratifications:

• • Newton stratification, which can be constructed by studying the associated F-isocrystals with additional structure.

• • Kottwitz-Rapoport (KR) stratification ${\mathscr {A}}_k=\coprod _{w\in \textrm {Adm}(\mu )_{K_p}}{\mathscr {A}}^{w}_k$ , which is induced from the local model diagram for ${\mathscr {A}}_k$ and the geometry of $\mathbb {M}^{\mathrm {loc}}_k$ , cf. [Reference Pappas and Rapoport49, Reference Levin40, Reference Haines and Richarz21].

• • Ekedahl-Kottwitz-Oort-Rapoport (EKOR) stratification ${\mathscr {A}}_k=\coprod _{x\in {}^{K_p}\textrm {Adm}(\mu )}{\mathscr {A}}^{x}_k$ , which can be constructed as in [Reference Shen, Yu and Zhang59], see the Appendix of the current paper; in particular, see Example A.11 for more information on EKOR strata in the Hilbert case.

While on ${\mathscr {A}}^{\mathrm {spl}}_k$ we have the following stratifications:

• • Newton stratification, which can be constructed by forgetting the splitting structures and considering the associated F-isocrystals with additional structure,

• • Hodge stratification, which is constructed in [Reference Bijakowski and Hernandez6],

• • Ekedahl-Oort stratification, which is constructed in this paper.

The Newton stratification on ${\mathscr {A}}^{\mathrm {spl}}_k$ naturally factors through ${\mathscr {A}}_k$ . In subsection 3.6 we will show that the Hodge stratification also factors through ${\mathscr {A}}_k$ . Moreover, the resulting stratification on ${\mathscr {A}}_k$ is coarser than the Kottwitz-Rapoport stratification. On the other hand, the Ekedahl-Oort stratification on ${\mathscr {A}}^{\mathrm {spl}}_k$ does not factor through ${\mathscr {A}}_k$ in general. Nevertheless, one can try to compare the EO strata of ${\mathscr {A}}^{\mathrm {spl}}_k$ with pullbacks of KR and EKOR strata of ${\mathscr {A}}_k$ under $\pi $ . On each KR stratum ${\mathscr {A}}_k^w$ , we have a $\mathcal {G}_0^{\textrm {rdt}}$ -zip of type $J_w$ , see [Reference Shen, Yu and Zhang59] and our Appendix. Pulling back to ${\mathscr {A}}^{\mathrm {spl}}_k$ , we get a family of $\mathcal {G}_0^{\textrm {rdt}}$ -zips of different types. It is curious to study the relationship with our $\mathcal {G}^{\mathrm {spl}}_0$ -zip of fixed type $\mu $ . In doing so, we prove the following results.

In particular, using a result of He-Nie [Reference He and Nie24], we can deduce that the maximal EO stratum coincides with the $\mu $ -ordinary locus, cf. Corollary 3.17. Thus by Theorem 1.1 we reprove the open density of the $\mu $ -ordinary locus of ${\mathscr {A}}^{\mathrm {spl}}_k$ , cf. [Reference Bijakowski and Hernandez6].

Once we have the morphism

$$\begin{align*}\zeta: {\mathscr{A}}^{\mathrm{spl}}_0\rightarrow \mathcal{G}^{\mathrm{spl}}_0\textrm{-Zip}^\mu_{\kappa},\end{align*}$$

we get Hasse invariants on $ {\mathscr {A}}^{\mathrm {spl}}_0$ by pulling back the group-theoretic Hasse invariants on the stack $\mathcal {G}^{\mathrm {spl}}_0\textrm {-Zip}^\mu _{\kappa }$ constructed by Goldring-Koskivirta in [Reference Goldring and Koskivirta20]. Note that in [Reference Bijakowski and Hernandez5] Bijakowski and Hernandez have constructed Hasse invariants for the $\mu $ -ordinary locus (which is also our maximal EO stratum) by an explicit method. It would be interesting to compare their construction with our construction here. On the other hand, in the Hilbert case, in subsection 4.4 we do compare the EO strata and Hasse invariants here with those constructed in [Reference Reduzzi and Xiao56].

To proceed as in [Reference Reduzzi and Xiao56] and [Reference Goldring and Koskivirta20] to get applications to Galois representations, we need arithmetic compactifications of splitting models and we have to extend the above construction to the compactifications. Fortunately, the arithmetic (toroidal and minimal) compactifications for splitting models with good properties have already been established by Lan in [Reference Lan37], based on his previous constructions in [Reference Lan33, Reference Lan35, Reference Lan36]. At this point, we have to slightly modify the integral model ${\mathscr {A}}^{\mathrm {spl}}={\mathscr {A}}^{\mathrm {spl}}_{K^p}$ by considering the open and closed subspace

$$\begin{align*}\mathsf{M}_K^{\mathrm{spl}}\subset {\mathscr{A}}^{\mathrm{spl}}_{K^p}\end{align*}$$

studied in [Reference Lan37]. Here $K=K^pK_p$ and $K_p=\mathcal {G}(\mathbb {Z}_p)$ . All previous constructions and results also hold for $\mathsf {M}_K^{\mathrm {spl}}$ by simple modifications. For projective and smooth compatible collections of cone decompositions $\Sigma $ , by [Reference Lan37] we have a toroidal compactification $\mathsf {M}_{K, \Sigma }^{\mathrm {spl},\mathrm {tor}}$ , together with a proper surjective morphism to the minimal compactification $\oint : \mathsf {M}_{K, \Sigma }^{\mathrm {spl},\mathrm {tor}}\rightarrow \mathsf {M}_{K}^{\mathrm {spl},\mathrm {min}}$ . As we are working with smooth $\mathsf {M}_K^{\mathrm {spl}}$ , the compactification $\mathsf {M}_{K, \Sigma }^{\mathrm {spl},\mathrm {tor}}$ is also smooth by [Reference Lan37] Proposition 3.4.14. We write their special fibers as $\mathsf {M}^{\mathrm {spl}}_{K,0}$ and $\mathsf {M}^{\mathrm {spl},\mathrm {tor}}_{K,\Sigma ,0}$ .

The extension of ${\mathcal {G}}^{\mathrm {spl}}_0$ -zip is easy to construct, as the canonical extension of the universal de Rham bundle together with its Hodge filtration has already been given by Lan in [Reference Lan37], so we just repeat the above construction. To prove the smoothness of $\zeta ^{\mathrm {tor}}$ , we adapt some ideas of Andreatta [Reference Andreatta2] in the unramified Hodge-type case. We also prove that EO strata here are well-positioned in the sense of Lan-Stroh [Reference Lan and Stroh38]. By the well-positionedness in fact we get another proof of the smoothness of $\zeta ^{\mathrm {tor}}$ (this is the approach taken in [Reference Lan and Stroh38] for the unramified PEL case).

Before talking about arithmetic applications, we note another advantage of the smooth splitting models $\mathsf {M}_K^{\mathrm {spl}}$ . As the modified local model diagram for $\mathsf {M}_K^{\mathrm {spl}}$ is similar to that in the unramified case, we get naturally automorphic vector bundles on $\mathsf {M}_K^{\mathrm {spl}}$ as in the classical procedure. In contrast, automorphic vector bundles do not extend to the singular integral model $\mathsf {M}_K$ , the corresponding open and closed subspace of ${\mathscr {A}}={\mathscr {A}}_{K^p}$ . We also get canonical and subcanonical extensions of automorphic vector bundles to $\mathsf {M}_{K, \Sigma }^{\mathrm {spl},\mathrm {tor}}$ , as the local model diagram extends. Moreover, pullbacks of canonical extensions of automorphic vector bundles satisfy similar properties as in the unramified case (cf. [Reference Lan34] Proposition 5.6). In particular, we have the relative vanishing of higher direct images for the projection $\oint : \mathsf {M}_{K, \Sigma }^{\mathrm {spl},\mathrm {tor}}\rightarrow \mathsf {M}_{K}^{\mathrm {spl},\mathrm {min}}$ , cf. Proposition 5.3 and [Reference Lan37] Theorem 4.4.9.

Theorems 1.1 and 1.3 allow us to deduce some further consequences on the mod p geometry and arithmetic related to PEL-type Shimura varieties with ramifications arising from Weil restrictions. As an illustration, we discuss congruences of mod p automorphic forms and Galois representations associated to torsion classes in coherent cohomology. For some other possible applications, see [Reference Diamond and Kassaei14] and [Reference Wedhorn and Ziegler66]. From Theorem 1.3, we get extensions of Hasse invariants to toroidal compactifications. Then we can apply Goldring-Koskivirta’s machinery of Hasse-regular sequences introduced in [Reference Goldring and Koskivirta20], which formalizes some key properties of the subschemes defined by Hasse invariants. Indeed, the key technical conditions 6.4.2 and 7.1.2 of [Reference Goldring and Koskivirta20] have been verified in our setting as above. Let S be the finite set of primes $\ell $ where $K_\ell $ is not hyperspecial. Let $P_\mu \subset G_F$ be the parabolic subgroup associated to the cocharacter $\mu : \mathbb {G}_{m,F}\rightarrow G_F$ in the usual way (see the paragraph above Corollary 2.4), $\mathbf {L}\subset P_\mu $ the Levi subgroup and $V_\eta \in \mathrm {Rep}_F\mathbf {L}$ an irreducible representation of highest weight $\eta $ . Using our modified local model diagram, we get the associated integral automorphic vector bundle $\mathcal {V}_\eta $ on $\mathsf {M}_K^{\mathrm {spl}}$ and the subcanonical extension $\mathcal {V}_\eta ^{\mathrm {sub}}$ of $\mathcal {V}_\eta $ to $\mathsf {M}_{K, \Sigma }^{\mathrm {spl},\mathrm {tor}}$ . For each integers $i\geq 0, n\geq 1$ , the Hecke algebra $\mathcal {H}^S$ acts on the cohomology groups

$$\begin{align*}H^i(\mathsf{M}_{K, \Sigma,{\mathcal{O}}_F/\varpi^n}^{\mathrm{spl},\mathrm{tor}}, \mathcal{V}_\eta^{\mathrm{sub}}),\end{align*}$$

where $\varpi $ is a uniformizer of ${\mathcal {O}}_F$ . Let $\mathcal {H}^{i,n}_\eta $ be its image in the endomorphism algebra. Fix a representation $r: {}^LG\rightarrow \mathrm {GL}_m$ of the Langlands dual group over $\mathbb {C}$ (and fix an isomorphism $\mathbb {C}\simeq \overline {\mathbb {Q}}_p$ ). With all the geometric ingredients at hand, one can prove the following theorem by the same arguments as in [Reference Goldring and Koskivirta20].

As [Reference Goldring and Koskivirta20], this theorem is a consequence of a result that the Hecke action on $H^i$ factorizes through $H^0$ with increased weight, which is in turn deduced by the machinery of Hasse invariants on toroidal compactifications, see Theorem 6.1.

For the unramified PEL-type case, Theorem 1.4 is due to [Reference Goldring and Koskivirta20] (the key technical conditions 6.4.2 and 7.1.2 there hold for the unramified PEL type case), which recovers the main result of [Reference Emerton, Reduzzi and Xiao16] (for the unramified Hilbert case), and completes the work of Boxer [Reference Boxer7] by a different approach. We refer to [Reference Goldring and Koskivirta20] subsection 10.1 for further discussions on the appearance of nondegenerate limit of discrete series in higher coherent cohomology of Shimura varieties. Recall that since $K_p\subset G(\mathbb {Q}_p)$ is a very special parahoric subgroup, irreducible smooth representations $\pi _p$ of $G(\mathbb {Q}_p)$ such that $\pi ^{K_p}_p\neq 0$ can be classified by their spherical parameters, see [Reference Zhu71] section 6. The significance of Theorem 1.4 lies in the flexibility it provides for the construction of automorphic Galois representations, in the sense that we allow the p-component of the automorphic representation $\pi $ to be ramified as $\pi ^{K_p}_p\neq 0$ . It reproves [Reference Reduzzi and Xiao56] Theorem 1.1 in the Hilbert case by a different method. We also mention that a similar result has been given in [Reference Pilloni and Stroh52] in the ramified setting, but there the integral structure is given by the strange integral models introduced by Scholze using perfectoid geometry. As all these works demonstrate, Theorem 1.4 should be useful when studying modularity lifting problems, cf. [Reference Calegari and Geraghty10].

In this paper we restrict to the PEL case, since so far splitting models for Shimura varieties have only been constructed in this setting ([Reference Pappas and Rapoport49]). It would be interesting and useful to construct splitting models in more general Hodge and abelian type cases. This would certainly require quite different ideas and independent treatments. Once more general splitting models are available, we do expect that our constructions in this paper would naturally extend.

This paper is organized as follows. In section 2, we fix the PEL datum and discuss the corresponding Pappas-Rapoport splitting models. In particular, under a basic assumption on the datum we prove the smoothness of splitting models by a modified local model diagram. In section 3, we construct the universal $\mathcal {G}^{\mathrm {spl}}_0$ -zip of type $\mu $ over the smooth special fiber ${\mathscr {A}}^{\mathrm {spl}}_0$ and study the induced Ekedahl-Oort stratification. As an application, we prove the open density of the $\mu $ -ordinary locus. Along the way we also show that the Hodge stratification constructed in [Reference Bijakowski and Hernandez6] descends to ${\mathscr {A}}_{0}$ . In section 4, we discuss Hasse invariants on ${\mathscr {A}}^{\mathrm {spl}}_0$ , by pulling back those on the zip stack constructed by Goldring-Koskivirta. We also discuss some concrete examples, in particular we compare the Hasse invariants and EO strata here with those constructed by Reduzzi-Xiao in [Reference Reduzzi and Xiao56] in the Hilbert case. In section 5, we first review Lan’s constructions of arithmetic compactifications for splitting models, then we extend the universal $\mathcal {G}^{\mathrm {spl}}_0$ -zip to the smooth toroidal compactifications $\mathsf {M}_{K, \Sigma ,0}^{\mathrm {spl},\mathrm {tor}}$ and prove the smoothness of the induced morphism $\zeta ^{\mathrm {tor}}$ . In section 6, we apply the method of Goldring-Koskivirta [Reference Goldring and Koskivirta20] to deduce similar consequences on Hecke algebras and Galois representations associated to torsion classes in coherent cohomology of smooth splitting models. Finally, in the Appendix we discuss related local model diagrams for general parahoric levels, and briefly review the construction of EKOR stratification on ${\mathscr {A}}_{0}$ .