The Global Structure of Moduli Spaces

of Polarized p-Divisible Groups

Eva Viehmann

Received: February 26, 2008 Revised: December 2, 2008

Communicated by Peter Schneider

Abstract. We study the global structure of moduli spaces of quasiisogenies of polarized p-divisible groups introduced by Rapoport and Zink. Using the corresponding results for non-polarized p-divisible groups from a previous paper, we determine their dimensions and their sets of connected components and of irreducible components.

2000 Mathematics Subject Classification: 14L05, 14K10 (Primary) 14G35 (Secondary)

1 Introduction

Let k be an algebraically closed field of characteristic p > 2. Let W = W (k) be its ring of Witt vectors and L = Quot(W). Let σ be the Frobenius automorphism on k as well as on W. By Nilp_{W} we denote the category of schemes S over Spec(W) such that p is locally nilpotent on S. Let \overline{S} be the closed subscheme of S that is defined by the ideal sheaf pO_{S}. Let (𝕏, λ_{𝕏}) be a principally polarized p-divisible group over k. If X is a p-divisible group, we denote its dual by X̂. Then the polarization λ_{𝕏} is an isomorphism 𝕏 → 𝕏̂. We consider the functor

M : Nilp_{W} → Sets,

which assigns to S ∈ Nilp_{W} the set of isomorphism classes of pairs (X, ρ), where X is a p-divisible group over S and ρ : 𝕏\overline{S} = 𝕏 ×_{Spec(k)} \overline{S} → X ×_{S} \overline{S} is a quasi-isogeny such that the following condition holds. There exists a principal polarization λ : X → X̂ such that ρ^{∨} ∘ λ\overline{S} ∘ ρ and λ_{𝕏,\overline{S}} coincide up to a scalar. Two pairs (X_{1}, ρ_{1}) and (X_{2}, ρ_{2}) are isomorphic if ρ_{1} ∘ ρ_{2}^{−1} lifts to an isomorphism X_{2} → X_{1}. This functor is representable by a formal scheme M which is locally formally of finite type over Spf(W) (see [RaZ], Thm. 3.25). Let M_{red} be its underlying reduced subscheme, that is the reduced subscheme of M defined by the maximal ideal of definition. Then M_{red} is a scheme over Spec(k). The analogues of these moduli spaces for p-divisible groups without polarization have been studied by de Jong and Oort in [JO] for the case that the rational Dieudonné module of 𝕏 is simple and in [V1] without making this additional assumption. There, the sets of connected components and of irreducible components, as well as the dimensions, are determined. In the polarized case, the moduli spaces M_{red} have been examined in several low-dimensional cases. For example, Kaiser ([Kai]) proves a twisted fundamental lemma for GSp_{4} by giving a complete description in the case that 𝕏 is two-dimensional and supersingular. An independent description of this case is given by Kudla and Rapoport in [KR]. In [Ri], Richartz describes the moduli space in the case of three-dimensional supersingular 𝕏. In this paper we derive corresponding results on the global structure of the mo duli space M_{red} for arbitrary 𝕏. The first main result of this paper concerns the set of connected components of M_{red}.

Theorem 1. Let 𝕏 be nontrivial and let 𝕏_{m} × 𝕏_{bi} × 𝕏_{et} be the decomposition into its multiplicative, bi-infinitesimal, and étale part. Then

π_{0}(M_{red}) ≅ (GL_{ht(𝕏_{m})}(ℚ_{p})/GL_{ht(𝕏_{m})}(ℤ_{p})) × ℤ.

Next we consider the set of irreducible components of M_{red}. Let (N, F) be the rational Dieudonné module of 𝕏. Here, N is an L-vector space of dimension ht(𝕏) and F : N → N is a σ-linear isomorphism. The polarization λ_{𝕏} induces an anti-symmetric bilinear perfect pairing ⟨·, ·⟩ on N. Let G be the corresponding general symplectic group of automorphisms of N respecting ⟨·, ·⟩ up to a scalar. Let

J = {g ∈ G(L) | g ∘ F = F ∘ g}.

It is the set of ℚ_{p}-valued points of an algebraic group over ℚ_{p} (see [RaZ], Prop. 1.12). There is an action of J on M_{red}.

Theorem 2. The action of J on the set of irreducible components of M_{red} is transitive.

We choose a decomposition N = ⨁_{j=1}^{l} N^{j} with N^{j} simple of slope λ_{j} = m_{j}/(m_{j} + n_{j}) with (m_{j}, n_{j}) = 1 and λ_{j} ≤ λ_{j′} for j < j′. Let

m = ⌊\frac{1}{2} ∑_{j} min\{m_{j}, n_{j}\}⌋ ,

Moduli Spaces of Polarized p-Divisible Groups 827

where ⌊x⌋ is the greatest integer less or equal x. As N is the isocrystal of a polarized p-divisible group, its Newton polygon is symmetric, i. e. λ_{l+1−j} = 1 − λ_{j}. Hence we obtain

m = ⌊ ∑_{\{j|m_{j}<n_{j}\}} m_{j} + \frac{1}{2} | \{j | m_{j} = n_{j} = 1\} |⌋ .  (1.1)

Theorem 3. M_{red} is equidimensional of dimension

dim M_{red} = \frac{1}{2} (∑_{j} \frac{(m_{j} − 1)(n_{j} − 1)}{2} + ∑_{j<j′} m_{j}n_{j′} + m) .  (1.2)

Note that the equidimensionality is already a consequence of Theorem 2. However, it also follows from the proof of the dimension formula without requiring additional work. Our results on the set of connected components and on the dimension of M_{red} are analogous to those for other affine Deligne-Lusztig sets for split groups where a scheme structure is known. We now define these affine Deligne-Lusztig varieties and give a brief overview over the general results in comparison to the results for the case treated in this paper. Let O be a finite unramified extension of ℤ_{p} or 𝔽_{p}[[t]] and let G be a split connected reductive group over O. Let F be the quotient field of O. Let K = G(O). Let L be the completion of the maximal unramified extension of F and let σ be the Frobenius of L over F. Let A be a maximal torus and B a Borel subgroup containing A. Let μ ∈ X_{∗}(A) be dominant and let b ∈ G(L). Let ε^{μ} be the image of p or t ∈ F^{×} under μ. Let

X_{μ}(b) = {g ∈ G(L)/K | g^{−1}bσ(g) ∈ Kε^{μ}K}  (1.3)

be the generalized affine Deligne-Lusztig set associated to μ and b. We assume that b ∈ B(G, μ) to have that X_{μ}(b) is nonempty (compare [Ra], 5). There are two cases where it is known that X_{μ}(b) is the set of k-valued points of a scheme. Here, k denotes the residue field of O_{L}. The first case is that F = ℚ_{p} and that X_{μ}(b) is the set of k-valued points of a Rapoport-Zink space of type (EL) or (PEL). In that case μ is always minuscule. Rapoport-Zink spaces without polarization were considered in [V1], in that case G = GL_{h}. For the moduli spaces considered in this paper let G = GSp_{2h}. We choose a basis {e_{i}, f_{i} | 1 ≤ i ≤ h} identifying N with L^{2h} and the symplectic form on N with the symplectic form on L^{2h} defined by requiring that ⟨e_{i}, e_{j}⟩ = ⟨f_{i}, f_{j}⟩ = 0 and ⟨e_{i}, f_{j}⟩ = δ_{i,h+1−j}. Let B be the Borel subgroup of G = GSp_{2h} fixing the complete isotropic flag (e_{1}) ⊂ (e_{1}, e_{2}) ⊂ ⋯ ⊂ (e_{1}, …, e_{h}). We choose A to be the diagonal torus. Let π_{1}(G) be the quotient of X_{∗}(A) by the coroot lattice of G. Then the multiplier G → 𝔾_{m} induces an isomorphism π_{1}(G) → π_{1}(𝔾_{m}) ≅ ℤ. Let μ ∈ X_{∗}(A) be the unique minuscule element whose image in π_{1}(G) is 1. Then p^{μ} is a diagonal matrix with diagonal entries 1 and p, each with multiplicity h. We write F = bσ with b ∈ G. Note that there is a bijection between M_{red}(k) and the set of Dieudonné lattices in N. Using the above notation, we have the bijection

X_{μ}(b) → M_{red}(k)
g ↦ g(W (k)^{2h}).

The second case is that F is a function field. Here X_{μ}(b) obtains its scheme structure by considering it as a subset of the affine Grassmannian G(L)/G(O_{L}). In this case we do not have to assume μ to be minuscule. The X_{μ}(b) are locally closed subschemes of the affine Grassmannian. The closed affine Deligne-Lusztig varieties X_{⪯μ}(b) are defined to be the closed reduced subschemes of G(L)/G(O_{L}) given by X_{⪯μ}(b) = ⋃_{μ′⪯μ} X_{μ′}(b). Here μ′ ⪯ μ if μ − μ′ is a nonnegative linear combination of positive coroots. Note that the two schemes X_{μ}(b) and X_{⪯μ}(b) coincide if μ is minuscule. The sets of connected components of the moduli spaces of non-polarized pdivisible groups are given by a formula completely analogous to Theorem 1 (compare [V1], Thm. A). For closed affine Deligne-Lusztig varieties in the function field case, the set of connected components is also given by a generalization of the formula in Theorem 1 (see [V3], Thm. 1). The sets of connected components of the non-closed X_{μ}(b) are not known in general. There are examples (compare [V3], Section 3) which show that a result analogous to Theorem 1 cannot hold for all non-closed X_{μ}(b). The only further general case where the set of irreducible components is known are the reduced subspaces of moduli spaces of p-divisible groups without polarization. Here, the group J also acts transitively on the set of irreducible components. There are examples of affine Deligne-Lusztig varieties in the function field case associated to non-minuscule μ where this is no longer true (compare [V2], Ex. 6.2). To discuss the formula for the dimension let us first reformulate Theorem 3. Let G = GSp_{2h} and μ be as above. Let ν = (λ_{i}) ∈ ℚ^{h} ≅ X_{∗}(A)_{ℚ} be the (dominant) Newton vector associated to (N, F) as defined by Kottwitz, see [Ko1]. Let ρ be the half-sum of the positive roots of G and ω_{i} the fundamental weights of the adjoint group G_{ad}. Then one can reformulate (1.2) as

dim M_{red} = ⟨2ρ, μ − ν⟩ + ∑_{i}⌊⟨ω_{i}, ν − μ⟩⌋.  (1.4)

In this form, the dimension formula proves a special case of a conjecture by Rapoport (see [Ra], Conjecture 5.10) for the dimension of affine Deligne-Lusztig varieties. Denote by rk_{ℚ_{p}} the dimension of a maximal ℚ_{p}-split subtorus and let def_{G}(F) = rk_{ℚ_{p}}G − rk_{ℚ_{p}}J. Note that def_{G}(F) only depends on the conjugacy class of F or equivalently on the σ-conjugacy class of b if we write F = bσ for some b ∈ G. In our case, it is equal to h−⌈l/2⌉ where l is the number of simple summands of N. Using Kottwitz’s reformulation of the right hand side of (1.4) in [Ko2], we obtain

dim M_{red} = ⟨ρ, μ − ν⟩ − \frac{1}{2}def_{G}(F).  (1.5)

For the case of moduli of p-divisible groups for G = GL_{h}, the analogous formula for the dimension is shown in [V1]. In the function field case, the dimension of the generalized affine Deligne-Lusztig variety has been determined in [V2], [GHKR]. The formula for the dimension is also in this case the analogue of (1.5). The dimension of the moduli spaces M_{red} is also studied by Oort and by Chai using a different approach. In [O2], Oort defines an almost product structure (that is, up to a finite morphism) on Newton strata of moduli spaces of polarized abelian varieties. It is given by an isogeny leaf and a central leaf for the pdivisible group. The dimension of the isogeny leaf is the same as that of the corresponding M_{red}. The dimension of the central leaf is determined by Chai in [C] and also by Oort in [O4]. The dimension of the Newton polygon stratum itself is known from [O1]. Then the dimension of M_{red} can also be computed as the difference of the dimensions of the Newton polygon stratum and the central leaf. We outline the content of the different sections of the paper. In Section 2 we introduce the necessary background and notation, and reduce the problem to the case of bi-infinitesimal groups. In the third and fourth section, we define the open dense subscheme S_{1} where the a-invariant of the p-divisible group is 1 and describe its set of closed points. This description is refined in Sections 5 and 6 to prove the theorems on the set of irreducible components and on the dimension, respectively. In the last section we determine the set of connected components. Acknowledgement. Part of this paper was written during a stay at the Université Paris-Sud at Orsay which was supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD). I thank the Université Paris-Sud for its hospitality. I thank the referee for very helpful remarks.

2 Notation and preliminary reductions

2.1 A decomposition of the rational Dieudonné module

The principal polarization λ_{𝕏} equips the rational Dieudonné module (N, F) of 𝕏 with a nondegenerate anti-symmetric bilinear pairing ⟨·, ·⟩. It satisfies

⟨v, F w⟩ = σ(⟨V v, w⟩)  (2.1)

for all x, y ∈ N. We assumed k to be algebraically closed. Then the classification of isocrystals shows that N has a decomposition into subisocrystals N_{i} of one of the following types. Let l be the number of supersingular summands in a decomposition of N into simple isocrystals. Then

N = \begin{cases}N_{0} ⊕ N_{1} & if l is even \\ N_{0} ⊕ N_{\frac{1}{2}} ⊕ N_{1} & otherwise, \end{cases}  (2.2)

satisfying the following three properties.

1. The slopes of N_{0} are smaller or equal to \frac{1}{2}.

2. The summand N_{\frac{1}{2}} is simple and supersingular.

3. N_{1} is the isocrystal dual to N_{0}, i.e.

⟨N_{0}, N_{0}⟩ = ⟨N_{1}, N_{1}⟩ = ⟨N_{0}, N_{\frac{1}{2}} ⟩ = ⟨N_{1}, N_{\frac{1}{2}} ⟩ = 0.

Note that if l > 1, then this decomposition is not unique and N_{0} and N_{1} also contain supersingular summands. For i ∈ \{0, \frac{1}{2}, 1\} we denote by p_{i} the canonical projection N → N_{i}. The moduli spaces M_{red} for different (𝕏, λ_{𝕏}) in the same isogeny class are isomorphic. Replacing 𝕏 by an isogenous group we may assume that

𝕏 = \begin{cases}𝕏_{0} × 𝕏_{1} & if l is even \\ 𝕏_{0} × 𝕏_{\frac{1}{2}} × 𝕏_{1} & otherwise. \end{cases}  (2.3)

Here, 𝕏_{i} is such that its rational Dieudonné module is N_{i}. Mapping (X, ρ) ∈ M_{red}(k) to the Dieudonné module of X defines a bijection between M_{red}(k) and the set of Dieudonné lattices in N that are self-dual up to a scalar. Here a sublattice Λ of N is called a Dieudonné lattice if φ(Λ) ⊆ Λ for all φ in the Dieudonné ring of k,

D = D(k) = W (k)[F, V]/(F V = V F = p, aV = V σ(a), F a = σ(a)F). (2.4)

All lattices considered in this paper are Dieudonné lattices. A lattice Λ ⊂ N is self-dual up to a scalar if the dual lattice Λ^{∨} satisfies Λ^{∨} = cΛ with c ∈ L^{×}. The following notion is introduced by Oort in [O3].

Definition 2.1. Let X be a p-divisible group over k and Λ_{min} be its Dieudonné module. Then X is a minimal p-divisible group if End(Λ_{min}) is a maximal order in End(Λ_{min}) ⊗_{W} L.

Remark 2.2. By Morita equivalence X is minimal if and only if Λ_{min} is the direct sum of submodules Λ_{min}^{i} such that N^{i} = Λ_{min}^{i} ⊗_{W} L is simple and that End(Λ_{min}^{i}) is a maximal order in End(N^{i}), which is Oort’s original definition. Note that in every isogeny class of p-divisible groups over k there is exactly one isomorphism class of minimal p-divisible groups (compare [O3], 1.1).

Lemma 2.3. There is a k-valued point (X, ρ) of M_{red} such that X is minimal.

Proof. Let N_{0} and N_{1} as in the decomposition above. Let Λ_{min,0} ⊂ N_{0} be the lattice of a minimal p-divisible group and let Λ_{min, \frac{1}{2}} ⊂ N_{\frac{1}{2}} be the Dieudonné module of 𝕏_{\frac{1}{2}} . There is only one isomorphism class of one-dimensional supersingular p-divisible groups and it consists of minimal p-divisible groups. Let c ∈ L^{×} with Λ_{min, \frac{1}{2}}^{∨} = cΛ_{min, \frac{1}{2}} . Let

Λ_{min,1} = {x ∈ N_{1} | ⟨x, cy⟩ ∈ W for all y ∈ Λ_{min,0}}.

Then Λ_{min,1} is also the Dieudonné module of a minimal p-divisible group. Furthermore, Λ_{min} = Λ_{min,0} ⊕ Λ_{min, \frac{1}{2}} ⊕ Λ_{min,1} satisfies Λ_{min}^{∨} = cΛ_{min}. Thus Λ_{min} corresponds to an element of M_{red}(k) and to a minimal p-divisible group.

Remark 2.4. There is the following explicit description of the Dieudonné module of a minimal p-divisible group: Let N = ⨁_{j} N^{j} be a decomposition of N into simple isocrystals. For each j we write the slope of N^{j} as m_{j}/(m_{j}+n_{j}) with (m_{j}, n_{j}) = 1. Then there is a basis e_{1}^{j}, …, e_{m_{j}+n_{j}}^{j} of N^{j} with F (e_{i}^{j}) = e_{i+m_{j}}^{j} for all i, j. Here we use the notation e_{i+m_{j}+n_{j}}^{j} = pe_{i}^{j}. For the existence compare for example [V1], 4.1. Furthermore, these bases may be chosen such that ⟨e_{i}^{j}, e_{i′}^{j′}⟩ = δ_{j,l+1−j′} · δ_{i,m_{j}+n_{j}+1−i′} for 1 ≤ i, i′ ≤ m_{j} + n_{j} = m_{l+1−j} + n_{l+1−j}. Then we can take the lattice Λ_{min} to be the lattice generated by these basis elements e_{i}^{j}.

2.2 Moduli of non-polarized p-divisible groups

For the moment let 𝕏 be a p-divisible group without polarization. Then associated to 𝕏 there is an analogous moduli problem of quasi-isogenies of p-divisible groups without polarization. If 𝕏 is polarized, we thus obtain two functors which are closely related. In this section we recall the definition of the moduli spaces of non-polarized p-divisible groups and relate them to M_{red}. Besides, we provide a technical result on lattices in isocrystals which we need in the following section. Let M_{𝕏}^{np} be the functor associating to a scheme S ∈ Nilp_{W} the set of pairs (X, ρ) where X is a p-divisible group over S and ρ a quasi-isogeny 𝕏\overline{S} → X\overline{S}. Two such pairs (X_{1}, ρ_{1}) and (X_{2}, ρ_{2}) are identified in this set if ρ_{1}∘ρ_{2}^{−1} lifts to an isomorphism X_{2} → X_{1} over S. This functor is representable by a formal scheme which is locally formally of finite type over Spf(W) (see [RaZ], Theorem 2.16). Let M_{𝕏,red}^{np} be its reduced subscheme. We always include 𝕏 in this notation, because we compare M_{red} to the two moduli spaces M_{𝕏,red}^{np} and M_{𝕏_{0},red}^{np}. Let J^{np} = {g ∈ GL(N) | g ∘ F = F ∘ g}. Then J ⊆ J^{np}. If 𝕏 is a principally polarized p-divisible group, then forgetting the polarization induces a natural inclusion as a closed subscheme

M_{red} ↪ M_{𝕏,red}^{np}.

Furthermore, there is a natural inclusion as a closed subscheme

M_{𝕏_{0},red}^{np} ↪ M_{red}  (2.5)

mapping an S-valued point (X_{0}, ρ_{0}) to (X_{0} × X_{0}^{∨}, (ρ, ρ^{∨})) if the number of supersingular summands of N is even and to (X_{0} × X_{\frac{1}{2}} × X_{0}^{∨}, (ρ, ρ_{\frac{1}{2}} , ρ^{∨})) otherwise. Here X_{\frac{1}{2}} = 𝕏_{\frac{1}{2} ,S} is the base-change of the unique one-dimensional supersingular p-divisible group over k and ρ_{\frac{1}{2}} = id. Let ṽ be the valuation on the Dieudonné ring D determined by

ṽ(aF^{i}V^{j}) = 2v_{p}(a) + i + j  (2.6)

for every a ∈ W (k).

Lemma 2.5. One can decompose each B ∈ D uniquely as B = LT(B)+B′ with ṽ(B′) > ṽ(B) and

LT(B) = ∑_{0≤i≤ṽ(B),2α+i=ṽ(B)} p^{α}([a_{i}]V^{i} + [b_{i}]F^{i}).

Here [a_{i}] and [b_{i}] are Teichmüller representatives of elements of k^{×} or 0 and [b_{0}] = 0.

Proof. The V^{i} with i ≥ 0 and the F^{i} with i > 0 together form a basis of the W (k)-module D. Besides, as k is perfect, every element of W (k) can be written in a unique way as x = ∑_{α≥0} p^{α}[x_{α}]. Hence we can write B = ∑_{i≥0} x_{i}V^{i} + ∑_{i>0} y_{i}F^{i} = ∑_{i≥0} ∑_{α≥0} p^{α}[x_{i,α}]V^{i} + ∑_{i>0} ∑_{α≥0} p^{α}[y_{i,α}]F^{i} where x_{i}, y_{i} are 0 for all but finitely many i. By the definition of ṽ(B), all x_{i,α}, y_{i,α} with i + 2α < ṽ(B) vanish. Let LT(B) be the sum of all terms p^{α}[x_{i,α}]V^{i} and p^{α}[y_{i,α}]F^{i} on which ṽ takes the value ṽ(B), i.e. those with 2α + i = ṽ(B). Then LT(B) is as in the lemma and ṽ(B − LT(B)) > ṽ(B).

Lemma 2.6. Let (N_{0}, b_{0}σ) be the rational Dieudonné module of some p-divisible group over k. Let m = v_{p}(det b_{0}) and n = dim_{L}(N_{0}) − m. Let v ∈ N_{0} be not contained in any proper sub-isocrystal of N_{0}.

1. Ann(v) = {φ ∈ D | φ(v) = 0} is a principal left ideal of D. There is a generating element of the form

A = aF^{n} + bV^{m} + ∑_{i=0}^{n−1} a_{i}F^{i} + ∑_{i=1}^{m−1} b_{i}V^{i}.

with a, b ∈ W^{×} and a_{i}, b_{i} ∈ W.

2. If N_{0} is simple (and thus of slope m/(m + n)), we have

LT(A) = \begin{cases}[a]F^{n} & if n < m \\ [b]V^{m} & if m < n \\ [a]F + [b]V & if m = n = 1 \end{cases}  (2.7)

for some a, b ∈ k^{×}.

3. Let N_{0} = ⊕_{j}N^{j} be a decomposition of N_{0} into simple summands. Then LT(A) = LT(∏_{j} L_{j}). Here each L_{j} is of the form (2.7) associated to some nonzero element in N^{j}.

Proof. We use induction on the number of summands in a decomposition of N_{0} into simple isocrystals. If N_{0} is simple, the lemma follows immediately from [V1], Lemma 4.12. For the induction step write N_{0} = N′ ⊕ N′′ where N′ is simple. Let A′ be as in the lemma and associated to N′ and p_{N′}(v) where p_{N′} : N → N′ is the projection. Note that an element of an isocrystal is not contained in any proper sub-isocrystal if and only if the Dieudonné module generated by the element is a lattice. Let Λ be the lattice generated by v. The Dieudonné module generated by A′(v) is equal to Λ ∩ N′′, and hence also a lattice. We may therefore apply the induction hypothesis to A′(v) and N′′ and obtain some A′′ generating Ann(A′(v)). Thus Ann(v) is a principal left ideal generated by A′′A′. Multiplying the corresponding expressions for A′′ and A′, the lemma follows.

2.3 Reduction to the bi-infinitesimal case

Let 𝕏 = 𝕏_{et}×𝕏_{bi}×𝕏_{m} be the decomposition of 𝕏 into its étale, bi-infinitesimal, and multiplicative parts.

Lemma 2.7. We have

M_{𝕏,red} ≅ \begin{cases}M_{𝕏_{et},red}^{np} × M_{𝕏_{bi},red} & if 𝕏_{bi} is nontrivial \\ M_{𝕏_{et},red}^{np} × ℤ & otherwise. \end{cases}

and

M_{𝕏_{et},red}^{np} ≅ GL_{ht(𝕏_{et})}(ℚ_{p})/GL_{ht(𝕏_{et})}(ℤ_{p}).

Proof. Consider the following morphism ι from the right to the left hand side of the first isomorphism. In the first case, an S-valued point ((X_{et}, ρ_{et}), (X_{bi}, ρ_{bi})) is mapped to (X_{et}×X_{bi}×X_{m}, (ρ_{et}, ρ_{bi}, ρ_{m})) where X_{m} = X̂_{et}. Furthermore, ρ_{m} is the dual isogeny of c · ρ_{et} and c is the scalar determined by the duality condition for ρ_{bi}. In the second case ((X_{et}, ρ_{et}), l) is mapped to (X_{et} ×X_{m}, (ρ_{et}, ρ_{m})) with X_{m} = X̂_{et} and ρ_{m} = (p^{l} ·ρ_{et})^{∨}. In both cases ι is a monomorphism, and to check that it is a closed immersion we verify the valuation criterion for properness. Let (X, ρ) be a k[[t]]-valued point of M_{𝕏,red} such that the generic point is in the image of ι. Let π_{X} : X ↠ X_{et} with X_{et} étale over Spec(k[[t]]) and X inf initesimal over X_{et}, as in [M], Lemma II.4.8. Our assumption implies that this map has a right inverse X_{et,k((t))} → X_{k((t))} after base change to Spec(k((t))). By [J1], Corollary 1.2, this morphism lifts to a morphism X_{et} → X over k[[t]]. Together with the inclusion of the kernel of π_{X} in X we obtain a morphism of the product of an étale and an infinitesimal p-divisible group over Spec(k[[t]]) to X. Its inverse is constructed similarly by lifting the projection morphism of X_{k((t))} to the kernel of π_{X} from k((t)) to k[[t]]. Hence X can be written as a product of an étale and an infinitesimal p-divisible group. As X is selfdual, it is then also the product of an étale, a bi-infinitesimal, and a multiplicative p-divisible group, thus of the form X_{et} × X_{bi} × X_{m} with X_{m} = X̂_{et}. The quasiisogeny ρ is compatible with this decompos ition, and the compatibility with the polarizations shows that the induced quasi-isogenies (ρ_{et}, ρ_{bi}, ρ_{m}) have the property that ρ_{bi} is selfdual up to some scalar c and ρ_{m} is the dual isogeny of c·ρ_{et}. Hence (X, ρ) is in the image of ι. This finishes the proof that ι is proper, hence a closed immersion.

To show that ι is an isomorphism it is thus enough to show that each k-valued point of the left hand side is contained in its image. From the Hodge-Newton decomposition (see [Kat], Thm. 1.6.1) we obtain for each k-valued point (X, ρ) a decomposition X = X_{et} × X_{bi} × X_{m} and ρ = ρ_{et} × ρ_{bi} × ρ_{m} into the étale, biinfinitesimal, and multiplicative parts. The compatibility with the polarization then yields that up to some scalar p^{l}, the quasi- isogenies ρ_{m} and ρ_{et} are dual. From this the first isomorphism follows. The second isomorphism is shown by an easy calculation (compare [V1], Section 3).

The lemma reduces the questions after the global structure of M_{red} to the same questions for M_{𝕏_{bi},red}. Thus from now on we assume that 𝕏 is bi-infinitesimal.

3 The dense subscheme S_{1}

In [V1], 4.2 we define an open dense subscheme S_{𝕏,1}^{np} of M_{𝕏,red}^{np}. Let Λ ⊂ N be the lattice associated to x ∈ M_{𝕏,red}^{np}(k). Then x ∈ S_{𝕏,1}^{np} if and only if a(Λ) = dim_{k}(Λ/(F Λ + V Λ)) = 1. As F and V are topologically nilpotent on Λ, this is equivalent to the existence of some v ∈ Λ such that Λ is the Dieudonné submodule of N generated by v. Note that a(Λ) can also be defined as dim_{k}(Hom(α_{p}, X)) where X is the p-divisible group associated to Λ. Let

S_{1} = S_{𝕏,1}^{np} ∩ M_{red} ⊆ M_{red}.

Then S_{1} is open in M_{red}.

Lemma 3.1. The open subscheme S_{1} is dense in M_{red}.

Proof. Recall that we assume 𝕏 to be bi-infinitesimal. Let (\overline{X}, \overline{ρ}) ∈ M_{red}(k) and let \overline{λ} be a corresponding polarization of \overline{X}. Note that by [M], Lemma II.4.16 (or by [J2], Lemma 2.4.4) there is an equivalence of categories between p-divisible groups over an adic, locally noetherian affine formal scheme Spf(A) and over Spec(A). From [O1], Corollary 3.11 we obtain a deformation (X, λ) of (\overline{X}, \overline{λ}) over Spec(k[[t]]) such that the generic fiber satisfies a = 1. Next we show that after a base change we may also deform \overline{ρ} to a quasi-isogeny ρ between (X, λ) and the constant p-divisible group (𝕏, λ_{𝕏}) that is compatible with the polarizations. From [OZ], Corollary 3.2 we obtain a deformation of \overline{ρ} to a quasi-isogeny between X and a constant p-divisible group Y after a basechange to the perfect hull of k[[t]]. As Y is constant it is quasi-isogenous to the base change 𝕏_{Spec(k[[t]]^{perf} )} of 𝕏. After composing the deformation of \overline{ρ} with a quasi-isogeny between Y and 𝕏_{Spec(k[[t]]^{perf} )} we may assume that Y is already equal to 𝕏_{Spec(k[[t]]^{perf} )}. Let x be the point of Spec(k[[t]]^{perf}) over the generic point of Spec(k[[t]]). Then we may further compose the quasi-isogeny with a self-quasi-isogeny of 𝕏_{Spec(k[[t]]^{perf} )} such that in x it is compatible with the polarizations of the two groups in this point. Thus we obtain a k[[t]]^{perf}-valued point of M_{𝕏,red}^{np} such that the image of x is in M_{red}. As M_{red} is closed, this has to be a k[[t]]^{perf}-valued point of M_{red}. Modifying the point by a suitable elemen t of J, we may assume that the special point is mapped to (\overline{X}, \overline{ρ}). In the generic point, the a-invariant of the p-divisible group X is 1. Thus this provides the desired deformation of (\overline{X}, \overline{ρ}) to a point of S_{1}.

To determine the dimension and the set of irreducible components of M_{red} it is thus sufficient to consider S_{1}. We proceed in the same way as for the moduli spaces of p-divisible groups without polarization. In contrast to the non-polarized case it turns out to be useful to use two slightly different systems of coordinates to prove the assertions on the dimension and on the set of irreducible components of M_{red}. Let us briefly recall the main steps for the moduli spaces M_{𝕏,red}^{np} of nonpolarized p-divisible groups. Their sets of irreducible components and their dimension are determined by studying S_{𝕏,1}^{np}. In [V1], 4 it is shown that the connected components of S_{𝕏,1}^{np} are irreducible and that J^{np} = {j ∈ GL(N)|j ∘ F = F ∘ j} acts transitively on them. The first step to prove this is to give a description of S_{𝕏,1}^{np}(k). One uses that each such point corresponds to a lattice in N with a-invariant 1. As Dieudonné modules, these lattices are generated by a single element and the description of the set of points is given by classifying these elements generating the lattices. The second step consists in the construction of a family in M_{𝕏,red}^{np} to show that a set of points which seems to parametrize an irreducible compo nent of S_{𝕏,1}^{np} indeed comes from an irreducible subscheme. More precisely, a slight reformulation of the results in [V1], Section 4 yields the following proposition.

Proposition 3.2. Let (N, F) be the isocrystal of a p-divisible group 𝕏 over k. Let m = v_{p}(det F). Let S = Spec(R) ∈ Nilp_{W} be a reduced affine scheme with pR = 0 and let j ∈ J^{np}. Let v ∈ N_{R} = N ⊗_{L} W (R)[\frac{1}{p}] such that in every x ∈ S(k), the reduction v_{x} of v in x satisfies that

v_{x} ∈ jΛ_{min}

and

v_{p}(det j) = max{v_{p}(det j′) | j′ ∈ J^{np} and v_{x} ∈ j′Λ_{min}}.

Here, Λ_{min} ⊂ N is the lattice of the minimal p-divisible group in Remark 2.4. Let R̃ = σ^{−m}(R) be the unique reduced extension of R such that σ^{m} : R̃ → R̃ has image R. Let ṽ ∈ NR̃ with σ^{m}(ṽ) = v. Then there is a morphism φ : Spec(R̃) → M_{𝕏,red}^{np} such that for every x ∈ Spec(R̃)(k), the image φ(x) corresponds to the Dieudonné module Λ_{x} in N generated by v_{x}. Assume in addition that 𝕏 is principally polarized and that for every x, the Dieudonné module Λ_{x} corresponds to a point of M_{red}. Then φ factors through M_{red}.

Note that the second condition on v_{x} (or more precisely the existence of the maximum) implies that the Dieudonné submodule of N generated by v_{x} is a lattice.

Proof. To prove the first assertion we may assume that j = id. Note that ṽ satisfies the same conditions as v. The conditions on the ṽ_{x} are reformulated in [V1], Lemma 4.8. The condition given there is exactly the condition needed in [V1], Section 4.4 to construct a display over S leading to the claimed morphism φ. It maps x to the Dieudonné lattice generated by σ^{m}(ṽ_{x}) = v_{x}. The second assertion is trivial as S is reduced and M_{red} a closed subscheme of M_{𝕏,red}^{np}.

Remark 3.3. We use the same notation as in the proposition. From [V1], 4.4 we also obtain that under the conditions of Proposition 3.2, the elements

v, V v, …, V^{v_{p}(det F)}v, F v, …, F^{dim N−v_{p}(det F)−1}v

are a basis of the free W (R̃)[\frac{1}{p}]-module NR̃. They are the images of the standard basis of N under some element of GL(NR̃). We apply the preceding to the situation of an isocrystal N = N_{0} and its dual, N_{1}. Then GL(N_{0}) × 𝔾_{m} is isomorphic to the Siegel Levi subgroup of GSp(N_{0} ⊕ N_{1}). Let v ∈ N_{0} = N be as in Proposition 3.2. Then there are elements y_{i} ∈ (N_{1})R̃ which form a basis of (N_{1})R̃ which is dual to the basis

(x_{1}, …, x_{dim N_{0}}) = (v, V v, …, V^{v_{p}(det F)}v, F v, …, F^{dim N_{0}−v_{p}(det F)−1}v)

with respect to ⟨·, ·⟩. In other words, the y_{i} ∈ (N_{1})R̃ are such that ⟨x_{i}, y_{j}⟩ = δ_{ij}.

4 Geometric points of S_{1}

4.1 N with an even number of supersingular summands

In this subsection we consider the case that N has an even number of supersingular summands. By (2.2) we have a decomposition N = N_{0} ⊕ N_{1}. Recall that by a lattice we always mean a Dieudonné lattice. Let Λ ⊂ N be the lattice corresponding to a k-valued point of M_{red}. Then Λ^{∨} = cΛ for some c ∈ L^{×}. Let Λ_{0} = p_{0}(Λ) and Λ_{1} = Λ ∩ N_{1}. For a subset M of N and δ ∈ {0, 1} let

(M)_{δ}^{∨} = {x ∈ N_{δ} | ⟨x, x′⟩ ∈ W for all x′ ∈ M}.  (4.1)

Then cΛ_{1} = (Λ_{0})_{1}^{∨}. Hence Λ_{0} and Λ_{1} correspond to dual p-divisible groups, which implies a(Λ_{0}) = a(Λ_{1}). The geometric points of S_{1} correspond to lattices Λ that in addition satisfy a(Λ) = 1. Especially, a(Λ_{0}) = a(Λ_{1}) = 1. In this subsection we classify a slightly larger class of lattices. We fix a lattice Λ_{0} ⊂ N_{0} with a(Λ_{0}) = 1 and c ∈ L^{×}. Then we consider all lattices Λ ⊂ N with

p_{0}(Λ) = Λ_{0} and Λ^{∨} = cΛ.  (4.2)

Note that we have a description of the set of lattices Λ_{0} ⊂ N_{0} with a(Λ_{0}) = 1 from [V1], see also Section 2.2. The considerations above show that Λ ∩ N_{1} = Λ_{1} = c^{−1}(Λ_{0})_{1}^{∨} is determined by Λ_{0} and c. Let v_{0} be an element generating Λ_{0} as a Dieudonné module. If v ∈ Λ with p_{0}(v) = v_{0}, then Λ is generated by v and Λ_{1}. Let A be a generator of Ann(v_{0}) as in Lemma 2.6. We write v = v_{0} + v_{1} for some v_{1} ∈ N_{1}. Then Av = Av_{1} ∈ Λ_{1}.

Remark 4.1. Let Λ_{0} and c be as above, and let Λ_{1} = c^{−1}(Λ_{0})_{1}^{∨}. Let Λ be a Dieudonné lattice with p_{0}(Λ) = Λ_{0}, Λ ∩ N_{1} ⊇ Λ_{1} and

Λ^{∨} ⊇ cΛ.  (4.3)

Let vol(·) denote the volume of a lattice, normalized in such a way that the lattice corresponding to the basepoint (𝕏, id) of M_{red} has volume 0. The conditions imply above that vol(Λ^{∨}) ≤ vol(cΛ) ≤ vol(c(Λ_{0} ⊕ Λ_{1})) = vol((Λ_{0} ⊕ Λ_{1})^{∨}). Dualizing the inequality for the first and last term, we see that all terms must be equal. Thus Λ satisfies (4.2) and Λ ∩ N_{1} = Λ_{1}.

The next step in the description of lattices with (4.2) is to reformulate (4.3). To do so, we fix a generator v_{0} of Λ_{0} and describe the set of all v_{1} ∈ N_{1} such that the lattice Λ generated by v = v_{0} + v_{1} and Λ_{1} as a Dieudonné lattice satisfies (4.3) and Λ ∩ N_{1} = Λ_{1}. Generators for Λ as a W -module are given by Λ_{1}, and all F^{i}v with i ≥ 0 and V^{i}v with i > 0. Let m, n be as in Lemma 2.6 (associated to the given N_{0}). Note that as N_{0} contains all simple summands of N with slope < 1/2 and half of the supersingular summands, m is the same as in (1.1) and n = h−m ≥ m. By Lemma 2.6, 1. applied to v_{0} ∈ N_{0}, the F^{i}v with i > n, and the V^{i}v with i ≥ m can be written as a linear combination of the F^{i}v with i ≤ n and the V^{i}v with i < m, and a summand in Ann(v_{0}) · v ⊂ Λ ∩ N_{1} = Λ_{1}. Hence Λ is already generated by Λ_{1}, the F^{i}v with 0 ≤ i ≤ n and the V^{i}v with 0 < i < m. The inclusion (4.3) is equivalent to ⟨x, y⟩ ∈ c^{−1}W for all x, y ∈ Λ. This is equivalent to the same condition for pairs (x, y) where x and y are among the generators of Λ described above. From the definition of Λ_{1} we see that the values on pairs of elements of Λ automatically satisfy this if one of the elements is in Λ_{1}. By (2.1) it is enough to consider the products of v with all other generators. Thus (4.3) is equivalent to

⟨v, F^{i}v⟩ ∈ c^{−1}W

and

⟨v, V^{i}v⟩ ∈ c^{−1}W  (4.4)

for n ≥ i > 0. Furthermore, the equations for V^{i} together with (2.1) imply those for F^{i}. If x and y are elements of the same of the summands N_{0} or N_{1}, then ⟨x, y⟩ = 0. Hence the decomposition of v together with (2.1) shows that (4.4) is equivalent to

⟨v_{0}, V^{i}v_{1}⟩ − ⟨V^{i}v_{0}, v_{1}⟩ = ⟨F^{i}v_{0}, v_{1}⟩^{σ^{−i}} − ⟨V^{i}v_{0}, v_{1}⟩ ∈ c^{−1}W.  (4.5)

For ϕ ∈ D let

ξ_{v_{1}}(ϕ) = ⟨ϕv_{0}, v_{1}⟩.  (4.6)

Then ξ_{v_{1}} is left-W -linear in ϕ. Let A be a generator of Ann(v_{0}) as in Lemma 2.6. Then

ξ_{v_{1}}(ψA) = 0  (4.7)

for all ψ ∈ D. Note that an element v_{1} ∈ N_{1} is uniquely determined by ⟨v_{1}, F^{i}v_{0}⟩ for i ∈ {0, …, n − 1} and ⟨v_{1}, V^{i}v_{0}⟩ for i ∈ {1, …, m}. We are looking for the set of v_{1} satisfying (4.5). In terms of ξ_{v_{1}}, this is

ξ_{v_{1}}(F^{i})^{σ^{−i}} − ξ_{v_{1}}(V^{i}) ∈ c^{−1}W.  (4.8)

Lemma 4.2. 1. Let M be the set of W -linear functions ξ : D → L with (4.7) and (4.8) for i ≤ n. Then (4.6) defines a bijection between M and the set of elements v_{1} ∈ N_{1} as above.

2. Let \overline{M} be the set of functions ξ : D → L/c^{−1}W with the same properties as in 1. Then (4.6) defines a bijection between \overline{M} and the set of equivalence classes of elements v_{1} as above. Here two such elements are called equivalent if their difference is in c^{−1}(Λ_{0})_{1}^{∨}.

Proof. Let ξ : D → L be given. An element v_{1} of N_{1} is uniquely determined by the value of ⟨·, v_{1}⟩ on v_{0}, F v_{0}, …, F^{h−m−1}v_{0}, V v_{0}, …, V^{m}v_{0}. These h values may be chosen arbitrarily. For the values of ⟨·, v_{1}⟩ on the other elements of Dv_{0}, a complete set of relations is given by ⟨ψAv_{0}, v_{1}⟩ = 0 for all ψ ∈ D. This is equivalent to (4.7). Furthermore, (4.8) is equivalent to the condition that the lattice generated by Λ_{1} and v_{0} + v_{1} satisfies all required duality properties. To prove 2., we want to lift ξ : D → L/c^{−1}W to a function with values in L. We lift the values of ξ at ϕ ∈ {V^{m}, V^{m−1}, …, 1, …, F^{h−m−1}} arbitrarily. Then the lifts of the remaining values are uniquely determined by (4.7). As (4.8) was satisfied before, it still holds (as a relation modulo c^{−1}W) for the lifted functions. Then 1. implies the existence of v_{1}. Let now w_{1} be a second element inducing ξ (mod c^{−1}W). Then ⟨ϕv_{0}, w_{1} − v_{1}⟩ ∈ c^{−1}W for all ϕ ∈ D. Hence w_{1} − v_{1} ∈ c^{−1}(Λ_{0})_{1}^{∨}.

4.2 N with an odd number of supersingular summands

As parts of this case are similar to the previous one, we mainly describe the differences. By (2.2) we have a decomposition N = N_{0} ⊕ N_{\frac{1}{2}} ⊕ N_{1}. We want to classify the lattices Λ ⊂ N corresponding to k-valued points of S_{1}. As before let Λ_{0} = p_{0}(Λ) and Λ_{1} = Λ ∩ N_{1}. Let c ∈ L^{×} with Λ^{∨} = cΛ. Then cΛ_{1} = (Λ_{0})_{1}^{∨}. Besides,

cΛ ∩ N_{\frac{1}{2}} = (p_{\frac{1}{2}} (Λ))∨_{\frac{1}{2}}.  (4.9)

Here we use (·)∨_{\frac{1}{2}} analogously to (4.1). Again we use the description of the Dieudonné lattices Λ_{0} ⊂ N_{0} with a(Λ_{0}) = 1. We have to classify the Λ corresponding to some fixed Λ_{0} and c, and begin by describing and normalizing the possible images under the projection to N_{0}⊕N_{\frac{1}{2}} . Let v ∈ Λ with Dv = Λ and write v = v_{0} + v_{\frac{1}{2}} + v_{1} with v_{i} ∈ N_{i}. Let A with ṽ(A) = m be a generator of Ann(v_{0}) as in Lemma 2.6.

Proposition 4.3. 1. There is a j ∈ J such that j(v) is of the form v_{0} + ṽ_{\frac{1}{2}} + ṽ_{1} with ṽ_{i} ∈ N_{i} and ADṽ_{\frac{1}{2}} = DAṽ_{\frac{1}{2}} .

2. Let j be as in the previous statement. Then p_{\frac{1}{2}} (jΛ) is the unique Dieudonné lattice in N_{\frac{1}{2}} with p_{\frac{1}{2}} (jΛ)^{∨} = (cp^{m})p_{\frac{1}{2}} (jΛ). Besides, (jΛ) ∩ N_{\frac{1}{2}} = p^{m}p_{\frac{1}{2}} (jΛ).

Proof. To prove 1. let ṽ_{\frac{1}{2}} ∈ N_{\frac{1}{2}} be such that v′_{\frac{1}{2}} = v_{\frac{1}{2}} − ṽ_{\frac{1}{2}} is in the kernel of A and ADṽ_{\frac{1}{2}} = DAṽ_{\frac{1}{2}} . We first reduce the assertion of 1. to the case where N_{0} and N_{1} are simple of slope \frac{1}{2}. Let A_{\frac{1}{2}} be a generator of Ann(v′_{\frac{1}{2}} ) as in Lemma 2.6. As A ∈ Ann(v′_{\frac{1}{2}} ),

we can write A = ÃA_{\frac{1}{2}} with Ã ∈ D. Then Ã generates Ann(A_{\frac{1}{2}} v_{0}). From the description of annihilators of elements of N_{0} in Lemma 2.6 we see that we may write v_{0} = v′_{0} + ṽ_{0} with A_{\frac{1}{2}} v′_{0} = 0 and ṽ_{0} lying in a proper subisocrystal Ñ_{0} of N_{0}. Then v′_{0} generates a simple subisocrystal N′_{0} of N_{0} of slope \frac{1}{2} and N_{0} = N′_{0} ⊕ Ñ_{0}. Let N′_{1} be the subisocrystal of N_{1} which is dual to N′_{0}. Then we want to show that the assertion of the proposition holds for some j ∈ J ∩ End(N′_{0} ⊕ N ′_{\frac{1}{2}} ⊕ N′_{1}). To simplify the notation, we may assume that N only consists of these three summa nds. We construct the inverse of the claimed element j ∈ J. Let j̃ ∈ {g ∈ GL(N_{0} ⊕ N_{1}) | g ∘ F = F ∘ g} be in the unipotent radical of the parabolic subgroup stabilizing the subspace N_{1}. We assume that j̃ ∉ J, i. e. that j̃ is not compatible with the pairing. Let v_{0} + v_{1} with v_{1} ∈ N_{1} be the image of v_{0}. Then Ann(v_{1}) = Ann(v_{0}) and f = ⟨v_{0} + v_{1}, F (v_{0} + v_{1})⟩ ≠ 0. Let A be a generator of Ann(v_{0}) as in Lemma 2.6. Then A = aF + a_{0} + bV for some a, b ∈ W^{×} and a_{0} ∈ W. We obtain

0 = ⟨v_{0} + v_{1}, A(v_{0} + v_{1})⟩^{σ} = a^{σ}f^{σ} − bf.

This is a ℚ_{p}-linear equation of degree p, thus the set of solutions is a onedimensional ℚ_{p}-vector space in L. As A also generates Ann(v′_{\frac{1}{2}} ), the number ⟨v′_{\frac{1}{2}} , F (v′_{\frac{1}{2}} )⟩ is also in this vector space. Hence there is an α ∈ ℚ_{p}^{×} wit h αf = ⟨v′_{\frac{1}{2}} , F (v′_{\frac{1}{2}} )⟩. By multiplying v_{1} by a suitable factor, we may assume that α = −1. Note that this does not change Ann(v_{1}). This implies that

⟨v_{0} + v′_{\frac{1}{2}} + v_{1}, F (v_{0} + v′_{\frac{1}{2}} + v_{1})⟩ = 0.

Besides, we have Ann(v_{0} + v′_{\frac{1}{2}} + v_{1}) = Ann(v_{0}). The element j^{−1} we are

constructing will map v_{0} to v_{0} + v′_{\frac{1}{2}} + v_{1}. Let Ñ_{0} = D(v_{0} + v′_{\frac{1}{2}} + v_{1}). Then we

can extend j^{−1} uniquely to a linear map from N_{0} to Ñ_{0} which is compatible with F. On N_{1}, we define j^{−1} to be the identity. Then one easily checks that j^{−1} : N_{0} ⊕ N_{1} → Ñ_{0} ⊕ N_{1} respects the pairing. It remains to define j^{−1} on N_{\frac{1}{2}} . Let Ñ_{\frac{1}{2}} be the orthogonal complement of Ñ_{0} ⊕ N_{1}. Then Ñ_{\frac{1}{2}} ⊆ N_{\frac{1}{2}} ⊕ N_{1} Let j^{−1}(v′_{\frac{1}{2}} ) = u where u ∈ Ñ_{\frac{1}{2}} is such that p_{N_{\frac{1}{2}}} (u) = v′_{\frac{1}{2}} . Then Ann(u) ⊆

Ann(v′_{\frac{1}{2}} ). As u is contained in a simple isocrystal of slope \frac{1}{2}, this inclusion has to be an equality. As ⟨N_{\frac{1}{2}} ⊕ N_{1}, N_{1}⟩ = 0, we have ⟨u, F u⟩ = ⟨v′_{\frac{1}{2}} , F v′_{\frac{1}{2}} ⟩. Hence we can extend j^{−1} to an element of J. Then p_{\frac{1}{2}} (j(v_{0}+v_{\frac{1}{2}} +v_{1})) = v_{\frac{1}{2}} −v′_{\frac{1}{2}} = ṽ_{\frac{1}{2}} . Thus j satisfies all properties of 1. It remains to prove 2. We may assume that j = 1. Note that there is exactly one Dieudonné lattice of each volume in N_{\frac{1}{2}} . Equivalently, for each α ∈ L^{×} there is exactly one Λ ⊂ N_{\frac{1}{2}} with Λ^{∨} = αΛ. (For the rest of the proof all dual lattices are the dual lattices inside the selfdual isocrystal N_{\frac{1}{2}} .) We have Λ ∩ N_{\frac{1}{2}} = c^{−1}(p_{\frac{1}{2}} (Λ))^{∨} ⊆ p_{\frac{1}{2}} (Λ). Let Λ_{\frac{1}{2}} be the lattice with c^{−1}(Λ_{\frac{1}{2}} )^{∨} = Λ_{\frac{1}{2}} . Then

Λ ∩ N_{\frac{1}{2}} ⊆ Λ_{\frac{1}{2}} ⊆ p_{\frac{1}{2}} (Λ)  (4.10)

and the lengths of the two inclusions are equal. We have to show that the length of the inclusions are both equal to m. The lattice p_{\frac{1}{2}} (Λ) also contains A(p_{\frac{1}{2}} (Λ)). As ṽ(A) = m, the length of this inclusion is m. Furthermore,

Λ ∩ N_{\frac{1}{2}} = Ann(Av_{1})Av = Ann(Av_{1})Av_{\frac{1}{2}} ⊆ DAv_{\frac{1}{2}} = ADv_{\frac{1}{2}} = Ap_{\frac{1}{2}} (Λ).

Note that here we only know that the length of the inclusion is ≥ m = ṽ(A_{1}) where A_{1} is a generator of Ann(Av_{1}). Thus we obtain a second chain of inclusions

Λ ∩ N_{\frac{1}{2}} ⊆ Ap_{\frac{1}{2}} (Λ) ⊆ p_{\frac{1}{2}} (Λ).

We compare this to (4.10). To show that the length of the first inclusion of this chain is not greater than the length of the second inclusion, we have to show that Ap_{\frac{1}{2}} (Λ) ⊆ Λ_{\frac{1}{2}} . By definition of Λ_{\frac{1}{2}} this is equivalent to Ap_{\frac{1}{2}} (Λ) ⊆ c^{−1}(Ap_{\frac{1}{2}} (Λ))^{∨}. To prove this last inclusion we use again the duality relation for Λ. Note that Ap_{\frac{1}{2}} (Λ) = DAv_{\frac{1}{2}} = p_{\frac{1}{2}} (Λ ∩ (N_{\frac{1}{2}} ⊕ N_{1})). Let x, y ∈ N_{\frac{1}{2}} ⊕ N_{1}. Then ⟨x, y⟩ = ⟨p_{\frac{1}{2}} (x), p_{\frac{1}{2}} (y)⟩. Thus the duality relation for Λ implies that Ap_{\frac{1}{2}} (Λ) ⊆ c^{−1}(Ap_{\frac{1}{2}} (Λ))^{∨}.

For both Theorem 2 and Theorem 3 it is enough to describe a locally closed subset of S_{1} whose image under the action of J is all of S_{1}. Thus we may assume that j = 1 and that v itself already satisfies the property of the proposition. Especially, p_{\frac{1}{2}} (Λ) is then determined by c. The element v_{\frac{1}{2}} may be modified by arbitrary elements in p_{\frac{1}{2}} (Λ ∩ (N_{\frac{1}{2}} ⊕ N_{1})) without changing Λ. Indeed, for each such element there is an element in Λ whose projection to N_{0} ⊕ N_{\frac{1}{2}} is the given element. Thus for fixed v_{0}, the projection of Λ to N_{0} ⊕ N_{\frac{1}{2}} is described by the element v_{\frac{1}{2}} varying in the W -module

p_{\frac{1}{2}} (Λ)/p_{\frac{1}{2}} (Λ ∩ (N_{\frac{1}{2}} ⊕ N_{1})) = p_{\frac{1}{2}} (Λ)/A(p_{\frac{1}{2}} (Λ)) of length m which is independent of Λ. To choose coordinates for v_{\frac{1}{2}} we use that this module is isomorphic to W/p^{⌊m/2⌋}W ⊕ W/p^{⌈m/2⌉}W. Under this isomorphism, the element v_{\frac{1}{2}} is mapped to an element of the form

∑_{i=1}^{⌊m/2⌋}[y_{i}]p^{i−1} ⊕ ∑_{i=⌊m/2⌋+1}^{m}[y_{i}]p^{i−⌊m/2⌋−1}.  (4.11)

Here we use that k is perfect, and [y_{i}] is the Teichmüller representative of an element y_{i} of k. Note that a(Λ) = 1 (or the condition that j = 1) implies that A(v_{\frac{1}{2}} ) is a generator of Λ ∩ N_{\frac{1}{2}} and not only an arbitrary element. This is an open condition on p_{\frac{1}{2}} (Λ)/p_{\frac{1}{2}} (Λ ∩ (N_{\frac{1}{2}} ⊕ N_{1})). More precisely, it excludes a finite number of hyperplanes (compare [V1], Lemma 4.8). Let now v_{\frac{1}{2}} also be fixed. It remains to determine the set of possible v_{1} such that Λ = D(v_{0} + v_{\frac{1}{2}} + v_{1}) is a lattice with Λ^{∨} = cΛ. The same arguments as in the previous case show that v_{1} can be chosen in an open subset of the set of v_{1} with

⟨v_{0}, ϕv_{1}⟩ + ⟨v_{1}, ϕv_{0}⟩ ≡ −⟨v_{\frac{1}{2}} , ϕv_{\frac{1}{2}} ⟩ (mod c^{−1}W).  (4.12)

for all ϕ ∈ D.

Remark 4.4. Let ϕ ∈ D with ṽ(ϕ) = 2m. Then ϕv_{\frac{1}{2}} ∈ p^{m}p_{\frac{1}{2}} (Λ) ⊂ c^{−1}Λ^{∨}. Especially, ⟨v_{\frac{1}{2}} , ϕv_{\frac{1}{2}} ⟩ is in c^{−1}W. This is later used in the form that a_{i} = −⟨v_{\frac{1}{2}} , F^{i}v_{\frac{1}{2}} ⟩ satisfies (6.3).

Analogously to the previous case we use (4.6) to define ξ_{v}. Then we also obtain the analogue of Lemma 4.2.

Lemma 4.5. 1. Let M be the set of W -linear functions ξ : D → L with (4.7) and (4.12) for i ≤ n. Then (4.6) defines a bijection between M and the set of elements v_{1} ∈ N_{1} as above.

2. Let \overline{M} be the set of functions ξ : D → L/c^{−1}W with the same properties as in 1. Then (4.6) defines a bijection between \overline{M} and the set of equivalence classes of elements v_{1} as above. Here two such elements are called equivalent if their difference is in c^{−1}(Λ_{0})_{1}^{∨}.

5 The set of irreducible components

Lemma 5.1. Let Λ ⊂ N_{0}⊕N_{1} be a lattice generated by a sublattice Λ_{1} ⊂ N_{1} and an element v with v = v_{0}+v_{1} for some v_{0} ∈ N_{0} and v_{1} ∈ N_{1}. Let Λ̃ be generated by Λ_{1} and v_{0} + ṽ_{1} for some ṽ_{1} ∈ N_{1}. If ξṽ_{1}(F^{i})^{σ^{−i}} − ξṽ_{1}(V^{i}) = ξ_{v_{1}}(F^{i})^{σ^{−i}} − ξ_{v_{1}}(V^{i}) for every i ∈ {1, …, h} then there is a j ∈ J with j(Λ) = Λ̃.

Proof. The assumption implies that ⟨v_{0} + ṽ_{1} − v_{1}, φ(v_{0} + ṽ_{1} − v_{1})⟩ = 0 for φ ∈ {1, V, …, V^{h}} (see the reformulation of (4.4) in Section 4.1). By (2.1), the same holds for φ ∈ {F, …, F^{h}}. As dim N = 2h, the φ(v_{0} + ṽ_{1} − v_{1}) for these elements φ ∈ D generate N′ = (D(v_{0} + ṽ_{1} − v_{1})) [1/p] ⊆ N as an L-vector space. Especially,

⟨v_{0} + ṽ_{1} − v_{1}, φ(v_{0} + ṽ_{1} − v_{1})⟩ = 0  (5.1)

for all φ ∈ D. Let A be a generator of Ann(v_{0}) as in Lemma 2.6. Then (5.1) for φ = φ′A implies that ⟨v_{0}, φ′A(ṽ_{1}−v_{1})⟩ = 0 for all φ′ ∈ D. Thus A(ṽ_{1}−v_{1}) = 0. Let j ∈ GL(N) be defined by v_{0} ↦ v_{0} +ṽ_{1} −v_{1}, j|_{N_{1}} = id, and j ∘F = F ∘j. To check that this is well-defined we have to verify that Aj(v_{0}) = j(Av_{0}) = 0. But A(j(v_{0})) = A(v_{0}+ṽ_{1}−v_{1}) = 0. By definition j commutes with F. Furthermore, (5.1) implies that j ∈ G(L). Hence j ∈ J.

For v_{1} as above and i ∈ {1, …, h} let

ψ_{i}(v_{1}) = ξ_{v_{1}}(V^{i}) − ξ_{v_{1}}(F^{i})^{σ^{−i}}.  (5.2)

Then the lemma yields the following corollary.

Corollary 5.2. Let Λ and Λ̃ be two extensions of Λ_{0} and Λ_{1} as described in the previous section (or, in the case of an odd number of supersingular summands, two extensions of Λ_{0} and Λ_{1} associated to the same v_{\frac{1}{2}} ) and let v = v_{0} + v_{1} and ṽ = v_{0} + ṽ_{1} (resp. v = v_{0} + v_{\frac{1}{2}} + v_{1} and ṽ = v_{0} + v_{\frac{1}{2}} + ṽ_{1}) be the generators. Then ψ_{i}(v_{1}) = ψ_{i}(ṽ_{1}) for all i implies that Λ and Λ̃ are in one J-orbit.

Let v_{0} ∈ N_{0} such that Dv_{0} is a lattice in N_{0}. Then the next proposition implies that for each (c_{1}, …, c_{h}) ∈ L^{h} there is a v_{1} ∈ N_{1} with ψ_{i}(v_{1}) = c_{i} for all i. We fix an irreducible component of S_{𝕏_{0},1}^{np}. Then [V1], 4 describes a morphism from a complement of hyperplanes in an affine space to this irreducible component that is a bijection on k-valued points. Let Spec(R_{0}) be this open subscheme of the affine space. One first defines a suitable element v_{0,R_{0}} ∈ N_{0} ⊗_{W} W (R_{0}). The morphism is then constructed in such a way that each k-valued point x of Spec(R_{0}) is mapped to the lattice in N_{0} generated by the reduction of σ^{m}(v_{0,R_{0}}) at x.

Proposition 5.3. Let R be a reduced k-algebra containing σ^{m}(R_{0}). Let c_{1}, …, c_{h} ∈ W (R)[1/p]. Then there is a morphism R → R′ where R′ is a limit of étale extensions of R and a v_{1} ∈ N_{1,R′} with ψ_{i}(v_{1}) = c_{i} for all i. Here, the ψ_{i} are defined with respect to the universal element σ^{m}(v_{0,R_{0}}) ∈ (N_{0})_{σ^{m}(R_{0})}.

For the proof we need the following lemma to simplify the occurring system of equations.

Lemma 5.4. Let R be an 𝔽_{p}-algebra and let m, n ∈ ℕ with m ≤ n. For 0 ≤ i ≤ m and 0 ≤ j ≤ n let P_{ij}(x) ∈ (W (R)[\frac{1}{p}])[x] be a linear combination of the σ^{l}(x) = x^{p^{l}} with l ≥ 0. Assume that the coefficient of x is zero for j < i and in W (R)^{×} for i = j. Consider the system of equations

∑_{j=0}^{n} P_{ij}(x_{j}) = a_{i}

with a_{i} ∈ W (R)[\frac{1}{p}] and i = 0, …, m. It is equivalent to a system of equations of the form ∑_{j} Q_{ij}(x_{j}) = b_{i} with b_{i} ∈ W (R)[\frac{1}{p}] such that the Q_{ij} satisfy the same conditions as the P_{ij} and in addition Q_{ij} = 0 if j < i.

Proof. We use a modification of the Gauss algorithm to show by induction on λ that the system is equivalent to a system of relations of the form ∑_{j} Q_{ij}^{λ}(x_{j}) = b_{i}^{λ} with b_{i}^{λ} ∈ L such that the Q_{ij}^{λ} satisfy the same conditions as the P_{ij} and in addition Q_{ij}^{λ} = 0 if j < i and j ≤ λ. For the induction step we have to carry out the following set of modifications for j = λ + 1 and each i > λ + 1. If Q_{ij}^{λ} vanishes, we do not make any modification. We now assume Q_{ij}^{λ} to be nontrivial. Let σ^{l_{i}}(x) and σ^{l_{j}}(x) be the highest powers of x occurring in Q_{ii}^{λ} and Q_{ij}^{λ}. If l_{i} < l_{j}, we modify the jth equation by a suitable multiple of σ^{l_{j}−l_{i}} applied to the ith equation to lower l_{j}. Else we modify the ith equation by a suitable multiple of σ^{l_{i}−l_{j}} applied to the jth equation to lower l_{i}. We proceed in this way as long as none of the two polynomials Q_{ii}^{λ} and Q_{ij}^{λ} becomes trivial. Note that the defining properties of the P_{ij} are preserved by these modifications. As (by induction) Q_{ij}^{λ} does not have a linear term, the linear term of Q_{ii}^{λ} remains unchanged. Thus this process of modifications ends after a finite number of steps with equations ∑_{j} Q_{ij}^{λ+1}(x_{j}) = b_{i}^{λ+1} which satisfy Q_{ij}^{λ+1} = 0 for j < i and j ≤ λ + 1. For λ + 1 = n, this is what we wanted.

Proof of Proposition 5.3. An element v_{1} ∈ N_{1,R′} is determined by the values of ξ_{v_{1}} at any h consecutive elements of …, F^{2}, F, 1, V, V^{2}, . . .. The other values of ξ are then determined by ξ_{v_{1}}(ϕA) = 0 for all ϕ ∈ D. Here A ∈ Ann(v_{0}) is as in Lemma 2.6. Indeed, each of these equations for ϕ = F^{i} or V^{i} for some i gives a linear equation with coefficients in L between the values of ξ_{v_{1}} at h + 1 consecutive elements of …, F^{2}, F, 1, V, V^{2}, . . .. For the proof of the proposition we take the values ξ_{v_{1}}(F^{i}) for i ∈ {1, …, h} as values determining v_{1}. Then all other values are linear combinations of these ξ_{v_{1}}(F^{i}). The definition of ψ_{v_{1}} in (5.2) yields

ξ_{v_{1}}(V^{i})^{σ^{i}} = ξ_{v_{1}}(F^{i}) + ψ_{i}(v_{1})^{σ^{i}}

for i ∈ {1, …, h}. On the other hand, ξ_{v_{1}}(V^{i})^{σ^{i}} is a linear combination of the ξ_{v_{1}}(F^{j})^{σ^{i}} for j ∈ {1, …, h}. From this we obtain a system of h equations for the ξ_{v_{1}}(F^{i}) with 1 ≤ i ≤ h of the same form as in Lemma 5.4. The resulting equations ∑_{j} Q_{ij}(ξ_{v_{1}}(F^{j})) = b_{i} may be reformulated as Q_{ii}(ξ_{v_{1}}(F^{i})) = c_{i} where c_{i} also contains the summands corresponding to powers of F larger than i. We can then consider these equations by decreasing induction on i. For each i, the polynomial Q_{ii}(x) is a linear combination of powers of x of the form x^{σ^{l}}, and its linear term does not vanish. Thus there is a limit R′ of étale extensions of R and ξ_{v_{1}}(F^{i}) ∈ W (R′) ⊗ ℚ with v_{p}(ξ_{v_{1}}(F^{i})) ≥ v_{p}(c_{i}) satisfying these equations. Note that R′ is in general an infinite extension of R, because the equations are between elements of W (R) ⊗ ℚ and not over R itself. Given ξ_{v_{1}}, Remark 3.3 shows that there is an element v_{1} ∈ (N_{1})_{R′} which induces ξ_{v_{1}}. Indeed, choose v_{1} to be a suitable linear combination of the dual basis defined there.

5.1 Proof of Theorem 2

We begin by constructing an irreducible subscheme of the subscheme of S_{1} where the height of the universal quasi-isogeny is 0. The k-valued points of this subscheme correspond to lattices Λ with a(Λ) = 1 and Λ^{∨} = Λ. There is a d ∈ ℕ such that for each (c_{1}, …, c_{h}) ∈ (p^{d}W)^{h}, the v_{1} constructed in Proposition 5.3 lies in the lattice Λ_{1} ⊂ N_{1}. Let R_{0} as above. In the case of an even number of supersingular summands let R_{1} = σ^{m}(R_{0}). Otherwise let σ^{m}(v_{0,R_{0}}) + v_{\frac{1}{2}} ∈ N_{σ^{m}(R_{0})[y_{1},...,y_{m}]} where v_{0,R_{0}} is as above and where v_{\frac{1}{2}} ∈ (N_{\frac{1}{2}} )_{σ^{m}(R_{0})[y_{1},...,y_{m}]} is identified with the element in (4.11). The open condition on Spec(σ^{m}(R_{0})[y_{1}, …, y_{m}]) that Av_{\frac{1}{2}} is a generator and not only an element of p_{\frac{1}{2}} (Λ ∩ N_{\frac{1}{2}} ⊕ N_{1}) is equivalent to DAv_{\frac{1}{2}} = ADv_{\frac{1}{2}} . This condition is satisfied by all y_{1} that do not lie in some finite-dimensional ℚ_{p}-subvector space of k determined by the kernel of A (compare the proof of Proposition 4.3 1.). In this case let R_{1} be the extension of σ^{m}(R_{0}) corresponding to this affine open subscheme. Let in both cases

R = R_{1}[x_{i,j} | i ∈ {1, …, h}, j ∈ {0, …, d − 1}].

For i ∈ {1, …, h} let c_{i} = ∑_{j=0}^{d−1}[x_{i,j}]p^{j} ∈ W (R). Let Spec(R′) and v_{1} ∈ N_{R′} be as in Proposition 5.3. Let v = σ^{m}(v_{0,R_{0}})+v_{1}, resp. v = σ^{m}(v_{0,R_{0}})+v_{\frac{1}{2}} +v_{1}. Let S = Spec(R) be an irreducible component of the affine open subscheme of Spec(R′) consisting of the points x with v_{1,x} ∈ (Dv_{x})_{1}^{∨} \(F (Dv_{x})_{1}^{∨} +V (Dv_{x})_{1}^{∨}). We denote the image of v in N_{R} also by v. A s we already know that S_{1} is dense, this open subset is nonempty. Let R̃ be the inverse image of R under σ^{h} as in Proposition 3.2. Note that v_{p}(det F) = h, whereas v_{p}(det F |_{N_{0}}) = m. Let S̃ = Spec(R̃). The next step is to define an associated morphism φ : S̃ → M_{red} such that in each k-valued point x of S̃, the image in M_{red}(k) corresponds to the lattice generated by the reduction v_{x} of v at x. By Proposition 3.2 it is enough to show that there is a j ∈ J such that for each x ∈ S̃(k), we have v_{x} ∈ jΛ_{min} and v_{p}(det j) = max{v_{p}(det j′) | v_{x} ∈ j′Λ_{min}}. Let η be the generic point of S̃ and let j_{η} ∈ J be such a maximizing element for η. Then the same holds for each k-valued point in an open and thus dense subscheme of S̃. As the property v_{x} ∈ j_{η}Λ_{min} is closed, it is true for each x ∈ S̃(k). In [V1], 4 it is shown that for lattices Λ ⊂ N with a(Λ) = 1, the difference vol(Λ) − max{v_{p}(det j′) | Λ ⊂ eqj′Λ_{min}} is a constant only depending on N. In our case, the duality condition shows that vol(Dv_{x}) is constant on S̃ and only depending on c and N. Thus the maximum is also constant. Hence in every k-valued point, v_{p}(det j_{η}) is equal to this maximum, which is what we wanted for the existence of φ : S̃ → M_{red}. We obtain an irreducible subscheme φ(S̃) of S_{1} ⊆ M_{red}. To show that J acts transitively on the set of irreducible components we have to show that for each x ∈ S_{1}(k) there is an element j ∈ J such that jx lies in the image of φ. Let Λ ⊂ N be the lattice corresponding to x. The first step is to show that there is a j ∈ J such that j(Λ) is selfdual (and not only up to a scalar c(Λ)). It is enough to show that there is a j ∈ J such that v_{p}(c(Λ)) = v_{p}(c(jΛ)) + 1. Such an element is given by taking the identity on

N_{1}, multiplication by p on N_{0}, and the map ei^{\frac{1}{2}} ↦ ei+1^{\frac{1}{2}} on N_{\frac{1}{2}} . Here we use the notation of Remark 2.4 for the basis of N. Next we want to apply an element of J modifying Λ_{0}. We have a(p_{0}(Λ)) = a(Λ ∩ N_{1}) = 1. From the classification of lattices with a = 1 in [V1], 4 we obtain that J_{𝕏_{0}}^{np} (which may be considered as a subgroup of J by mapping j ∈ J_{𝕏_{0}}^{np} to the map consisting of j and its dual on N_{1}) is acting transitively on the set of irreducible components of M_{𝕏_{0},red}^{np}. Thus by possibly multiplying with such an element we assume that Λ_{0} lies in the fixed irreducible component chosen for Proposition 5.3. Recall from Section 4.2 that in the case of an odd number of supersingular summands, there is a j ∈ J mapping the element v_{\frac{1}{2}} to the irreducible family described there. It remains to study the possible extensions of the lattices Λ_{0} and Λ_{1} (or in the second case of the sublattice of N_{0} ⊕ N_{\frac{1}{2}} determined by Λ_{0} and v_{\frac{1}{2}} and of Λ_{1}). They are given by the associated elements v_{1}. Fix a generating element σ^{m}(v_{0}) of Λ_{0} (in the second case also an element v_{\frac{1}{2}} ) and let v_{1} be an element associated to the e xtension Λ with a(Λ) = 1. Then Lemma 5.1 and the construction of S show that there is an element of J mapping Λ to a lattice associated to a point of S inducing the same ψ_{i} as Λ. Thus the image of φ(S̃) under J is S_{1}, which proves the theorem.

6 Dimension

We use the same notation as before, namely Λ is the lattice corresponding to a point of S_{1}, generated by an element v = v_{0} + v_{\frac{1}{2}} + v_{1} with v_{i} ∈ N_{i}. Again, A is a generator of Ann(v_{0}) and Λ_{0} = p_{0}(Λ) and Λ_{1} = Λ ∩ N_{1}. To determine the dimension of S_{1} and of M_{red} we have to classify the elements v_{1} of Section 4 up to elements in c^{−1}Λ_{1} and not up to the (locally finite) action of J which we used in Section 5. To do so, it is more useful to use the values of ξ_{v_{1}} as coordinates instead of the values of ψ_{v_{1}}. We investigate the set of possible values ξ(ϕ) ∈ L/c^{−1}W for ϕ ∈ D using decreasing induction on ṽ(ϕ) ≥ 0. Here, ṽ is as in (2.6). Recall from Lemma 4.2 2. that the use of functions ξ with values in L/c^{−1}W instead of L corresponds to considering v_{1} as an element of N_{1}/Λ_{1}. But as v_{1} and v_{1} + δ with δ ∈ Λ_{1} lead to the same lattice Λ, this is sufficient to determine the set of possible extensions of Λ_{0} and Λ_{1}. Instead of equations (4.7) and (4.8) we consider the following slightly more general problem to treat at the same time the case of an odd number of supersingular summands. There, (4.8) is replaced by (4.12). We want to consider W -linear functions ξ : D → L/c^{−1}W with

ξ(F^{i}) − ξ(V^{i})^{σ^{i}} ≡ a_{i} (mod c^{−1}W)  (6.1)
ξ(ψA) ≡ 0 (mod c^{−1}W)  (6.2)

for all ψ ∈ D. Here a_{i} ∈ L are given elements satisfying

a_{i}p^{j_{i}} ∈ c^{−1}W if 2j_{i} + i ≥ 2m.  (6.3)

Let D^{i} = {ϕ ∈ D | ṽ(ϕ) ≥ i}. We call a W -linear function

ξ^{i_{0}} : D^{i_{0}} → L/(c^{−1}W)

satisfying (6.1) and (6.2) a partial solution of level i_{0}. Then the induction step consists in determining the possible partial solutions ξ^{i_{0}} of level i_{0} leading to a fixed solution of level i_{0} + 1. Note that the assumption on a_{i} implies that there exists the trivial partial solution ξ^{2m} ≡ 0 of level 2m inducing partial solutions of all higher levels. Recall that we assumed F and V to be elementwise topologically nilpotent on N. Thus for each function ξ with (6.1) and (6.2) there is a level i such that ξ induces the trivial partial solution of level i. Assume that we already know the ξ(ϕ) for ṽ(ϕ) > i_{0} and want to determine its possible values for ṽ(ϕ) = i_{0}. Then we know in particular ξ(pϕ) = pξ(ϕ) ∈ L/c^{−1}W, or ξ(ϕ) ∈ L/p^{−1}c^{−1}W. We want to determine the possible liftings modulo c^{−1}W. A basis of the k-vector space D^{i_{0}}/D^{i_{0}+1} is given by the i_{0} + 1 monomials

F^{i_{0}}, V F^{i_{0}−1} = pF^{i_{0}−2}, …, V^{i_{0}−1}F = pV^{i_{0}−2}, V^{i_{0}}.

Equation (6.1) leads to ⌊i_{0}/2⌋ relations between the values of ξ on these monomials. Recall that ṽ(A) = m. Thus if ṽ(ϕ) = i_{0} −m for some ϕ ∈ D, (6.2) leads to a relation between the value of ξ on LT(ϕA) ∈ D^{i_{0}} and values on D^{i_{0}+1}. As the ξ are linear, it is sufficient to consider the max{0, i_{0} − m + 1} relations for ϕ ∈ {F^{i_{0}−m}, pF^{i_{0}−m−2}, …, V^{i_{0}−m}} ∩ D^{i_{0}−m}. This count of relations leads to the notation

r(i_{0}) = ⌊i_{0}/2⌋ + max{0, i_{0} − m + 1}.

Then i_{0} + 1 ≤ r(i_{0}) is equivalent to i_{0} ≥ 2m. The following proposition is the main tool to prove Theorem 3 on the dimension of the moduli spaces.

Proposition 6.1. 1. Let i_{0} ≥ 2m. Then there is a partial solution ξ^{i_{0}} of (6.1) and (6.2) of level i_{0}. If we fix ξ^{i_{0}} and an l ∈ ℕ with l ≥ i_{0}, there are only finitely many other partial solutions ξ̃^{i_{0}} of level i_{0} such that ξ^{i_{0}} − ξ̃^{i_{0}} induces the trivial partial solution of level l of the associated homogenous system of equations.

2. Let i_{0} + 1 > r(i_{0}) and let ξ^{i_{0}+1} be a partial solution of (6.1) and (6.2) of level i_{0} + 1. Then to obtain a partial solution ξ^{i_{0}} of level i_{0} inducing ξ^{i_{0}+1}, one may choose the lifts to L/c^{−1}W of the values of ξ^{i_{0}} at the first i_{0} + 1 − r(i_{0}) monomials p^{α}V^{β} with 2α + β = i_{0} and β ≤ 2(i_{0} − r(i_{0})) + 1 arbitrarily. Each of the remaining values lies in some finite nonempty set depending polynomially on the values on the previous monomials.

Proof. Note that the existence statement in the first assertion is satisfied as the condition on the a_{i} yields that there is the trivial solution of level 2m. We show the two assertions simultaneously. Let ξ^{i_{0}+1} be a fixed partial solution of level i_{0} + 1 for any i_{0}. It is enough to show that for a lift ξ^{i_{0}}, the values of the first max{0, i_{0} + 1 − r(i_{0})} variables can be chosen arbitrarily, and that the remaining values then lie in some finite set depending polynomially on the values on the previous variables. If i_{0} + 1 > r(i_{0}), we have to show that this finite set is nonempty. We investigate the relations (6.1) and (6.2) more closely. The first set of relations shows that ξ^{i_{0}}(p^{a}F^{b}) with 2a + b = i_{0} is determined by ξ^{i_{0}}(p^{a}V^{b}). Thus it is sufficient to consider this latter set of values. Besides, we have to consider (6.2) for ψ ∈ {V^{i_{0}−m}, pV^{i_{0}−m−2}, …, F^{i_{0}−m}}. For B ∈ D let LT(B) as in Lemma 2.6. Then the equations for the values of ξ^{i_{0}} relate ξ^{i_{0}}(LT(ψA)) to something which is known by the induction hypothesis. Let us recall the description of LT(A) from Lemma 2.6 3. Let h′ be the number of supersingular summands of N_{0}. Let j ≥ 0 with i_{0} − m − j ≥ 0. Then LT(V^{i_{0}−m−j}F^{j}A) is a linear combination of V^{i_{0}−j}F^{j}, …, V^{i_{0}−j−h′}F^{j+h′} whose coefficients are Teichmüller representatives of elements of k. Furthermore, the coefficients of ξ^{i_{0}}(V^{i_{0}−j}F^{j}) and ξ^{i_{0}}(V^{i_{0}−j−h′}F^{j+h′}) are units in W. Using (6.1) we may replace values of ξ^{i_{0}} at monomials in F by σ-powers of the values of the corresponding monomials in V. We thus obtain a relation between a polynomial in the remaining ⌈(i_{0}+1)/2⌉ values of ξ^{i_{0}} and an expression which is known by induction. For 2j ≤ i_{0}, the first summand ξ^{i_{0}}(V^{i_{0}−j}F^{j}) remains the variable associated to the highest power of V which occurs linearly in this polynomial. In the following we ignore all equations for 2j > i_{0}. They only occur for i_{0} > 2m, a case where we only want to prove the finiteness of the set of solutions. The system of equations with 2j ≤ i_{0} is of the form considered in Lemma 5.4. The proof of this Lemma for coefficients in L/c^{−1}W is the same as for coefficients in L. Thus we obtain that the lifts of the values at the i_{0} + 1 − r(i_{0}) variables associated to the largest values of j can be chosen freely and the other ones have to satisfy some relation of the form Q_{ii}(x) ≡ b_{i} for some given b_{i}. As the Q_{ii} have a linear term they are nontrivial. This implies that the set of solutions of these equations is nonempty and finite and dep ends polynomially on the previous values.

6.1 Proof of Theorem 3

By Lemma 3.1 it is enough to show that S_{1} is equidimensional of the claimed dimension. From [V1], 4 we obtain that the connected components of S_{𝕏,1}^{np} are irreducible. The discrete invariant with values in J^{np}/(J^{np} ∩ Stab(Λ_{min})) distinguishing the components is given by Λ ↦ j_{Λ} with Λ ⊆ j_{Λ}Λ_{min} and

v_{p}(det j_{Λ}) = max{v_{p}(det j) | j ∈ J^{np}, Λ ⊆ jΛ_{min}}.

Especially, j_{Λ} is constant on each connected component of S_{1} ⊆ S_{𝕏,1}^{np}. Besides, p_{0}(j_{Λ}Λ_{min}) determines the connected component of p_{0}(Λ) = Λ_{0} inside S_{𝕏_{0},1}. Thus we may fix an irreducible component of S_{𝕏_{0},1} and determine the dimension of the union of connected components of S_{1} such that Λ_{0} is in this fixed component. Let R_{0} and R_{1}, v_{0} and v_{\frac{1}{2}} be as in the proof of Theorem 2. Again we use the functions ξ defined with respect to σ^{m}(v_{0}) instead of v_{0}. Fix an arbitrary partial solution ξ^{2m} of (6.1) and (6.2) of level 2m. Let

R_{2} = R_{1}[x_{iβ} | i ≥ 0, 1 ≤ β ≤ i + 1 − r(i)].

We use decreasing induction on i to lift ξ^{2m} to a partial solution of level i over an étale extension R_{2}^{i} of R_{2}. Let R_{2}^{2m} = R_{2}. Assume that a lift ξ^{i+1} is given. Then Proposition 6.1 shows that the values at i + 1 − r(i) monomials with ṽ = i may be lifted arbitrarily to a value of ξ^{i}. If p^{α}V^{β} with 2α + β = i and β ≤ i + 1 − r(i) is such a monomial we write (using the induction hypothesis) ξ^{i+1}(p^{α+1}V^{β}) = ∑_{i<v_{p}(c^{−1})}[a_{i}]p^{i} with a_{i} ∈ R_{2}^{i+1}. Then we choose

ξ^{i}(p^{α}V^{β}) = ∑_{i<v_{p}(c^{−1})}[a_{i}]p^{i−1} + [x_{iβ}]p^{v_{p}(c^{−1})−1}.

Let R_{2}^{i} be the extension of R_{2}^{i+1} given by adjoining further variables x_{iβ} for larger β parametrizing the other values of the lift of ξ^{i+1} to ξ^{i} and with relations as in Proposition 6.1, 2. and its proof. More precisely, R_{2}^{i} is obtained from R_{2}^{i+1} by a finite number of extensions given by equations of the form Q_{jj}(x) ≡ b_{j} (mod c^{−1}W) where Q_{jj}(x) is a polynomial that is a finite linear combination of the monomials x^{p^{l}} with l ≥ 0 such that the coefficient of x is in W (R_{2}^{i+1})^{×}. This implies that R_{2}^{i} is a finite étale extension of R_{2}^{i+1}. Let R_{3} = R_{2}^{0}. Let v_{1,R_{3}} ∈ N_{1,R_{3}} be such that ξ_{v_{1,R_{3}}} = ξ^{0}. Its existence follows again from the existence of the dual basis in Remark 3.3. Let v = σ^{m}(v_{0,R_{0}}) + v_{1,R_{3}}, or v = σ^{m}(v_{0,R_{0}}) + v_{\frac{1}{2}} + v_{1,R_{3}}. As in the proof of Theorem 2 let S = §pec(R) be an irreducible component of the affine open subscheme of Spec(R_{3}) over which Dv contains (Dv)_{1}^{∨}. As we want to compute the dimension of S_{1}, we only have to consider these subschemes. Let R̃ = σ^{−m}(R) as in Proposition 3.2. The same argument as in the proof of Theorem 2 shows that there is a morphism φ : Spec(R̃) → M_{red} mapping x ∈ Spec(R̃)(k) to the lattice generated by v_{x}. The finiteness statements in Proposition 6.1 imply that for each given y ∈ S_{1} (and thus given ξ) there is an open neighborhood in S_{1} which only contains points of φ(Spec(R̃)) for a finite number of choices of an irreducible component of S_{𝕏_{0},1} and a corresponding component S. Besides, the construction of R_{3} together with the description of the k-valued points of S_{1} shows that for each y ∈ S_{1}(k) there is exactly one irreducible component of S_{𝕏_{0},1}, one corresponding component S, and one point x ∈ Spec(R̃)(k) such that φ(x) = y. Thus dim M_{red} = dim S_{1} is the maximum of dim Spec(R̃) for all irreducible components S. It remains to show that this is equal to the right hand side of (1.2). Note that R_{i} is equidimensional for i = 0, 1, 2, 3. From the construction of S we see that in case of an even number of supersingular summands,

dim Spec(R̃) = dim S = dim R_{3} = dim R_{2}
= dim M_{𝕏_{0},red}^{np} + ∑_{i≥0} max{0, i + 1 − r(i)}.  (6.4)

In the other case,

dim Spec(R̃) = dim S = dim R_{3} = dim R_{2}
= dim M_{𝕏_{0},red}^{np} + ∑_{i≥0} max{0, i + 1 − r(i)} + m.  (6.5)

The last summand corresponds to the choice of v_{\frac{1}{2}} . From the decomposition of N into l simple summands N^{j} we obtain a decomposition N_{0} = ⨁_{j=1}^{l_{0}} N^{j} with l_{0} = ⌊l/2⌋. Let again λ_{j} = m_{j}/(m_{j} + n_{j}) be the slope of N^{j}. Recall from [V1], Theorem B that

dim M_{𝕏_{0},red}^{np} = ∑_{j=1}^{l_{0}} \frac{(m_{j} − 1)(n_{j} − 1)}{2} + ∑_{\{j,j′|j<j′≤l_{0}\}} m_{j}n_{j′}.

We denote the right hand side of (1.2) by D. Let us first consider the case of an even number of supersingular summands. Then by the symmetry of the Newton polygon we obtain

D − dim M_{𝕏_{0},red}^{np} = \frac{1}{2} ∑_{j<j′≤l} m_{j}n_{j′} + \frac{m}{2} − ∑_{j<j′≤l_{0}} m_{j}n_{j′}.

Again by the symmetry of the Newton polygon this is equal to

= ∑_{j=1}^{l_{0}} ∑_{j′=l_{0}+1}^{l} \frac{m_{j}n_{j′}}{2} + \frac{m}{2}

= ∑_{j,j′=1}^{l_{0}} \frac{m_{j}m_{j′}}{2} + \frac{m}{2}

= \frac{m(m + 1)}{2}.

In the other case, the same calculation shows that

D − dim M_{𝕏_{0},red}^{np} = \frac{m(m + 1)}{2} + 2 ∑_{j=1}^{l_{0}} \frac{m_{j}n_{l_{0}+1}}{2}

= \frac{m(m + 1)}{2} + m.

In the last step we used that N^{l_{0}+1} is supersingular, hence n_{l_{0}+1} = 1. On the other hand (6.4) implies that in the case of an even number of supersingular summands,

dim M_{red} − dim M_{𝕏_{0},red}^{np} = ∑_{i≥0} max{0, i + 1 − r(i)}

= ∑_{i=0}^{m−1} (⌊\frac{i}{2}⌋ + 1) + ∑_{i=m}^{2m−1} (⌊\frac{i}{2}⌋ − i + m)

= m + ∑_{i=0}^{2m−1} ⌊\frac{i}{2}⌋ − ∑_{i=0}^{m−1} i

= \frac{m(m + 1)}{2}.

The same calculation with (6.5) shows that for an odd number of supersingular summands

dim M_{red} − dim M_{𝕏_{0},red}^{np} = \frac{m(m + 1)}{2} + m.

Together with the calculation of D − dim M_{𝕏_{0},red}^{np}, this implies Theorem 3.

7 Connected components

In this section we determine the set of connected components of M_{red}. The reduction to the bi-infinitesimal case in Section 2.3 shows that Theorem 1 follows from the next theorem.

Theorem 7.1. Let 𝕏 be bi-infinitesimal and non-trivial. Then

κ : M_{red}(k) → ℤ
Λ ↦ v_{p}(c(Λ)),

where Λ^{∨} = c(Λ) · Λ, induces a bijection

π_{0}(M_{red}) ≅ ℤ.

Proof. From Theorem 2 we obtain a J-equivariant surjection π : J ↠ π_{0}(M_{red}). Besides, the map κ induces a surjection π_{0}(M_{red}) → ℤ. We choose the base point of M to be a minimal p-divisible group. Let Λ_{min} be the corresponding lattice in N. An element jΛ_{min} with j ∈ J is in the kernel of κ if and only if (jΛ_{min})^{∨} = jΛ_{min}. This is equivalent to jΛ_{min} = j′Λ_{min} for some j′ ∈ J ∩ Sp_{2h}(L). Thus we have to show that J ∩ Sp_{2h}(L) is mapped to a single connected component of M_{red}. Our choice of the base point implies that Stab(Λ_{min}) = K. Thus the surjection π maps J ∩ K to the component of the identity. Note that J ∩ Sp_{2h}(L) = J_{der}(ℚ_{p}) where J_{der} is the derived group of J. Hence the elements of (J ∩ Sp_{2h}(L))/(J ∩ K) correspond to vertices in the building of J_{der}. The building of J_{der} is connected. To show that all vertices correspond to points in one connected component of M_{red}, it is thus enough to show that if Λ_{0}, Λ_{1} are the lattices corresponding to two such vertices such that Λ_{0} ∩ Λ_{1} = Λ is of colength 1 in Λ_{0} and Λ_{1}, then the two lattices correspond to points in the same connected component of M_{red}. As a W -module Λ_{0} is generated by Λ and v_{0} for some v_{0} ∈ N. As the slopes of F are in (0, 1) we have F v_{0}, V v_{0} ∈ Λ. Similarly Λ_{1} is generated by Λ and some v_{1} with F v_{1}, V v_{1} ∈ Λ. For a ∈ 𝔸^{1}(k) let Λ_{a} = ⟨Λ, v_{0}+a(v_{1}−v_{0})⟩. As Λ_{0} and Λ_{1} are selfdual one easily sees that Λ_{a} is selfdual for each a. By [V1], Lemma 3.4 there is a morphism 𝔸^{1} → M_{red}^{np}(𝕏) mapping each point a as above to the point of M_{red}^{np}(𝕏) corresponding to Λ_{a}. As all Λ_{a} are selfdual, this induces a corresponding morphism f : 𝔸^{1} → M_{red}. Hence f(0) and f(1), the points corresponding to Λ_{0} and Λ_{1}, are in the same connected component of M_{red}.

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Eva Viehmann

Mathematisches Institut der Universität Bonn Beringstrasse 1 53115 Bonn Germany viehmann@math.uni-bonn.de