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Special cycles on unitary Shimura varieties I. unramified local theory

The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n-1) over Q is uniformized by a formal scheme \Cal N. In the case when p is inert, we define special cycles Z(x) in \Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional -- in which case it reduces to a single point -- we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.


Introduction
A relation between a generating series constructed from arithmetic cycles on an integral model of a Shimura curve and the derivative of a Siegel Eisenstein series of genus 2 was established by one of us in [10]. There, the hope is expressed that such a relation should hold in greater generality for integral models of Shimura varieties attached to orthogonal groups of signature (2, n − 2) for any n. The case of Shimura curves corresponds to n = 3; the relation for n = 2 is established in [16], and the cases n = 4 and n = 5 are considered in [13,15]. However, the case of arbitrary n seems out of reach at the present time, since these Shimura varieties do not represent a moduli problem of abelian varieties with additional structure of PEL-type, so that it is difficult to define and study integral models of them. For the low values of n mentioned above, the analysis depends on exceptional isomorphisms between orthogonal and symplectic groups.
In the present series of papers, we take up an idea already mentioned in a brief remark at the very end of section 16 of [10] and consider integral models of Shimura varieties attached to unitary similitude groups of signature (1, n − 1) over Q. These varieties are moduli spaces for abelian varieties for arbitrary n, and there are good integral models for them, at least away from primes of (very) bad reduction, [8], [19], [20], [21], [22]. In a sequel to the present paper, we define special arithmetic cycles in a modular way and study the classes determined by such cycles, together with suitable Green currents, in the arithmetic Chow groups. The ultimate goal is to relate the generating series defined by the height pairings, or arithmetic intersection numbers, of such classes to special values of derivatives of Eisenstein series for the group U(n, n). It should be noted that the complex points of our cycles coincide with the cycles defined in [9], [11], and [12], where the modular properties of the generating functions for the associated cohomology classes are established.
In the present paper, as a first step in this study, we consider the local analogue, in which the Shimura variety is replaced by a formal moduli space of p-divisible groups, the special arithmetic cycles are replaced by formal subvarieties, and the special values of the derivative of the Eisenstein series are replaced by the derivatives of representation densities of hermitian forms. The link between this local situation and the global one is provided by the uniformization of the supersingular locus by the formal schemes introduced in [23]. A similar local analogue occurs in our earlier work [14], where the intersection numbers of formal arithmetic cycles on the Drinfeld upper halfspace, which uniformizes the fibers of bad reduction of Shimura curves, are related to the derivative of representation densities of quadratic forms. The results of [14] are an essential ingredient in the global theory of arithmetic cycles on Shimura curves established in [17]. The results of the present paper will play a similar role in the unitary case.
It turns out that when two global cycles are disjoint on the generic fiber, their intersections are supported in the fibers at non-split primes. Thus, for primes not dividing the level, there are two cases, depending on whether the prime is inert or ramified in the imaginary quadratic field. As indicated in the title, in the present paper we handle the unramified primes.
We now describe our results in more detail.
Let k = Q p 2 be the unramified quadratic extension of Q p and let O k = Z p 2 be its ring of integers. Let F =F p and write W = W (F) for its ring of Witt vectors. There are two embeddings ϕ 0 and ϕ 1 = ϕ 0 • σ of k into W Q = W ⊗ Z Q. Let Nilp W be the category of W -schemes S on which p is locally nilpotent. For any scheme S over W , letS = S × Spec W Spec F be its special fiber.
Up to isogeny, there is a unique such a triple (X, ι, λ X ) over F such that X is supersingular 1 . Fixing (X, ι, λ X ), we denote by N the formal scheme over W which parametrizes the quadruples (X, ι, λ X , ρ X ) over schemes S in Nilp W , where (X, ι, λ X ) is as above, and where ρ X : X × SS → X × FS is a quasi-isogeny which respects the auxilliary structures imposed. (See section 2.1 and [23], section 1, for the precise definition of N .) Then N is formally smooth of relative dimension n − 1 over W , and the underlying reduced scheme N red is a singular scheme of dimension [(n − 1)/2] over F.
In order to explain our results, we need to recall some of the results on the structure of N red due to Vollaard [29], as completed by Vollaard and Wedhorn [31]. To the polarized isocrystal N of X there is associated a hermitian vector space (C, { , }) of dimension n over k satisfying the parity condition ord det(C) ≡ n + 1 mod 2.
Here det(C) ∈ Q × p /N (k × ) is the coset determined by det(({c i , c j })) for any k basis {c i } for C. Note that this condition determines (C, { , }) up to isomorphism. A vertex of level i is an O k -lattice Λ in C with is the dual lattice. Such lattices correspond to the vertices of the building B(U(C)) of the unitary group U (C), hence the terminology. The type of a vertex Λ is the index t(Λ) of p i+1 Λ ∨ in Λ. In fact, t(Λ) is always an odd integer between 1 and n. To every vertex Λ of level i, Vollaard and Wedhorn associate a locally closed irreducible subset V(Λ) o of N red of dimension 1 Recall that this means that X is isogenous to the nth power of the p-divisible group of a supersingular elliptic curve.
where Λ ′ runs over all vertex lattices of level i contained in Λ. Note that for such vertices t(Λ ′ ) ≤ t(Λ).
b) The union of V(Λ) o , as Λ ranges over all vertices of level i, is a connected component N i of N red , and as i varies, all connected components of N red arise in this way.
Thus the combinatorics of the stratification of N red are controlled by the building B(U(C)), just as the (much simpler) stratification of the special fiber of the formal model of the Drinfeld half-space is controlled by the tree B(GL 2 (Q p )).
We next define special cycles on N . Let (Y, ι, λ Y ) be the basic object over F used in the definition of the signature (1, n − 1) moduli space N in the case n = 1. Thus Y is a supersingular p-divisible group over F of dimension 1 with O k -action ι which satisfies the signature condition (1, 0) with its natural p-principal polarization λ Y . Next, let (Y, ι, λ Y ) be the triple obtained from (Y, ι, λ Y ) by changing ι to ι • σ. The O k -action on Y satisfies the signature condition (0, 1), and, again, the triple (Y, ι, λ Y ) is unique up to isogeny. Since Y has height 2, the pair (Y, ι) has a canonical lift (Ȳ , ι) over W , [3].
The space of special homomorphisms is the k-vector space with k-valued hermitian form given by Hereŷ is the dual of y. The parity of ord det(V) is always odd.
For a pair of integers (i, j) and a special homomorphism x ∈ V, there is an associated special cycle Z i,j (x) where, for any S ∈ Nilp W , Z i,j (x)(S) is the subset of points (X, ι, λ X , ρ X ) in N j (S) where the composition, a quasi-homomorphism, from the canonical lift of Y to X. Then Z i,j (x) is a relative (formal) divisor in N j . More generally, for an m-tuple x = [x 1 , . . . , x m ] of special homomorphisms x r ∈ V, the associated special cycle Z i,j (x) is the intersection of the special cycles associated to the components of x.
For a collection x of special homomorphisms, we define the fundamental matrix We now assume that m = n. This will be the situation that arises from the global setting when one considers the arithmetic intersections of cycles in complementary dimensions. First, we show that, if T / ∈ Herm n (O k ), then Z i,j (x) is empty. Next, assume that det T (x) = 0, as will be the case in the global setting when the cycles do not meet on the generic fiber. When T is integral, our main result describes the structure of the cycle Z i,j (x) in terms of the Jordan block structure of T .
(i) (compatibility with the stratification) Z i,j (x) red is a union of strata V(Λ) o where Λ ranges over a finite set of vertices of level j which can be explicitly described in terms of x.
(ii) (dimension) Let r 0 ( T ) = n − rank(red( T )) be the dimension of the radical of the hermitian form red( T ). Then Z i,j (x) red is purely of dimension be a Jordan decomposition ofT and let n + even = i≥2 even n i and Then Z i,j (x) red is irreducible if and only if max(n + even , n + odd ) ≤ 1. Moreover its dimension is then 1 2 (n 1 + n + odd − 1).
(iv) (zero-dimensional case) Z i,j (x) red is of dimension zero if and only ifT is O k -equivalent to diag(1 n−2 , p a , p b ), where 0 ≤ a < b and where a + b is odd. In this case, Z i,j (x) red consists of a single point ξ, and the length of the local ring O Z i,j (x),ξ is finite and given by The right hand side of the last identity can be expressed in terms of representation densities of hermitian forms. Recall that, for nonsingular hermitian matrices S ∈ Herm m (O k ) and T ∈ Herm n (O k ), with m ≥ n, the representation density α p (S, T ) is defined as The density depends only on the GL m (O k )-equivalence class of S (resp. the GL n (O k )-equivalence class of T ). An explicit formula for α p (S, T ) has been given by Hironaka, [5]. For r ≥ 0, let S r = diag(S, 1 r ). Then for a polynomial F p (S, T ; X) ∈ Q[X], as can be seen immediately from Hironaka's formula.
If m = n and ord(det(S)) + ord(det(T )) is odd, then α p (S, T ) = 0. In this case, we define the derivative of the representation density The right hand side of the identity in (iv) of Theorem 1.1 is now expressed in terms of hermitian representation densities, as follows.
In this form, the formula in (iv) should hold for any nonsingular fundamental matrix, that is, the relation between derivatives of representation densities and intersection multiplicities should continue to hold even in the case of improper intersections. More precisely: The Euler-Poincaré characteristic appearing here is indeed finite, since it can be shown that O Z i,j (x) is annihilated by a power of p. In the case that Z i,j (x) red is of dimension zero, it can be shown that there are no higher Torterms on the LHS of the above identity [27], so that indeed the statement (iv) of the above theorem confirms the conjecture in this case. Note that the analogue of Conjecture 1.3 in the case of improper intersections of cycles on the Drinfeld space was proved in [14]. The case of improper intersections of arithmetic Hirzebruch-Zagier cycles is considered by U. Terstiege in [26] and [27]. 3 The layout of the paper is as follows. In section 2, we define the moduli space N and recall the results of Vollaard and Wedhorn concerning its structure.
In section 3, we introduce special cycles Z i,j (x) on N . In section 4, we prove the statements on Z i,j (x) red in our main theorem. The rest of the paper is concerned with the determination of the length of O Z i,j (x),ξ in the situation of (iv) of the main theorem. In section 5, this problem is reduced to a deformation problem on 2-dimensional formal p-divisible groups. In section 6, we introduce the analogue in our context of Gross's quasi-canonical divisors. In section 7, we solve a lifting problem of homomorphisms analogous to the one solved by Gross in the classical case. In section 8, we use the results of the previous two sections to solve the deformation problem of section 5. Finally, in section 9, we relate the RHS in (iv) of the main theorem to representation densities.
There are two important ingredients of algebraic-geometric nature that are used in our proofs. The first is the results of I. Vollaard and T. Wedhorn on the structure of N red , cf. Theorem 2.7. The second is the determination, due to Th. Zink, of the length of a certain specific deformation space in equal characteristic, cf. Proposition 8.2. Both are based on Zink's theory of displays and their windows, [35,36].
We thank I. Vollaard, T. Wedhorn, and Th. Zink for their contributions and for helpful discussions concerning them. We also thank B. Howard, U. Terstiege and the referee for their remarks on the text. We also gratefully acknowledge the hospitality of the Erwin-Schrödinger Institute in Vienna in 2007, where part of this work was done.
2. Structure of the moduli space N In this section we recall some facts about the moduli space N from [29]. We write k = Q p 2 for the unramified quadratic extension of Q p and O k = Z p 2 for its ring of integers. We also write F =F p and W = W (F) for its ring of Witt vectors. There are two embeddings ϕ 0 and 2.1. Definition of the moduli space. Let (X, ι) be a fixed supersingular p-divisible group of dimension n and height 2n over F with an action ι : O k → End(X) satisfying the signature condition (r, n − r), Let λ X be a p-principal polarization of X for which the Rosati involution * satisfies ι(α) * = ι(α σ ). The data (X, ι, λ X ) is unique up to isogeny.
Let Nilp W be the category of W -schemes on which p is locally nilpotent. The functor N = N (r, n − r) associates to a scheme S ∈ Nilp W the set of isomorphism classes of data (X, ι, λ X , ρ X ). Here X is a p-divisible group over S with O k -action ι satisfying the signature condition (r, n − r), and λ X is a p-principal polarization of X/S, such that the Rosati involution determined by λ X induces the Galois involution σ on O k . Finally, Here, two such data (X, ι, λ X , ρ X ) and (X ′ , ι ′ , λ X ′ , ρ X ′ ) are said to be isomorphic if there is a O k -linear isomorphism α : X → X ′ with (α× W F)•ρ X = ρ X ′ , such that α ∨ •λ X ′ •α is a Z × p -multiple of λ X . This functor is represented by a separated formal scheme N , locally formally of finite type over W , [23]. Furthermore, N is formally smooth of dimension r(n − r) over W , i.e., all completed local rings of points in N (F) are isomorphic to W [[t 1 , . . . , t r(n−r) ]]. Moreover, there is a disjoint sum decomposition where N i is the formal subscheme where the quasi-isogeny ρ X has height ni.
We next review the structure of the underlying reduced subscheme N red of N , following [29]. Note that N red (F) = N (F). Let M be the (covariant) Dieudonné module of X and let N = M ⊗ Z Q be the associated isocrystal.
Then N has an action of k and a skew-symmetric W Q -bilinear form , and with αx, y = x, α σ y , for α ∈ k.
and  Since the isocrystal N is supersingular, the operator τ = V −1 F = pV −2 is a σ 2 -linear automorphism of degree 0 and has all slopes 0. Let be the space of τ -invariants, so that C is a n-dimensional Q p 2 -vector space and where δ ∈ Z × p 2 with δ σ = −δ is fixed once and for all. Note that the corresponding form in [29] is taken without the scaling by δ. By (2.2), Thus, { , } defines a Q p 2 -valued hermitian form on N τ =1 and, in particular, on C. Note that, since the polarization form , is non-degenerate on N with N 0 and N 1 isotropic subspaces, and since F N 0 = N 1 , the hermitian form { , } is non-degenerate on C.
Proof. The condition M = p i M ⊥ is equivalent to the condition Since V F = F V = p, this is, in turn, equivalent to F M 1 , F M 0 = p i+1 W , by (2.2). This last identity can be rewritten as as claimed.
Note that the lattice M 0 in N 0 determines the lattice M 1 in N 1 , either by (2.8) or by the condition Remark 2.4. As observed in [29], Lemma 1.16, for an O k -lattice A in C with there is a O k -basis for A for which the hermitian form {, } has matrix diag(1 r , p1 n−r ).
Remark 2.5. We record the two simplest examples for later use. We first consider the case of signature (1, 0). Let us call Y the base point used to define N above. In this case we have C = Q p 2 · 1 0 with hermitian form given by {1 0 , 1 0 } = 1, and N 0 = W Q · 1 0 . Moreover, N i (F) ≃ D i is empty for i odd, and N 2i (F) ≃ D 2i consists of a single point corresponding to the , and the polarization , 0 is given by as required. Also note that End(Y) can be identified with the ring of integers O D in the quaternion division algebra over Q p , and that the endomorphism In particular, N 2i = Spf W for every i.
In the case of signature (0, 1), again C = Q p 2 ·1 0 but now with hermitian form given by {1 0 ,1 0 } = p. Moreover, N i (F) ≃ D i is empty for i odd, and N 2i (F) ≃ D 2i consists of a single point corresponding to the W -lattice A = and the polarization is given by Note that the formal O k -module Y is obtained from Y by identifying the underlying p-divisible groups, but changing the O k -action by precomposing it with the Galois automorphism σ of O k . On the level of Dieudonné modules, this identification is given by switching the roles of M 0 0 and M 0 1 . In particular, the canonical liftȲ of Y to W is isomorphic to the canonical lift Y , with the conjugate O k -action.
2.2. The case of signature (1, n−1). From now on, we assume that r = 1 so that our signature is (1, n − 1).
The structure of D i is best described in terms of strata associated to vertices of the building for the unitary group U(C) of the hermitian space (C, {, }).
Here we are identifying lattices in C with τ -invariant lattices in N 0 . The type of a vertex Λ is the index t = 2d + 1 of p i+1 Λ ∨ in Λ (which is always an odd integer between 1 and n).
and the index of p i+1 Λ ∨ in Λ, the type of Λ, is 2d + 1.
As Λ ranges over the vertices of level i, the subsets V(Λ) of D i are organized in a coherent way, as follows.
For two vertices Λ and Λ ′ of level i, cf. [29], Prop. 2.7. In this case we have t(Λ) ≤ t(Λ ′ ) for the types. Let Then by [29], Prop. 3.5 where the (finite) union on the right hand side is disjoint and runs over all vertex lattices of level i contained in Λ (these are then of type t(Λ ′ ) ≤ t(Λ)).
The subsets V(Λ) and V(Λ) o have an algebraic-geometric meaning.
Vollaard [29] has proved this for signature (1, s) with s = 1 or 2. The general case is in [31]. In [31] it is also proved that the variety corresponding to V(Λ) is smooth. We note that N i is always non-empty if n is even; if n is odd, then N i is non-empty if and only if i is even, cf. [29], Prop. 1.22.
We conclude this section with the following observation about scaling, which will be useful later.
Proof. Here, in cases (ii) and (iii), note that 3. Cycles in the moduli space N .
Let (Y, ι) be the p-divisible group of dimension 1 and height 2 over F, with an action ι : O k → End(Y) of O k and with principal polarization λ Y satisfying the signature condition (0, 1), cf. Example 2.5. Let N 0 = N (0, 1) be the corresponding moduli space as in section 1. Recall that, by Remark 2.5, Definition 3.1. The space of special homomorphisms is the k-vector space We note that N 0 2i has been identified with Spf W , via the canonical liftȲ of Y, with its O k -action. Hence Z ij (x) can be identified with a closed formal subscheme of N j . (ii) The definition of the special cycles is compatible with intersections. Specifically, the intersection of Z(x) and Z(y) is the locus where the whole Remark 3.4. We note that N 0 2i has been identified with N 0 0 . Explicitly, Here the point is that the compositions ρ −1 Recall that a relative divisor is a closed formal subscheme, locally defined by one equation, which is neither a unit nor divisible by p.
Proof. The first statement follows from [23], Proposition 2.9. To prove the second statement, it suffices to prove that Z i,j is a relative divisor in N j (via the second projection in N 0 × N ). By following the proof of the corresponding statement in [26], Prop. 4.5, we see that, in order to prove that Z i,j (x) is locally defined by the vanishing of one equation, it suffices to prove the following statement. Let A be a W -algebra and let A 0 = A/I, where the ideal I satisfies I 2 = 0. We equip I with trivial divided powers. We assume given a morphism φ : Spec A → N j whose restriction to Spec A 0 factors through Z i,j . Then the obstruction to factoring the given morphism through Z i,j is given by the vanishing of one element in I.
The value DȲ of the Dieudonné crystal ofȲ × Spec W Spec A on (A, A 0 ) is given by M ⊗ W A, and is equipped with its Hodge filtration where FȲ is generated by the element1 0 . Similarly, the Dieudonné crystal D X of the pullback to A of the universal object (X, ι, λ) over N comes with its Hodge filtration The fact that the restriction of φ to Spec A 0 factors through Z i,j (x) implies that there is an O k -linear homomorphism α : DȲ → D X of A-modules, which respects the filtrations (3.2) and (3.3) after tensoring with A 0 . We need to show that the condition that α respect the filtrations (3.2) and (3.3) is locally defined by one equation. However, this condition is obviously that α(1 0 ) ∈ F X , i.e., that the image of α(1 0 ) in the degree zero component (Lie X) 0 vanishes. However, thanks to the signature condition, (Lie X) 0 is a locally free A-module of rank 1. After choosing a local generator of (Lie X) 0 , we may identify α(1 0 ) with an element in A with zero image in A 0 . Hence the condition is described by the vanishing of one element in the ideal I.
We still have to show that this element is non-trivial, and that it is not divisible by p. We first note the following simple fact.
Then V is open and closed in X red .
Proof. Consider the ideal sheaf Hence we obtain an identity in O X,x ′ , Now we prove that a local equation for Z i,j (x) is not divisible by p. Otherwise, by the previous lemma, and since, by Theorem 2.
We now distinguish the cases n ≥ 3 and n ≤ 2.
In the case n ≥ 3 we appeal to Proposition 3.10 below (of course, the proof of this proposition does not use the statement we are in the process of proving). According to this proposition, the inclusion where Λ runs through all vertices of level j. By the next lemma, the intersection (3.4) is trivial.
Here the intersection runs over all vertices of level j.
Proof. By the reduction argument right after Corollary 3.11 below, we may assume j = 0. First let n be odd. By the parity condition of [29] recalled in (4.1) below, we may choose a basis of C as follows. Let n − 1 = 2k with k ≥ 1. We choose a basis e, f ±1 , . . . , f ±k with Hence Λ is a vertex of level 0. By varying a ∈ Z k , the intersection of these vertices is zero, which implies the assertion in this case. Now let n be even. Let n − 1 = 2k + 1, with k ≥ 1. We choose a basis e, f ±1 , . . . , f ±k , g with Then for a = (a 1 , . . . , a k ) ∈ Z k , the lattice is a vertex of level 0, and the assertion follows as before.
This proves the assertion for n ≥ 3. For n = 1, the assertion is trivial. Now let n = 2. In this case N red is a discrete set of points, but the local rings are two-dimensional, hence the assertion is non-trivial in this case. Let y ∈ Z i,j (x)(F). By the uniformization theorem [23], Thm. 6.30, the complete local ring O N j ,y is isomorphic to the completion of the local ring of a closed point of the integral model of the Shimura variety attached to GU(1, 1). Hence by Wedhorn's theorem [32], the pullback of the universal p-divisible group X to the generic point of Spec Hence there is no non-trivial homomorphism fromȲ into this p-divisible group, hence the closed subscheme of Spec O N j ,y cut out by an equation of Z i,j (x) does not contain the special fiber, as was to be shown.
To study the set Z(x)(F) of F-points of a special cycle Z(x), we apply the construction of the previous section and reformulate things in terms of the hermitian space C.
We begin by describing the space V of special homomorphisms. Recall from corresponds to a homomorphism, which we also denote by x, fromN 0 to N . This homomorphism has degree zero with respect to the grading given by the O k -action, and so we may write The hermitian form on V defined by (3.1) can be written as where y * is the adjoint of y with respect to the polarizations, i.e., for u 0 ∈N 0 and u ∈ N , Lemma 3.8. For x = x 0 + x 1 and y = y 0 + y 1 in V, The component y * 0 • x 1 is found in the same way.
In summary, we have proved the following.
Lemma 3.9. There is an isomorphism The hermitian forms on the two spaces are related by Then the image of the projection of We call the hermitian matrix There is a variant of it which will also be useful in the sequel. Namely, when considering Z i,j (x), where the fundamental matrix of x is T , we will call the matrixT = p 2i−j T the scaled fundamental matrix attached to Z i,j (x).
Note that the matrix determined by the components of x 0 (1 0 ) is then We note various 'scaling relations' among the cycles Z i,j (x). First, there is an isomorphism Next, note that These two relations are simply the translation into the language of lattices of the scaling relations on the level of formal schemes in Remark 3.4. Finally, note that the set of lattices D j for the hermitian space (C, {, }) coincides with the set of lattices D 0 for the space (C, p j {, }). Both sets are empty if n and j are both odd.

Hermitian lattices
In this section, we consider the cycle Z i,j (x) determined by an n-tuple x of special homomorphisms with nonsingular fundamental matrix T = h(x, x) ∈ Herm n (k). For global reasons (cf. the Introduction), the cycle Z i,j (x) has empty generic fiber in this case, i.e., p is locally nilpotent on Z i,j (x). We give a description of Z i,j (x)(F) as a union of the strata in N 0 2i × N j defined in [29].
Recall [29] that the hermitian space (C, {, }) is determined up to isomorphism by its dimension, since We first note that W i,j (L) is a union of strata V(Λ).
where the Λ's are vertices of level j.
Proof. The lattice Λ(A) associated to A ∈ D j by Lemma 2.6 is the smallest τ -stable W -lattice containing A. By duality, Our main results about the structure of W i,j (L) are the following. In the rest of the section,T = p 2i−j T will denote the corresponding scaled fundamental matrix.
, let red(T ) be the image ofT in Herm n (F p 2 ), and let t 0 = t 0 (T ) be the largest odd integer less than or equal to n − rank( red(T ) ).
By Theorem 2.7, (ii) we deduce from this theorem:   Remark 4.8. Is it true that W i,j (L) is always connected?
Before proving these results, we make a few simple observations. First of all, since W i,j (L) = W 0,j (p i L), it suffices to consider the case where i = 0. Second, for a lattice A ⊂ C, note that p j A ∨ is the dual lattice with respect to the scaled hermitian form p −j {, }. Thus, if we denote by C (j) the scaled hermitian space (C, p −j {, }), we have D j = D (j) 0 in the obvious notation. Thus, Thus it suffices to consider the case where i = j = 0. It is important to note that for n odd, N j (F) is empty unless j is even. Thus, in all cases, the space C (j) again satisfies the parity relation (4.1).
From now on, in this section, we assume that i = j = 0 and write W(L) for W 0,0 (L). Also, all vertices will have level 0, and we assume thatT = T = h(x, x) ∈ Herm n (O k ).
First note that if L ⊂ pΛ ∨ for some vertex Λ, then the inclusions show that Λ/pΛ ∨ is a subquotient of pL ∨ /L and, in particular, the type t = t(Λ) is constrained by the structure of pL ∨ /L. More precisely, let D = D(L) = pL ∨ /L with k/O k -valued hermitian form determined by h( , ) = p −1 {, }. Note that pL ∨ is the dual lattice of L with respect to h, so that the resulting hermitian form on D is nondegenerate.
Next, we record a few more general facts. For any Λ ∈ Vert(L), let B ∈ GrD(L) be the O k -submodule of D = D(L) associated to Λ. Since the In particular, note that D[p] ⊂ (pB) ⊥ and that the quotient (pB) ⊥ /B ⊥ is killed by p. Since the pairing on D induced by h is perfect, we have where r = dim(Λ ∨ ∩ pL ∨ )/Λ. But the inclusion on the right implies that the subspace B[p] ⊂ D[p] has codimension at most r, so that we obtain Lemma 4.12. If Λ ∈ Vert(L) with t(Λ) < t 0 , then either or the special case (⋆⋆) m = t 0 , t = m − 2, r = 1, and dim B[p] = m − 1 holds.
Proof. By assumption, t 0 − t ≥ 2 is even so that r ≥ 2, as claimed, except in the special case.
Lemma 4.13. In the special case (⋆⋆), the lattice Proof. In this case, we have the picture.
For any The previous lemmas show that every Λ ∈ Vert(L) is contained in some Λ 0 ∈ Vert(L) with t(Λ 0 ) = t 0 . On the other hand, by Lemma 4.9, the type of any lattice Λ ∈ Vert(L) is at most t 0 . This completes the proof of Theorem 4.2.
Corollary 4.14. Suppose that Λ ∈ Vert(L) with t 0 = t(Λ) maximal and let B ∈ GrD(L) be the associated O k -submodule. Then Proof. If m is odd, then t = t 0 = m in Proof of Theorem 4.5. Suppose that T has the given Jordan decomposition with respect to some basis e 1 , . . . , e n of L, and note that pL ∨ has basis p −a i e i , where h(e i , e i ) = p a i . If max(n + even , n + odd ) ≥ 2, i.e., if (4.4) n + even = i≥2 i even n i ≥ 2 (resp. n + odd = i≥3 i odd n i ≥ 2 ), we can scale the e i 's to a basis f 1 , . . . , f n of C for which the hermitian form h has matrix T 0 = diag(1 n ′ , p 1 n ′′ , p a 1 2 ) with a = 2 (resp. a = 3). Let L 0 be the lattice spanned by f 1 , . . . , f n , and note that . Also note that L ⊂ L 0 so that Vert(L 0 ) ⊂ Vert(L). Take u ∈ Z × p 2 with uu σ = −1, and let These are isotropic vectors in L 0 with h(g 1 , g 2 ) = 2p a . Let and Then . . , f n ′ +n ′′ , p 1−a g 1 , g 2 ], so that Λ 1 and Λ 2 are vertices in Vert(L 0 ) of type n ′′ + 2. Suppose that Λ 1 and Λ 2 were contained in a common vertex Λ ∈ Vert(L). Then we would have But h(p −a g 2 , p −a g 1 ) = 2 p −a , so this is not the case. Thus there is more than one vertex Λ ∈ Vert(L) with t(Λ) = t 0 when (4.4) holds. Now we prove the converse. Suppose that max(n + even , n + odd ) ≤ 1. Then there are several cases for the Jordan decomposition of T . First suppose that n + even = n + odd = 1, so that T has Jordan decomposition diag(1 n 0 , p1 n 1 , p a , p b ) with 2 ≤ a < b and a + b odd. Thus, L = [e 1 , . . . , e n ] and pL ∨ = [e 1 , . . . , e n 0 , p −1 e n 0 +1 , . . . , p −1 e n−2 , p −a e n−1 , p −b e n ].
Thus, we may assume that n = 2 and that T = diag(p a , p b ) where 2 ≤ a, b and a is even. We proceed by explicit computation. Suppose that L = [e 1 , e 2 ] and write pΛ ∨ = [e 1 , e 2 ]S for S ∈ GL 2 (k) unique up to right multiplication by an element of GL 2 (O k ). Then we have Λ = [e 1 , e 2 ]T −1tS−1 and pL ∨ = [e 1 , e 2 ]T −1 , and the various inclusions amount to the following conditions: Moreover, Λ has type t(Λ) = t 0 = 1 if and only if ord(det tS T S) = 1.
Assuming that this is the case, we may modify S on the right by an element of GL 2 (O k ) so that T 1 := tS T S = diag(1, p). Note that the last of the above conditions is then immediate. Write S = diag(p −a/2 , p −(b−1)/2 )S 0 . Then tS 0 T 1 S 0 = T 1 so that u = det(S 0 ) has norm 1 and hence is a unit. After replacing S 0 by S 1 = diag(1,ū)S 0 , so that det(S 1 ) = 1, a short calculation shows that for α and β ∈ k with 1 = αᾱ + p −1 ββ. Since the two terms on the right side of this last identity have ord's of opposite parity, we must have ord(α) = 0 and ord(β) ≥ 1, so that S 1 and S 0 lie in GL 2 (O k ). Thus Λ = [p −a/2 e 1 , p −(b+1)/2 e 2 ] is the unique vertex in Vert(L) with t(Λ) = 1. Of course, this argument just amounts to the fact that the isometry group of T 1 is anisotropic so that the building of this group reduces to a single point.
This completes the proof of Theorem 4.5.

Intersection multiplicities
In this section, we fix i, j and consider a non-empty special cycle Z i,j (x), where we assume that x is an n-tuple of special homomorphisms whose scaled fundamental matrixT = p 2i−j h(x, x) = p 2i−j−1 {x, x}, which is still assumed to be non-degenerate, satisfies rank(red(T )) ≥ n − 2.
By Corollary 4.7, this implies that the cycle Z i,j (x) in N 0 2i × N j has underlying reduced scheme of dimension 0 which, in fact, reduces to a single point. Write Z i,j (x) = Spec R(x) for a local W -algebra R(x) with residue field F. The arithmetic degree of Z i,j (x) is then, by definition, where length W R(x) is the length of R(x) as a W -module.
Note that a + b ≡ ord(det(T )) mod 2 is odd, cf. (4.2) and the remarks after Corollary 4.7, so that, in fact, 0 ≤ a < b. As in the previous section, we may reduce to the case i = j = 0, which we assume from now on. Accordingly we write Z(x) for Z 0,0 (x).
The first step in the proof is to reduce to the case n = 2.
Lemma 5.2. Suppose that y = xg for g ∈ GL n (O k ) has matrix of inner products h(y, y) = diag(1 n−2 , p a , p b ) where a is even and b is odd (but a and b are not ordered by size). Then Z(x)(F) = Z(y)(F) corresponds to the lattice A = τ (A) given by 5 A = W y 1 + · · · + W y n−2 + W p −a/2 y n−1 + W p −(b+1)/2 y n .
If (X, ι, λ X , ρ X ) is the corresponding p-divisible group, then there is an isomorphism where inc i denotes the inclusion into the i-th factor of the product.
Here Π denotes the fixed uniformizer in O D = End(Y), cf. Remark 2.5.
Proof. Let a = 2r and b = 2s + 1. For the given lattice A, we have pA ∨ = W y 1 + · · · + W y n−2 + W p −r y n−1 + W p −s y n .
so that pA ∨ ⊂ A with index 1. Moreover, since τ (y i ) = y i , we have τ (A) = A so that A = Λ(A) is a vertex and Z(x) = V(Λ(A)) as claimed. For convenience, we let u i = y i , for i ≤ n−2, u n−1 = p −r y n−1 and u n = p −s−1 y n . The Dieudonné module M = M 0 + M 1 associated to A in Proposition 2.3 has M 0 = A and M 1 = W v 1 + · · · + W v n , where v i = F −1 u i , for i ≤ n − 1 and v n = pF −1 u n . Here recall that F M 1 = pA ∨ . Then, since we have u i , v j = δ δ ij for all i and j.
Recalling that Y has Dieudonné module M 0 = W1 0 + W1 1 , with F1 0 = p1 1 , as polarized Dieudonné modules. The inclusion maps are then given by Finally, recalling that we have already identified the isocrystal of X with that of X via ρ X , we see that the morphism y i : Y → X does indeed yield the given elements y i = y i0 (1 0 ) of N 0 . On the other hand, y i1 F = F y i0 , so that py i1 (1 1 ) = y i1 (F1 0 ) = F y i0 (1 0 ) = F y i . Hence y i : M 0 → M is given bȳ Thus, for i = n, we have y n = p s inc n • Π where Π :1 0 → p1 1 ,1 1 → 1 0 and is W -linear. The cases i ≤ n − 1 are clear.
Let Def (X, ι, λ X ; x) be the universal formal deformation ring of the collection (X, ι, λ X ; x), where x is our given n-tuple of special homomorphisms x i : Y → X. Since, as in the previous lemma, it is equivalent to deform the linear combination y = xg for g ∈ GL n (O k ), we have (5.1) Def (X, ι, λ X ; x) = Def (X, ι, λ X ; y).
We now want to calculate the length of this deformation ring.
Thus, it suffices to compute the length of the deformation ring (5.1) in the case n = 2, where y = [y 1 , y 2 ]. By the previous lemma, we have where a is even and b is odd (but a and b are not ordered by size).
Let M denote the universal deformation space of (X, ι, λ X ). Then M ≃ Spf W [[t]] and the locus Z(y 1 ) (resp. Z(y 2 )) where y 1 (resp. y 2 ) deforms is a (formal) divisor on this 2-dimensional regular (formal) scheme. The problem is now to determine the length of the intersection of these two formal subschemes of M.
This problem is solved in section 8 as an application of a variant of Gross's theory of quasicanonical liftings described in the next two sections.

Quasi-canonical liftings
In this section, we first explain a general construction of extending the endomorphism ring of a p-divisible group. We then apply this construction to quasi-canonical lifts, and show how this can be used to construct liftings of (X, ι, λ X ) to finite extensions of W .
For a p-divisible group X, we define the p-divisible group O k ⊗ X in the standard way as an exterior tensor product, e.g. [2], p. 131. Explicitly, after choosing a Z p -basis e = (e 1 , e 2 ) of O k , we define O k ⊗ X to be X × X. Any other choice e ′ of a Z p -basis differs from the first choice by a matrix g ∈ GL 2 (Z p ). Then g defines an automorphism α e,e ′ : X × X → X × X.
Hence we obtain in an unambiguous way a p-divisible group O k ⊗ X, unique up to unique isomorphism, with isomorphisms β e : O k ⊗X → X ×X, for any choice of a basis e, such that β e ′ = α e,e ′ • β e . The action by left translation of O k on itself defines an action O k → End(O k ⊗X). We obtain in this way a functor from the category of p-divisible groups to the category of p-divisible groups with O k -action. It is compatible with base change, We mention the following properties of this functor.
Similarly, for the p-adic Tate modules, There is an analogous statement for the Dieudonné module, if X is a pdivisible group over a perfect field.
(iii) For two p-divisible groups X and Y , and Proof. The first two assertions follow immediately from the definitions. For the first isomorphism in (iii), choose a Z p -basis of O k which identifies O k ⊗X and O k ⊗ Y with X 2 and Y 2 . For the second isomorphism, note that the left hand side is then identified with the matrices in M 2 (Hom(X, Y )) = Hom(X 2 , Y 2 ) that commute with the matrix for multiplication by δ.
Another useful fact is the following.
Lemma 6.2. Let p = 2. If (X, ι) is a p-divisible group with O k -action, then there is an isomorphism HereX denotes the group X with O k -action given by ι•σ. This isomorphism Proof. We simply observe the following facts. First, Finally, we will need the following construction of polarizations on O k ⊗ X.
We consider the standard hermitian form on O k , with {1, 1} = 1, and the associated perfect alternating Z p -valued form Using this form on the first factor, we have a canonical perfect pairing satisfying the identity on points α x, y = x, α σ y .
Using this pairing, we may identify the Cartier dual Hence, starting with a polarization λ : X → X ∨ , we obtain a O k -linear polarization such that the associated Rosati involution induces the Galois automorphism σ on O k .
We now apply this construction to Gross's quasi-canonical lifts. Let us recall briefly a few facts from Gross's theory, [3], [1], that we will use in what follows.
Let G be a p-divisible formal group of height 2 and dimension 1 over F. For an integer s ≥ 1, let be the order of conductor s in O k . Quasi-canonical lifts of level s always exist; they are not unique, but any two are conjugate under the Galois group Gal(M s /M ), and, in fact, this Galois group acts in a simply transitive way on all quasi-canonical lifts of level s. Quasi-canonical lifts are isogenous to the canonical lift. More precisely, there exists an isogeny ψ s : F 0 → F s of degree p s defined over W s . In terms of the Tate modules of the generic fibers, the isogeny ψ s can be described as follows. There exists a generator t of the free O k -module T p (F 0 ), such that We note that when s is even, the isogeny ψ s is compatible with the embedding of O k in O D = End(G), whereas when s is odd, it is compatible with ι • σ. More precisely, Lemma 6.4. Let Π be a uniformizer of O D which normalizes O k , as above. Given a quasi-canonical lift F s , there exists a unique O k -linear isogeny 6 The analogous groups for a ramified extension k are isomorphic. 7 More precisely, the induced embedding of O k,s coincides with the restriction to O k,s ⊂ O k of i, when s is even, or i • σ, when s is odd. This definition differs in fact from that given in loc.cit, where it is required that the induced embedding of O k,s in End(G) is equal to the restriction of the fixed embedding i of O k into OD. However, this condition is too strong since it would not allow quasi-canonical lifts for odd s, cf. for example Lemma 6.4. ψ s : F 0 → F s which induces the endomorphism Π s on the special fiber G. Moreover, the set H 0,s ⊂ O D of homomorphisms from F 0 ⊗ F = G to F s ⊗ F = G that lift to homomorphisms from F 0 to F s is precisely Π s O k .
We will refer to the canonical lift F 0 as a quasi-canonical lift of level 0. the signature condition (1, 1). Let λ be a p-principal polarization of F s . Note that λ is unique up to a scalar in Z × p , since it induces a perfect alternating form on the 2-dimensional Z p -module T p (F s ). By the construction outlined above, we obtain a p-principal O k -linear polarization λ (s) = id O k ⊗ λ on X (s) for which the Rosati involution induces the Galois conjugation on O k . The special fiber of X (s) is equal to O k ⊗ (F s ⊗ Ws F). Now F s ⊗ Ws F is equal to G; however, as a formal O k -module it is equal to Y when s is even, and equal to Y when s is odd. Applying Lemma 6.2, we obtain identifications Recall from (5.2) that X = Y × Y. We now identify X (s) ⊗ Ws F with X by using the switch isomorphism Y × Y ≃ Y × Y when s is even, resp. the identity map on Y× Y when s is odd. In all cases we have therefore obtained canonical O k -linear isomorphisms By pulling back the polarization λ X to X (s) ⊗ Ws F, we obtain a O k -linear p-principal polarization on X (s) ⊗ Ws F which differs from λ (s) by a scalar in Z × p (same argument as above, using the Dieudonné module instead of the Tate module). Hence we may change λ (s) by this scalar such that these polarizations coincide. Hence (X (s) , ι, λ (s) ) is a deformation of (X, ι, λ X ) to W s . We therefore obtain a morphism (6.5) ϕ s : Spf W s → M.
Lemma 6.5. The morphism ϕ s is a closed immersion.
Proof. Denoting by R the affine ring of M, we have to show that the mor- , and W s is a finite totally ramified extension of W . To show that ϕ * : R → W s is surjective, it therefore suffices to show that ϕ * (t) is a uniformizing parameter, i.e., that W s ⊗ R F = F. By the universal property of M, this says that the locus in Spec W s /pW s where there exists an O k -linear isomorphism α : O k ⊗ F s → O k ⊗ G is equal to Spec F. According to Lemma 6.1 we can write α = 1 ⊗ α 0 + δ ⊗ α 1 , where α 0 and α 1 are homomorphisms from F s to G. By identifying the special fibers of F s and G, α becomes a unit in O D . But then α 0 or α 1 is a unit in O D , and hence one of them defines an isomorphism from F s −→ F 0 over this locus. By [33], Cor. 4.7, this implies that the locus in question is reduced to the special point. Definition 6.6. Let Z s be the divisor on M defined by the image of ϕ s .

Deformations of homomorphisms
We consider the following lifting problem. As in [33], suppose that A is a finite extension of W with uniformizer λ, and let A m = A/λ m+1 . We also let e be the ramification index of A over W and denote by ord A the discrete valuation on A with ord A (λ) = 1/e. For integers r, s ≥ 0 let F r and F s be quasi-canonical liftings defined over A. Suppose that a homomorphism Also, as in the first part of (iii) of Lemma 6.1, write Our aim is to prove the following theorem.
Theorem 7.1. Write µ as in (7.1), and suppose that Proof. Note that µ lifts to A m if and only if the components µ i all lift to homomorphisms F 0 ⊗ W A m → F s ⊗ Ws A m . Recall from [34], section 1.4, that for a homomorphism ψ : F r ⊗ Wr F −→ F s ⊗ Ws F, n r,s (ψ) is defined to be the maximum m such that ψ lifts to a homomorphism F r ⊗ Wr A m → F s ⊗ Ws A m . Thus, m s (µ) = min So, we are reduced to determining the quantities n 0,s (µ i ). To this end, we prove the next Proposition, a slight extension of [34], Proposition 1.2.
As in [34], let H r,s ⊂ D be the subset of elements φ that lift to homomorphisms from F r to F s . For example, H s,s = O k,s . In general, if s ≥ r, with the conventions introduced in the previous section, H r,s = Π s−r O k,r and there is an isomorphism Now suppose that s > r and that ψ ∈ O D \ H r,s with l = l r,s (ψ) ≥ s − r.
Since, for 0 ≤ k < s, e s /e s−k = p k and e 0 = 1, we obtain the expressions claimed in Proposition 7.2. Theorem 7.1 follows immediately.

Computation of intersection multiplicities
We now return to the situation at the end of section 5. To compute the intersection number (5.3), we first decompose the divisors Z(y i ). Here Z s is the divisor on M given in Definition 6.6.
Proof. We begin by showing that each Z s for s in the given range lies in Z(y i ). Suppose that F s is a quasi-canonical lift of level s. There is then a unique isogeny ψ s : F 0 → F s of degree p s such that the reduction is equal to Π s . Here recall that, for any quasicanonical lift F s , an identification F s ⊗ Ws F = G is given so thatψ s is identified with an element of O D . We also write ψ s for the corresponding isogeny There is an isomorphism given by composing the isomorphism Y ×Ȳ ∼ −→ X (0) of Lemma 6.2 with the switch of factors. We then obtain an O k -linear isogeny Note that the diagram is an O k -linear homomorphism whose reduction is Then, for s of the correct parity,ỹ i :Ȳ → X (s) is a lift of y i . This shows that the divisor Z s is a component of Z(y 1 ) (resp. Z(y 2 ) ) for 0 ≤ s ≤ a even (resp. 1 ≤ s ≤ b, odd The assertion now follows from the following proposition. Let y : Y → X be an O k -linear homomorphism with y * • y = 0. Consider the universal deformation of (X, ι, λ X ) in equal characteristic over , and the maximal closed formal subscheme Z(y) p of M p , where y deforms into a homomorphism y : Y × Spec F Z(y) p → X × Mp Z(y) p . The proof of the following proposition is due to Th. Zink. Proof. (Zink): We are going to use the theory of displays [35] and their windows [36].
. We extend the Frobenius automorphism σ on W to A by setting σ(t) = t p . For any a ≥ 1, we set R a = F[[t]]/t a and A a = A/t a . Then A is a frame for R, resp. A a is a frame for R a , with augmentation ideal generated by p.
We consider the category of p-divisible groups over R which have noétale part modulo t or, in other words, the category of formal groups over R which are p-divisible modulo t. For simplicity of expression we call them formal pdivisible groups over R.
Formal p-divisible groups over R are classified by A−R-windows (M, M 1 , φ, φ 1 ), which satisfy a nilpotence condition, [36], Thm. 4. Recall that an A− Rwindow consists of a 4-tuple (M, M 1 , φ, φ 1 ), where M is a free A-module of finite rank and M 1 is a submodule containing pM such that M/M 1 is a free R-module. Furthermore, φ : M → M is a σ-linear endomorphism such that φ(M 1 ) ⊂ pM and such that φ(M 1 ) generates pM as an A-module (this last condition is easily seen to be equivalent to condition (ii) in [36], Def. 2).
There is a base change functor from A−R-windows to W −F-windows. This is compatible with base-changing formal p-divisible groups. (To see this, one first passes to the display associated to the window, then applies the base change property of displays, cf. [35], Definition 20, and then passes back to the associated window. We are using here that the frames for R and for F are chosen in a compatible way.) The category of W −F-windows is isomorphic to the category of ordinary Dieudonné modules over k. The nilpotence condition says that V is topologically nilpotent on M F . Since we will only consider deformations of formal p-divisible groups, the nilpotence condition is always automatically satisfied and we will therefore ignore it.
We denote by Proof. Choosing a normal decomposition, we have The assertion follows since we see that φ ♯ 1 induces a surjection between free A-modules of the same rank.
We obtain from (M, M 1 , φ, φ 1 ) the free A-module M 1 and the A-linear homomorphism α : as the composition In this way, the category of formal p-divisible groups over R becomes equivalent to the category of pairs (M 1 , α), consisting of a free A-module of finite rank and an A-linear injective homomorphism α : 1 , such that Coker α is an R-module which is free. An analogous description holds for the category of formal p-divisible groups over R a . Under this equivalence the category of formal p-divisible groups with an O k -action becomes equivalent to the category of pairs (M 1 , α), such that M 1 is Z/2Z-graded and α is a homogeneous morphism of degree 1.
Consider the p-divisible group Y over F with its action ι of O k . It corresponds to the pair 8 (N, β), where N is the Z/2-graded free W -module of rank 2 with and β(n 0 ) = p ⊗ n 1 , β(n 1 ) = 1 ⊗ n 0 .
By base change W → A we obtain the pair (N , β) over A with the same defining relations which corresponds to the constant p-divisible group Y over R.
The p-divisible group X over F corresponds to the and to the graded map α : M → M (σ) given by We consider the deformation of (X, ι) given by the free A-module M with the same generators as for M and the homomorphism α t (in terms of the ordered basis f 0 , e 0 , e 1 , f 1 ), One checks that this deformation respects the polarization λ X of X (rewrite the deformation in terms of the original display (M, M 1 , φ, φ 1 ) and use [32], Cor. 3.29). In fact, (M, α t ) defines the universal deformation of (X, ι, λ X ), cf. [32]. Now let y correspond to the graded A 1 -linear homomorphism, Then the length ℓ of the deformation space of γ is the maximal a such that there exists a liftγ making the diagram commute.
To calculate ℓ, we distinguish cases. First let v = 2r be even. Applying Lemma 5.2, in this case we may postcompose y with an automorphism of X such that y = p r · inc 1 , i.e. γ is given by γ = (γ 0 , γ 1 ) = (X(0), Y (0)), with In order to lift γ mod t p , we search for matrices Note that σ can be viewed as a map A 1 → A p . Since σ(X(1)) = σ(X(0)) and σ(Y (1)) = σ(Y (0)), we obtain the identities Since A p has no p-torsion, we obtain as unique solution provided that the matrices on the RHS have coefficients which are integral, i.e. which lie in W [t]/t p . More precisely, we obtain ℓ ≥ p if these coefficients are integral; otherwise ℓ is the maximum power t a such that the coefficients mod t a are integral.
Inserting the values for X(0) and Y (0), an easy calculation shows If follows that ℓ = 1 if r = 0 and ℓ ≥ p if r ≥ 1. This proves the assertion for v = 0.
In the next step we try to lift γ from A p to A p 2 , in the next step from A p 2 to A p 3 and inductively from A p n to A p n+1 for any n ≥ 1. At each step we use the map σ : A p n → A p n+1 . This gives at each step the recursive identities which can be solved after inverting p for every n.
The claim shows that γ deforms to A p 2r , but not to A p 2r+1 . In fact the upper right coefficient of Y (2r + 1) shows that γ deforms precisely to A p 2r +...+p+1 , which proves the proposition in this case.
Next we consider the case when v = 2r + 1 is odd. In this case, Lemma 5.2 shows that we may postcompose y with an automorphism of X such that y = p r · inc 2 • Π. Hence in this case In this case an easy calculation using the identities (8.4) shows that
By Proposition 8.1, Let m s (y) be the maximum m such that the O k -linear homomorphism Then m s (y) = Z s · Z(y 2 ) for s even, Z(y 1 ) · Z s for s odd.
We can write (8.6) as where we have simply taken the conjugate linear O k -action on the second factor of the sourceȲ ×Ȳ. The matrix for µ(y) is unchanged, and m s (y) is the maximum m such that this map lifts to a map To apply Theorem 7.1, we need to write µ(y) in the form (7.1). We take 1 and δ as Z p -basis for O k , and hence, for any p-divisible group X, we have an identification O k ⊗ X = X × X. If X is a p-divisible group with O k -action, then the isomorphism of Lemma 6.2 Thus, the matrix for µ(y) is if s is even, and 1 2 if s is odd. Now if s ≤ a is even, then Π a ∈ Π s O k so that l = b in Theorem 7.1. If s ≤ b is odd, then Π b ∈ Π s O k and l = a. This yields the following result.
Corollary 8.5. Let r = s. Then Z s · Z t = e min{s,t} .
By summing the Z s · Z(y 2 )'s (resp. the Z(y 1 ) · Z s 's) of Proposition 8.4 as in (8.5), we obtain the expression in Theorem 5.1.

Representation densities of hermitian forms
In this section, we show that the expression given in Theorem 5.1 for the intersection multiplicity in the case where the scaled fundamental matrixT is GL n (O k )-equivalent to diag(1 n−2 , p a , p b ) coincides, up to an elementary factor, with the derivative of a certain representation density associated tõ T . As explained in the introduction, this relation is the component at p of an identity between a global arithmetic intersection number or height, and a Fourier coefficient of the derivative of an Eisenstein series on U(n, n). To avoid introducing additional notation, we continue to suppose that k = Q p 2 is the unramified quadratic extension of Q p .
First recall that, for nonsingular matrices S ∈ Herm m (O k ) and T ∈ Herm n (O k ), with m ≥ n, the representation density α p (S, T ) is defined as The density depends only on the GL m (O k )-(resp. GL n (O k )-) equivalence class of S (resp. T ). An explicit formula for α p (S, T ) has been given by Hironaka, [5].
Let ℓ(T ) be the smallest ℓ such that p ℓ T −1 ∈ Herm n (O k ). In fact, for k > ℓ(T ), the quantity (p −k ) n(2m−n)) |A p k (S, T )| is constant and is non-zero if and only if there exists an x ∈ M m,n (O k ) such that S[x] = T . For r ≥ 0, let S r = diag(S, 1 r ). Then for a polynomial F p (S, T ; X) ∈ Q[X], as can be seen immediately from Hironaka's formula.
Recall that the isometry class of a non-degenerate hermitian space V of dimension n over k is determined by its determinant det(V ) ∈ Q × p /N (k × ). Thus, if S and T ∈ Herm n (O k ) are non-singular with ord(det(S))+ord(det(T )) odd, then α p (S, T ) = 0. In this case, we define the derivative of the representation density The main result of this section is the following. (1 − (−1) ℓ p −ℓ ).
Comparing this expression with that given in Theorem 5.1, we find the following relation between the derivative of the hermitian representation density and the arithmetic intersection multiplicity.
Proof of Proposition 9.1. The first step is the following reduction formula, which is the hermitian analogue of Corollary 5.6.1 in [7]. For the convenience of the reader, we will sketch the proof below.
Proposition 9.4 (Nagaoka). Suppose that S ′ = 1 2 and that T ′ = diag(p a , p b ) with 0 ≤ a ≤ b, but with no condition on the parity of a + b. Then We need the following preliminary results. Lemma 9.7. If ϕ : L → M with (ϕ(x), ϕ(y)) ≡ (x, y) mod p k , for some sufficiently large k, then there is an isometry η : L → ϕ(L) ⊂ M .
Lemma 9.8. Suppose that ϕ 1 and ϕ 2 are two homomorphisms satisfying the conditions of the previous lemma. Also suppose that ϕ 1 ≡ ϕ 2 mod p k , for some sufficiently large k. Then there is an isometry γ ∈ U(M ) such that ϕ 2 (L) = γ(ϕ 1 (L)).
For given L and M and a sublattice N ⊂ M such that N is isometric to L, we let In this last setφ ∈Ĩ k (L, M ) is a preimage of ϕ. By the preliminary lemmas, these sets are well defined for k sufficiently large.
Write ϕ 2 = ψ 1 + ψ 2 with ψ 1 : L 2 → p k N ∨ i /p k+a N ∨ i and ψ 2 : L 2 → N ⊥ i /p k N ⊥ i . Since we are assuming that p k N ∨ i ⊂ N i , we have (ψ 1 (x), ψ 1 (y)) ∈ (N i , p k N ∨ i ) ⊂ p k O k . Thus the condition on ϕ 2 in (9.5) just amounts to the condition (ψ 2 (x), ψ 2 (y)) ≡ (x, y) mod p k , with no restriction on ψ 1 ∈ Hom O k (L 2 , p k N ∨ i /p k+a N ∨ i ). This yields the claimed expression, once the various lattice indices are taken into account.  This proves Proposition 9.3.
We take this occasion to close a gap in [17], where we inadvertently omitted the proof of Lemma 7.7.3. This lemma is also implicitly used in [24] (the equality of divisors right after Lemma 3.1). We formulate here the lemma and give the proof. where W s (ϕ) denotes the quasi-canonical divisor of level s (relative to k).
Proof. All quasi-canonical divisors W s (ϕ) are prime divisors which are pairwise distinct and with W s (ϕ) ⊂ T for 0 ≤ s ≤ c. Hence we have an inequality of relative divisors Here e s = [W s : W ] denotes the absolute ramification index.
To determine M p · T , we first note that ϕ ∈ (Z p + Π ℓ O D ) \ (Z p + Π ℓ+1 O D ), where ℓ = 2c in case k is unramified over Q p , and ℓ = 2c + 1 in case k is ramified over Q p , comp. [34], 1.2. Now we use the result of Keating [6], Theorem 1.1 (see also [30], Theorem 2.1), which gives as the length of the deformation locus of ϕ in M p exactly the expression on the RHS of equation (9.6) above.