Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series

Let V be a rational quadratic space of signature (m,2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m+1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimedean and non-archimedian places.


Introduction
The classical Siegel-Weil formula connects the arithmetic of quadratic forms with Eisenstein series for symplectic groups [Si], [We], [KRal1]. In particular, it yields explicit formulas for the representation numbers of integers by the genus of a quadratic form in terms of generalized divisor sum functions.
The Siegel-Weil formula also has important geometric applications. For instance, it leads to formulas for the degrees of special cycles on orthogonal Shimura varieties in terms of Fourier coefficients of Eisenstein series. To describe this, we let (V, Q) be a rational quadratic space of signature (m, 2). To simplify the exposition, we assume throughout the introduction that m is even, the general case is treated in the body of this paper. Denote by H = SO(V ) the special orthogonal group of V , and let D be the corresponding hermitian symmetric space, realized as the Grassmannian of oriented negative definite planes in V (R).
For a compact open subgroup K ⊂ H(A f ) we consider the Shimura variety We consider arithmetic cycles in the sense of Gillet-Soulé (see [GiSo], [SABK]), which are given by pairs consisting of a cycle on an integral model of X K and a Green current for the cycle. For x ∈ V (R), Kudla constructed a Green function z → ξ 0 (x, z) on D. It has a logarithmic singularity along the special divisor determined by x, see (5.1). More generally, if x = (x 1 , . . . , x n ) ∈ V n (R) such that the moment matrix T = Q(x) = 1 2 ((x i , x j )) i,j is invertible, one obtains a Green current for the special codimension n cycle D x = {z ∈ D | (z, x 1 ) = . . . (z, x n ) = 0} by taking the star product ξ n 0 (x, z) = ξ 0 (x 1 , z) * · · · * ξ 0 (x n , z). It satisfies the current equation dd c [ξ n 0 (x, z)] + δ Dx = [ϕ n KM,0 (x, z)], where ϕ n KM,0 (x, z) = ϕ n KM (x, z) · e 2π tr Q(x) is essentially the Poincaré dual form considered above, and δ Dx is the Dirac current given by integration over D x . For the rest of this introduction we assume that T ∈ Sym n (Q) is invertible. Then we obtain a Green current for the cycle Z(T, ϕ) on X K by where z ∈ D and h ∈ H(A f ).
To describe the integral models of X K and the special cycles we are working with, we assume for convenience that V contains a unimodular even lattice L. This assumption can and will be relaxed when considering local integral models later on, see Remark 1.3. We let K = SO(L) be the stabilizer ofL = L ⊗Ẑ in H(A f ), and let ϕ = ϕ L = char(L n ) be the characteristic function ofL n . By work of Kisin, Vasiu, and Madapusi Pera, the Shimura variety X K has a canonical integral model X K , which is a smooth stack over Z, see [Ki], [MP], [AGHM,Theorem 4.2.2]. There is a polarized abelian scheme A of dimension 2 m+1 over X K , which is equipped with an action of the Clifford algebra C(L) of L. For an S-valued point of X K there is a space of special endomorphisms on the pull-back A S of A, which is endowed with a positive definite even quadratic form Q, see [AGHM,Section 4]. It can be used to define an integral model of Z(T, ϕ) of Z(T, ϕ) as the sub-stack of X K whose S-valued points have an x ∈ V (A S ) n with Q(x) = T . The pair Z(T, ϕ, v) = Z(T, ϕ), G (T, ϕ, v) determines a class in an arithmetic Chow group. Through the Green current it depends on v = ℑ(τ ). In analogy with the geometric situation described earlier, we would like to understand the classes of these cycles and their relations.
As before we focus on the case of top degree cycles, which is here the case n = m + 1. If T is not positive definite, then Z(T, ϕ) vanishes, but the arithmetic cycle Z(T, ϕ, v) has non-trivial current part. On the other hand, if T is positive definite, then Z(T, ϕ, v) has trivial current part, and the cycle is entirely supported in positive characteristic. In fact, if it is non-trivial then it is supported in the fiber above one single prime p. The dimensions of the irreducible components were recently determined by Soylu [So]. In particular, he showed that Z(T, ϕ)(F p ) is finite if and only if the reduction of T modulo p is of rank n−1, n − 2, or of rank n − 3 (plus a technical condition). We refer to Theorem 6.3 for details. Throughout this paper we consider the cases when either T is not positive definite, or T is positive definite and Z(T, ϕ) has dimension 0.
According to [MP,Theorem 7.4], there exists a regular toroidal compactification X K of X K with generic fiber X K . The cycle Z(T, ϕ, v) defines a class in Ch where for an arithmetic cycle (Z, G) the local degrees are defined as Here ht p (x) denotes the length of theétale local ring O Z,x of Z at the point x. Kudla conjectured the following description of the arithmetic degrees of special cycles in terms of derivatives of Siegel Eisenstein series of genus n, see [Ku2], [Ku5].
Note that for T positive definite with Z(T, ϕ) of higher dimension the arithmetic degree has to be defined more carefully as in [Te], but we do not consider this here. The conjecture can be further generalized to include the cases where T is singular, leading to an identity between the generating series of the arithmetic degrees of the Z(T, ϕ, v) and the central derivative of the Eisenstein series E(τ, s, λ(ϕ)⊗Φ κ ) analogous to (1.1), which can be viewed as an arithmetic Siegel-Weil formula. The full conjecture is known for m = 0 and for the m = 1 case of Shimura curves, see [KRY1], [KRY2].
To state our results on Conjecture 1.1, we let C = p≤∞ C p be the incoherent quadratic space over A for which C f = ⊗ p<∞ C p ∼ = V (A f ) and C ∞ is positive definite of dimension m + 2. The Eisenstein series appearing in Conjecture 1.1 is naturally associated with the Schwartz function on S(C n ) given by the tensor product of ϕ and the Gaussian on C n ∞ via the intertwining operator λ. Hence it is incoherent and vanishes at s = 0. The conjecture gives a formula for the leading term of the Taylor expansion in s at s = 0. Following Kudla [Ku2], define the Diff set associated with C and T as (1.2) Diff(C, T ) = {p ≤ ∞ : C p does not represent T }.
Then Diff(C, T ) is a finite set of odd order, and ∞ ∈ Diff(C, T ) if and only if T is not positive definite.
Theorem 1.2. Assume that T ∈ Sym n (Q) is invertible. Then Conjecture 1.1 holds in the following cases: (1) If | Diff(C, T )| > 1. In this case both sides of the equality vanish.
(2) If Diff(C, T ) = {∞}. In this case T is not positive definite, and the only contribution comes from the archimedian place, i.e., If Diff(C, T ) = {p} for a finite prime p = 2 and Z(T, ϕ)(F p ) has dimension 0. In such a case, the only contribution comes from the prime p, i.e, Remark 1.3. Since the cycle Z(T, ϕ, v) is supported in a single fiber, all assertions of Theorem 1.2 can be reformulated in terms of 'local' models of X K . We will actually prove the local analogues in much greater generality. Generalizing (2) we will show that if ∞ ∈ Diff(C, T ) then Since this is an assertion only about the complex fiber X K , we will be able to prove it for any compact open subgroup K ⊂ H(A f ) (in particular V does not have to contain an even unimodular lattice) and any K-invariant Schwartz function ϕ ∈ S(V (A f ) n ), see Theorem 7.1.
To generalize (3) we consider the canonical integral model X K,(p) of X K over the localization Z (p) . In this setting we will show that if p ∈ Diff(C, T ) for a finite prime p = 2 and Z(T, ϕ) is 0-dimensional, then This will be proved under the assumption that K is the stabilizer of a Z p -unimodular lattice L ⊂ V and for ϕ = char(L n ), see Theorem 7.3.
To prove Theorem 1.2, we decompose the Fourier coefficients of the Eisenstein series into local factors. If Φ = ⊗ v Φ v is a factorizable section of the induced representation, then where W T,v (g, s, Φ v ) is the local Whittaker function given by (2.1). It is a basic fact that the local Whittaker function W T,p (g p , 0, λ p (ϕ p )) vanishes for every p ∈ Diff(C, T ). This implies assertion (1) of Theorem 1.2 in a rather direct way. Indeed, if | Diff(C, T )| > 1 then the right hand side of the conjectured identity is automatically zero. To prove that the left hand side also vanishes, we consider for a prime p ≤ ∞ the neighboring global quadratic space V (p) at p associated with C, which is the quadratic space over Q with local components V (p) q ∼ = C q for all q = p and such that V (p) p and C p have the same dimension and quadratic character but different Hasse invariants (for p = ∞ we also require that V (∞) ∞ has signature (m, 2), and hence V (∞) = V ).
When Z(T, ϕ)(F p ) is non-empty for a prime p < ∞, then one can show (see the proof of Proposition 7.2 for example) that V (p) represents T . This implies Diff(C, T ) = {p}. Similarly, the proof of Theorem 7.1 shows that if the Green current G(T, ϕ, v) is nonvanishing, then V (∞) represents T and hence Diff(C, T ) = {∞}.
In the situation of part (2) of Theorem 1.2, when Diff(C, T ) = {∞}, the local Whittaker function W T,∞ (g, 0, Φ κ ) vanishes, and hence The derivative of the archimedian Whittaker function is given by the following arithmetic local Siegel-Weil formula for the archimedian local height function on V n (R), which is our second main result (see also Theorem 5.2). The contributions from the non-archimedian places can be computed by means of the local Siegel-Weil formula (see Propositions 2.2 and 2.3) .
Theorem 1.4. Let x ∈ V n (R) such that the Q(x) = T is invertible. Then we have In the special case m = 0 Theorem 1.4 was proved in [KRY1], for m = 1 in [Ku2], and for m = 2 in [YZZ]. For the related case of Shimura varieties associated to unitary groups of signature (m, 1) it was proved in [Liu]. But the argument of [Liu] does not seem to generalize to the case of orthogonal groups considered in the present paper. Recently, Garcia and Sankaran employed Quillen's theory of super-connections to obtain a different proof of Theorem 1.4, see [GaSa].
In all these works it is first noticed that because of the equivariance of ξ n 0 (x, z) with respect to the action of H(R), the local height function ht ∞ (x) only depends on T = Q(x). Then a crucial step consists in proving that ht ∞ (T ) := ht ∞ (x) is invariant under the action of SO(n) on Sym n (R) (respectively U(n) on Herm n (C)) by conjugation. Hence it suffices to prove the claimed identity for diagonal T . In this case the star product reduces to a single integral, which can be related to the derivative of the Whittaker function by a direct (but rather involved) computation.
Our approach is different. For general nonsingular T , we consider the recursive formula for the star product (see (5.5)) and compute its 'main term' by means of the classical archimedian local Siegel Weil formula (see Theorem 5.4). The result turns out to be the sum of a main term, which is the desired right hand side of (1.5), plus a boundary term, given by the derivative of a genus n − 1 Whittaker function. By an inductive argument, the boundary term cancels against the remaining terms of the star product. This approach does not require proving SO(n)-invariance of the local height function at the outset. We obtain this invariance a posteriori from the obvious invariance of the Whittaker function.
Finally, we describe our approach to part (3) of Theorem 1.2. When Diff(C, T ) = {p} for a finite prime p = 2, the local Whittaker function W T,p (g, 0, Φ κ ) vanishes, and hence The derivative of the local Whittaker function at p is given by the following arithmetic local Siegel-Weil formula, which is our third main result paralleling Theorem 1.4. The terms away from p can again be computed by means of the local Siegel-Weil formula. Recall that ϕ = ϕ L ∈ S(V n (A f )) is the characteristic function ofL n .
Theorem 1.5. Let p = 2 be a prime number and assume that Z(T, ϕ)(F p ) is finite. Then for x ∈ Z(T, ϕ)(F p ), the local height ht p (x) is independent of the choice of x and is given by where T u is any unimodular matrix in Sym n (Z p ) (i.e., det T u ∈ Z × p ). This theorem will be restated and proved as Theorem 6.9. As in the archimedian case the proof is given by an inductive argument. According to Soylu's condition mentioned above, Z(T, ϕ)(F p ) being finite implies that T is Z p -equivalent to diag(T 1 , T 2 ) where T 1 is Z p -unimodular of rank n − 3. On the Whittaker function side, we will prove the following recursive formula (see Proposition 6.8): .
Here T u and T u 2 are unimodular symmetric Z p matrices of order n and 3 respectively, and L 2,p is a unimodular Z p -quadratic lattice of rank 4 with L p = M 1,p ⊕ L 2,p for a unimodular Z p -quadratic lattice M 1,p whose quadratic matrix is given by T 1 .
This suggests a similar recursion for the local height function ht p (x). Soylu proved that the abelian variety associated with x is supersingular. The local height function depends only on the associated p-divisible group, and it can be computed using the p-adic uniformization of the supersingular locus by a Rapoport-Zink space (see Section 6). The required recursion formula for the local height function is proved by employing recent work of Li and Zhu ([LZ,Lemma 3.1.1], see Corollary 6.5).
By the recursion formulas, the proof of Theorem 1.5 is reduced to the case n = 3 in the local situation, where L = L 2,p is a unimodular Z p -lattice of rank 4 and T = T 2,p is a symmetric Z p -matrix of rank 3, and x = (x 1 , x 2 , x 3 ) determines a point in Z(T, ϕ)(F p ) on the associated Rapoport-Zink space. This turns out to be exactly the local case considered by Kudla and Rapoport in their work on (twisted) Hilbert modular surfaces [KRap1]. This paper is organized as follows. Section 2 contains some preliminaries and basic facts about Whittaker functions. Moreover, we state the classical local Siegel-Weil formula with an explicit formula for the constant of proportionality. In Section 3 we derive a variant of the archimedian local Siegel Weil formula for integrals of certain Schwartz functions over the hermitian symmetric space of the orthogonal group, again with explicit constant of proportionality. The main result Theorem 3.1 is one of the key ingredients in the proof of Theorem 1.4. In Section 4 we investigate the asymptotic behavior of the archimedian Whittaker function as one of the radial parameters goes to infinity. Our analysis relies on Shimura's work on confluent hypergeometric functions [Shi]. The main result, Theorem 4.8, which is of independent interest, is the second main ingredient in the proof of Theorem 1.4. Section 5 is devoted to the proof of the archimedian arithmetic Siegel-Weil formula Theorem 1.4. In Section 6 we recall some facts about the Rapoport-Zink space for GSpin groups from [HP] and [So] and prove the non-archimedian local arithmetic Siegel-Weil formula, Theorem 1.5. Finally, Section 7 contains the proofs of our main global results, Theorem 1.2 and the refinements described in Remark 1.3.
We thank Jose Burgos Gil, Stephan Ehlen, Jens Funke, Ben Howard, Steve Kudla, Chao Li, Cihan Soylu, and Torsten Wedhorn for many helpful comments and conversations related to this paper. We also thank the referee for his/her careful reading of our manuscript and for the insightful comments.

The local Siegel-Weil Formula
In this section we introduce the basic local setup and recall the local Siegel Weil formula, see Theorem 2.1. We make the involved constant explicit in Proposition 2.2.
2.1. The basic local set-up and local Whittaker functions. Let F be a local field or the ring of adeles of a number field, and let ψ be a non-trivial additive character of F (or adele class character). Let P = NM be the standard Siegel parabolic subgroup of the symplectic group Sp n (F ) given by We also denote w = 0 −I n I n 0 .
Let (V, Q) be a non-degenerate quadratic space over F of dimension l. Then there is a Weil representation ω = ω V,ψ of Mp n,F on S(V n ) given by [Ku1,Page 400]. In particular, where d ψ y is the self-dual Haar measure on V with respect to ψ, and γ(V n ) = γ(ψ • det Q) −n . Here γ(ψ) and γ(a, ψ) (for a ∈ F × ) are the local Weil indices defined in [Rao, Appendix], and for an F -basis {e 1 , . . . , e l } of V . Finally, is the quadratic character associated to V . It is well-known that the Weil representation factors through Sp n (F ) when l is even. Since γ(a, 1 2 ψ) 2 = (a, −1) F , the formula for ω(m(a))φ above works for both even and odd l. From now on, let G = Sp n (F ) or Mp n,F depending on whether n is odd or even, and let P be the standard Siegel parabolic subgroup or the preimage of the standard Siegel parabolic subgroup. If F = R, we let K G ⊂ G be the maximal compact subgroup given by either or the inverse image of U(n) under the covering map (when G = Mp n,R ). For a character χ of F × , let I(s, χ) = Ind G P χ| det | s be the induced representation. A section Φ ∈ I(s, χ) satisfies For a symmetric matrix T ∈ Sym n (F ), the Whittaker function of Φ with respect to T is defined to be where d ψ b is the self-dual Haar measure on Sym n (F ) with respect to the pairing (b 1 , b 2 ) → ψ(tr(b 1 b 2 )). It has the transformation behavior (2.2) Here we have shortened γ(a, 1 2 ψ) = γ(det a, 1 2 ψ) (and similarly for χ(a)). We remark that γ(a, 1 2 ψ) = 1 when det a > 0 and F = R. Let s l,n = l−n−1 2 . Then there is a G-equivariant map λ : We will also denote by λ(φ) the associated standard section in I(s, χ V ) with λ(φ)(g, s l,n ) = λ(φ)(g). Assume that l = n + 1. Then a formal unfolding suggests that there is a Haar measure dh on H = SO(V ) such that for all φ ∈ S(V n ) where C is some constant which is independent of T and φ, and if there is x ∈ V n with Q(x) = T (otherwise set the orbital integral to be zero). This is the content of the so-called local Siegel-Weil formula which we will describe in next two subsections. In particular, we will determine the constant C.

2.2.
Kudla's local Siegel-Weil formula. In this subsection we review the local Siegel-Weil formula given in [KRY2,Section 5.3], following a general result in [Ra,Chapter 4]. Let the notation be as in Section 2.1, and assume dim(V ) = n + 1. Let be the moment map. Let V n reg be the subset of x ∈ V n with det Q(x) = 0, and let Sym reg n (F ) be the subset of T ∈ Sym n (F ) with det T = 0. Then Q induces a regular map from V n reg to Sym reg n (F ). Put a(n) = n(n+1) 2 . We let α be a gauge form on V n , that is, a generator of (∧ 2a(n) V n ) * (a top level differential of the topological vector space V n ), and let β be a gauge form on Sym n (F ), i.e., a generator of ∧ a(n) (Sym n (F )) * .
Fix an x = (x 1 , . . . , x n ) ∈ V n reg with Q(x) = T . If we identify the tangent space T x (V n reg ) with V n , then the differential dQ x is given by
Then dQ x • j x (u) = u, and we have the decomposition Now choose any u = (u 1 , . . . , u a(n) ) ∈ (Sym n (F )) a(n) with β(u) = 0. We define an a(n)form ν ∈ (∧ a(n) V n ) * on V n reg as follows: for any t = (t 1 , . . . , t a(n) ) ∈ (V n ) a(n) , we put This quantity is independent of the choice of u. Then [KRY2,Lemma 5.3.1] asserts that for h ∈ SO(V ) and g ∈ GL n , where SO(V ) × GL n acts on V n via (h, g)x = hxg −1 . Moreover, ν defines a gauge form on Q −1 (T ) if we identify ker dQ x with the tangent space T x (Q −1 (T )) of Q −1 (T ). Finally, using the isomorphism (here dim V = n + 1 is critical to insure that the pointwise stabilizer H x of x is trivial), we obtain a gauge form i * x (ν) on SO(V ), which we will still denote by ν for simplicity. The key point (see [KRY2,Lemma 5.3.2]) is that this gauge form ν does not depend on T or x, which can be seen by (2.8).
This gauge form ν gives a Haar measure dh = d ν h on SO(V ). Let d α x be the Haar measure on V n associated to α and d β T be the Haar measure on Sym n (F ) associated to β, and let d ψ x and d ψ T be the self-dual Haar measures on V n and Sym n (F ) with respect to ψ, respectively. Then there are constants c(α, ψ) and c(β, ψ) such that Finally, we can state Kudla's local Siegel-Weil formula, which is [KRY2,Proposition 5.3.3] (although only stated for n = 2 there, the proof goes through for general n without any change).
Theorem 2.1 (Local Siegel-Weil formula). Given a gauge form α on V n and a gauge form β on Sym n (F ), let d ν h be the Haar measure on H(F ) associated to α and β as above. Then one has for any φ ∈ S(V n ), T ∈ Sym reg n (F ), and g ∈ G, and γ(V n ) = γ(V ) n by [Ku1,Lemma 3.4].
We remark that our C(V, α, β, ψ) is the reciprocal of the same notation in [KRY2].
2.3. Explicit construction. Let e = (e 1 , . . . , e n+1 ) be an ordered basis of V and put J = Q(e) = 1 2 ((e i , e j )) ∈ Sym n+1 (F ). When F is p-adic, let L = ⊕O F e j be the associated O F -lattice. Using this basis, we identify V with F n+1 (column vectors) and V n with M n+1,n .
Let E ij denote a matrix whose (ij)-entry is one and all other entries are zero (we do not identify the size of the matrix). Then {E ij , 1 ≤ i ≤ n + 1, 1 ≤ j ≤ n} is a basis of V n . Let de ij be its dual basis, and let α = ij de ij be the gauge form on V n (up to sign, which does not affect the associated Haar measure) with (2.11) α((E ij )) = α(E 11 , E 12 , . . . , E n+1,n ) = 1.
Proposition 2.2. Let J, α, and β be given as above, and let d ν h be the associated Haar measure on H(F ) = SO(V )(F ). We take ψ(x) = e(x) = e 2πix when F = R and assume that ψ is unramified when F is p-adic. Then for all φ ∈ S(V n ) and g ∈ G. Here Finally, when F = R and V has signature (p, q), then γ(V n ) = e( n(q−p)
By the proposition, we see that d ν h depends only on | det(2J)| F . For this reason, we will sometime denote d ν h by d J h or d L h in the p-adic case. We also write C(L) = C(J) in the p-adic case as det(2J) = det L.
Proof. First assume that F is p-adic. Let O F be ring of integers of F . Let L = ⊕O F e i = O n F ⊂ V = F n and f = char(L n ) = char(M n+1,n (O F )) ∈ S(V n ). Then the Fourier transforms of f with respect to d α x and d ψ x are given bŷ where L ′ is the dual lattice of L with respect to ψ. Since d ψ x is the self-dual Haar measure on V with respect to ψ, one has On the other hand, if d ψ t is the self-dual Haar measure on Sym n (F ) with respect to ψ, then . Now it is clear that C(V, α, β, ψ) = C(J) as claimed. Now assume that F = R and ψ(x) = e(x). To compute the quantity c(α, ψ), we write J = t P diag(a 1 , . . . , a n+1 )P and denote |J| = t P diag(|a 1 |, . . . , |a n+1 |)P . We consider the Schwartz function on M n+1,n (R) given by where we write P x = (x ij ). Then its Fourier transformation with respect to d α x = dx ij isf and so c(α, ψ) = | det 2J| − n 2 as claimed. To compute c(β, ψ), notice that d β T = 2 −n i dt ii i<j dt ij for T = (t ij ) ∈ Sym n (R), and consider the Schwartz function on Sym n (R) given by Then its Fourier transformation with respect to d β T iŝ This shows the equality c(β, ψ) = 2 −n− n(n−1) 4 . We have again C(V, α, β, ψ) = C(J) as claimed. The formula for γ(V n ) is given by β V (w) in [Ku1,(3.4)].
The following proposition shows how to compute the Haar measure d ν h in some cases and will be used in Section 7.
Proposition 2.3. Let F be a p-adic local field with p = 2 and a uniformizer π, and let ψ be an unramified additive character of F . For a lattice L over (1) When L is unimodular of rank n + 1, we have In both cases, C(L) = C(J) is given by Proposition 2.2.
Proof. We prove (2) using Proposition 2.2 with φ = φ L , and leave the slightly easier (1) to the reader. Choose a basis {e n , e n+1 } of L 0 so that Q(ae n + be n+1 ) = π(a 2 + ǫb 2 ) for some ǫ ∈ O × F . Let e = (e 1 , . . . , e n ) ∈ L n , then Q(e) = T . We claim that Clearly, K L ⊂ K T . We just need to prove that he n+1 ∈ L for h ∈ K T . In this case, with ǫ 1 ǫ 2 = ǫ and non-negative integers a i satisfying a 1 + a 2 = 2. Since we have x, y ∈ O F and π = N(e n ) = N(he n ) = ǫ 1 π a 1 x 2 + ǫ 2 π a 2 y 2 , which implies a 1 = a 2 = 1. Now write i.e., We remark that the Whittaker functions involved in the above proposition have explicit formulas, see Section 7. Now we describe ν and d ν h = d J h more explicitly by choosing the basis e and thus J nicely, i.e., we assume J = diag(a 1 , . . . , a n+1 ). It will be used in next section.
Let h = so(V ) be the Lie algebra of SO(V ). In terms of coordinates with respect to the basis e, one has X = (x ij ) ∈ h if and only if t XJ + JX = 0, i.e., a i x ij + a j x ji = 0. Hence we have the following lemma.

The local Siegel-Weil formula on a hermitian symmetric domain
Let V be a quadratic space over R of signature (m, 2), and let H = SO(V ). Let D be the corresponding hermitian domain, which we realize as the Grassmannian of oriented negative 2-planes in V . The purpose of this section is to prove Proposition 3.1, a variant of the archimedian local Siegel-Weil formula involving an integral over D. Throughout this section we fix the additive character ψ(x) = e(x) of R and assume that n = m + 1. Recall that ρ n = n+1 2 . Let e, f ∈ V be isotropic vectors such that (e, f ) = 1, and let V 0 = (Re + Rf ) ⊥ ⊂ V . Then V 0 has signature (m − 1, 1) and we have the Witt decomposition V = V 0 + Re + Rf . The hermitian symmetric domain D can also be realized as the tube domain Then H(R) acts on H by linear fractional transformations, characterized by The map z → w(z) can be viewed as a section of the tautological bundle over D. The Petersson norm of this section is − 1 2 (w(z), w(z)) = −(y, y).
Hence Ω = dd c log(−(y, y)) (3.2) defines an invariant (1, 1)-form on H ∼ = D, the first Chern form of the dual of the tautological bundle on D equipped with the Petersson metric. Here d c = 1 4πi (∂ −∂). According to [Ku4,Proposition 4.11], in the coordinates of H, it is given by Moreover, it can be obtained from the Kudla-Millson form ϕ KM (x, z) (see (5.2)) by an identity which we will only need in Section 5. Notice that −Ω is a Kähler form, and therefore (−Ω) m is a positive invariant top degree form on H.
In particular, one has B 2, The basic idea of the proof is simple and natural: we relate the gauge form on the tangent space p of D with the differential form Ω m precisely. The actual calculation is a little long and technical, and can be skipped on first reading. We will also provide an alternative proof in Section 5.5.
3.1. The differential Ω m and the gauge form ν. Let e = (e 1 , e 2 , . . . , e m , e n , e n+1 ) be an ordered basis of V with quadratic matrix We write V + (respectively V − ) for be the subspace generated by the e i with 1 ≤ i ≤ m (respectively i = n, n+1). Let K ± = SO(V ± ), then K ∞ = K + ×K − is a maximal connected compact subgroup of H(R). In the notation of the last section, a i = 1 for 1 ≤ i ≤ m and −1 for i = n, n + 1.
It is easy to see that the gauge form ν given in Proposition 2.5 has the following decomposition (up to sign): where ν + , ν − , and ν p are the gauge forms on K + , K − , and on H(R)/K ∞ , which are characterized by ν + (X 12 , . . . , X m−1,m ) = 1, ν − (X n,n+1 ) = 1, and (3.7) ν p (X 1n , X 1,n+1 , . . . , X mn , X m,n+1 ) = 1. Now we deal with the relation between Ω m and ν p . We use a tube domain realization for D as above. To this end we define a different basis e ′ of V as follows. Let e = 1 2 (e 1 + e n+1 ), f = 1 2 (e 1 − e n+1 ), and e ′ = (e n , e 2 . . . , e m , e, f ). Its associated matrix is We put Rf is a Witt decomposition as considered before. We write H for the corresponding tube domain realization of D as in (3.1). We will also identity V with R n+1 and V n with M n+1,n (R) with respect to the basis e Similarly, we will use [v] e and [x] e to denote the coordinates of v and x with respect to the basis e when necessary. For γ ∈ H(R), we denote [γ] e and [γ] e ′ for its coordinates with respect to the bases e and e ′ respectively. Then one has (3.9) [ We now compute the action of H(R) on H more explicitly. For h ∈ H(R), write where all theH ij are matrices, allh ij s are numbers, andH 22 is a square matrix of order Fixing the base point z = ie n ∈ H, we have the isomorphism This induces an isomorphism between p and the tangent space T z (H) ∼ = V 0,C (extending to the tangent bundle, too): wherex i andX ij are the coordinates of X with respect to e ′ just as for h. In terms of the coordinates with respect to e, one has and by a direct direct calculation using (3.9) we obtain So we have proved the following lemma.
Lemma 3.3. Using the above notation, we have Proof. Using the coordinates of (3.8), one sees where α and β have the obvious meanings. Notice that Since α(i, j, k, l) = −α(k, j, i, l), we have α 2 = 0. This implies It is easy to check that and therefore So we obtain the following proposition from the above two lemmas. Proposition 3.4. Let the notation be as above and z = ie n ∈ H. Then Proof of Proposition 3.1. First, let z = ie n ∈ H as Proposition 3.4, and let ν = ν + ∧ ν − ∧ ν p be as in (3.6), and let d ν h, dh + , dh − , and d p h be the associated Haar measures. Then, by Proposition 3.4, we have On the other hand, Proposition 2.2 gives Consequently, Applying Proposition 2.2 to K + = SO(V + ) and φ ∞ = e −2π tr Q + (x) , one sees by Proposition 4.3 that where Γ n (s) is given by (4.1). We obtain .
We remark that the above calculation of vol(K + , dh + ) has the following well-known formula as a consequence.
Corollary 3.5. Let l ≥ 1 be an integer, and let be the standard special orthogonal group. Let ν l be the gauge form defined as ν + for K + = SO m (R), and let dh l be the associated Haar measure on SO l (R). Then .

Asymptotic properties of Whittaker functions
Throughout this section we consider the local field F = R, the additive character ψ(x) = e(x), and the group G = Sp n (R) or Mp n,R . We investigate the asymptotic behavior of the archimedian Whittaker function for G as one of the radial parameters of the Levi subgroup M goes to ∞. The main results are Theorem 4.8 and Corollary 4.11. Our analysis is based on Shimura's work on confluent hypergeometric functions [Shi]. We fix a quadratic character χ of R × and an half integer κ ∈ ρ n + Z (not necessarily equal to ρ n ) satisfying the compatibility condition . We also fix a matrix T ∈ Sym n (R). 4.1. Basic properties of archimedian Whittaker functions. Let Φ = Φ κ ∈ I n (s, χ) be the weight κ standard section, that is, the unique function in I n (s, χ) whose restriction to K G is the character det(k) κ . Then the Whittaker integral (2.1) can be expressed in terms of Shimura's confluent hypergeometric function. As in [Ku2,Lemma 9.3], the following result can be proved.
Lemma 4.1. Assume that det(T ) = 0. If a ∈ GL + n (R) and y = a t a, then denotes Shimura's confluent hypergeometric function of matrix argument [Shi,(1.25) , and C n,∞ = 2 The normalizing factor C n,∞ comes from comparing the measures dn and dx. Recall that the Siegel gamma function of genus n is defined by Following Shimura, we define another special function by For all regular T , by [Shi,Remark 4.3], the integral converges when ℜ(α) > ρ n − 1 and ℜ(β) > ρ n − 1. According to [Shi,(1.29 and therefore Proof. The first assertion follows from (3.1.K) of [Shi]. The second assertion follows from this by means of (4.4).
The special values of Eisenstein series and Whittaker functions at s = 0 will be of particular interest. Here we collect the facts that we will require. Proposition 4.3. Assume that det(T ) = 0 and that κ = ρ n . ( Proof. According to [Shi,Theorem 4.2], the function is holomorphic for (α, β) ∈ C 2 . Hence, in view of (4.4), W T (m(a), s, Φ κ ) is equal to a holomorphic function in a neighborhood of s = 0 times the gamma factor Therefore, the first assertion follows from (4.1) by working out the vanishing order of this gamma factor. To prove (ii), we use (4.35.K), (4.12.K), and (4.6.K) of [Shi] to see that for sig(T ) = (n, 0) we have η(g, h, ρ n , β) = Γ n (β) det(g) −β e − tr(gh) .
By means of (4.4) we get where y = a t a. This proves the proposition.
Remark 4.4. Assume that sig(T ) = (n, 0) and that κ = ρ n . Then with the constant B n,∞ of Proposition 3.1 we have Later we will also need the following lemmas.
Lemma 4.5. If f is a measurable function on Sym n (R) and a ∈ GL n (R), we have Here dv denotes the Lebesgue measure on R n .
Lemma 4.7. Let u = u 1 u 12 t u 12 u 2 be a symmetric block matrix. Then the following are equivalent: (1) u > 0, In this case we have Proof. This is direct consequence of the Jacobi decompositions , whenever the inverses make sense. See also [Shi,Lemma 2.1].

Asymptotic properties.
Here we investigate the asymptotic behavior of the Whittaker function W T (g, s, Φ κ ). We assume that T ∈ Sym n (R) with det(T ) = 0, and a ∈ GL + n (R). We put y = a t a. Recall that We write T and the variable of integration u in block form as with T 1 ∈ R, T 2 ∈ Sym n−1 (R), and T 12 ∈ R 1×(n−1) , and analogously for u.
Here we have added a superscript to η to indicate in which genus it is considered.
Remark 4.9. In the case n = 1 the function η (0) is to be interpreted as the constant function with value 1. Then the theorem states On the other hand, for T ∈ R × and y ∈ R >0 we have where U(a, b, z) denotes Kummer's confluent hypergeometric function, see [AS,(13.1.3)].
Proof of Theorem 4.8.
By means of Lemma 4.5 we rewrite the integral as follows: Here and throughout the proof of the theorem we briefly write |u| for the determinant of u. In view of Lemma 4.7, we obtain η(y, T, α, β) (4.7) = y −β 1 e − tr T y u 2 +2T 2 >0 u>0 χ(u, y 1 , T )e − tr(uy ′ ) u 1 /y 1 u 12 /y 1/2 1 t u 12 /y where χ(u, y 1 , T ) denotes the characteristic function We now compute the desired limit as y 1 → ∞ assuming that the integration can be interchanged with the limit. After that we will come back to the justification of the interchange. We have lim y 1 →∞ e T 1 y 1 y β 1 · η (n) (y, T, α, β) = e −2T 12 t y 12 −tr T 2 y 2 × u 2 +2T 2 >0 u>0 lim y 1 →∞ χ(u, y 1 , T )e − tr(uy ′ ) u 1 /y 1 u 12 /y 1/2 1 t u 12 /y If T 1 ≤ 0, then the domain of integration is empty and the integral vanishes as claimed.
Step 3. We now show that the formulas of the theorem actually hold for all α, β ∈ C. If T is positive definite, we use the functional equation of η under (α, β) → (ρ n − β, ρ n − α) and argue as in [Shi,pp. 281]. For general T , we then apply the integral representation (4.24) in [Shi,pp. 289] to deduce the assertion.
If a ∈ R × , we let Theorem 4.8 implies the following asymptotic behavior of the Whittaker function.
Corollary 4.10. Let κ = ρ n . For general invertible T we have Here the Whittaker function on the left hand side is in genus n and the one on the right hand side in genus n − 1 (which is clear from the size of the matrices in the subscripts).
Consequently, we obtain This concludes the proof of the corollary.

The archimedian arithmetic Siegel-Weil formula
Here we use the archimedidean local Siegel-Weil formula (see Proposition 3.1), the asymptotic behavior of Theorem 4.8, and some computations in the induced representation to prove Theorem 1.4.
We use the same setup and notation as in Section 3. In particular, V is a quadratic space over R of signature (m, 2), and κ = m+2 2 . Moreover, D is the hermitian domain associated with H = SO(V ), realized as the Grassmannian of oriented negative 2-planes in V .

Green currents and local heights.
For z ∈ D the orthogonal complement z ⊥ is positive definte of dimension m. If x ∈ V , we denote the orthogonal projetion of x to z and z ⊥ by x z and x z ⊥ , respectively. The quadratic form is positive definite on V , the so called majorant associated with z. We also put Then D x is a non-trivial analytic divisor of D if Q(x) > 0, and it is empty if Q(x) ≤ 0 (which we will view as the zero divisor). Following [Ku2] we define the Kudla Green function ξ(x, z) = − Ei(−2πR(x, z)) · e −π(x,x) , (5.1) where Ei(u) = u −∞ e t dt t is the exponential integral, see [AS,Chapter 5]. If x ∈ V is fixed, then ξ(x, z) is a smooth function on D \ D x with a logarithmic singularity along D x . It has the equivariance property ξ(gx, gz) = ξ(x, z) for g ∈ H(R). The differential form extends to a smooth (1, 1)-form on all of D, where d c = 1 4πi (∂ −∂). It is the Kudla-Millson Schwartz form which is Poincaré dual for the cycle e −π(x,x) D x , see [KM1], [Ku2]. More precisely, as currents on D, we have the identity Because of the H(R)-equivariance of ξ(x, z), the (1, 1) form In fact, it is equal to the invariant differential form defined earlier in (3.2). For x = (x 1 , . . . , x n ) ∈ V n and z ∈ D we also define the Kudla-Millson Schwartz form in genus n as ϕ n KM (x, z) = ϕ KM (x 1 , z) ∧ · · · ∧ ϕ KM (x n , z). With respect to the action of G through the Weil representation it transforms under the maximal compact subgroup K G with the character det(k) κ . If Q(x) ∈ Sym n (R) is invertible, the form ϕ n KM (x, z) is Poincaré dual for the special cycle e −π tr(x,x) D x , where D x = {z ∈ D | z ⊥ x i for i = 1, . . . , n}, see [KM1]. We define a Green current for the cycle e −π tr(x,x) D x by taking the star product ξ n (x, z) = ξ(x 1 , z) * · · · * ξ(x n , z) (5.3) in the sense of [GiSo]. As a current on compactly supported smooth differential forms it satisfies the equation When D x is compact, it follows from the growth estimates in [KM1, Section 6] that ξ n (x, z) is rapidly decaying and extends to a current on forms of moderate growth with (5.4) still holding. A recursive formula for the star product is given by where ϕ 0 KM has to be interpreted as 1 and ξ 0 as 0. The current ξ n (x, z) is invariant under permutations of the components of x.

Then the current equation becomes
Note that the current equation (5.4) together with Proposition 4.3 implies the following geometric local Siegel-Weil formula, which is the local archimedian version of (1.1).

Proposition 5.1. Assume that n = m and that T = Q(x) is invertible. Then
Moreover, in both cases this is equal to where Φ κ ∈ I n (s, χ V ) is the weight κ standard section, that is, the unique standard section whose restriction to K G is the character det(k) κ .
Proof. The first statement is a direct consequence of the current equation (5.4) applied to the constant function with value 1. The second statement can be deduced from the first one by means of the formulas of [Shi]. Since we do not need it here, we omit the proof.
Throughout the rest of this subsection we assume that n = m + 1. Then ξ n 0 (x, z) is a top degree current, which can be evaluated at the constant function 1. For x ∈ V n we define the archimedian local height function by In this section we prove the archimedian arithmetic local Siegel Weil formula Theorem 1.4, relating ht ∞ (x) to the derivative of a Whittaker function in genus n. It can be viewed as an arithmetic analogue of Proposition 5.1. We restate the Theorem for convenience.
Theorem 5.2. Let x ∈ V n (R) such that the moment matrix T = Q(x) is invertible. Then we have where B n,∞ is the constant in Proposition 3.1, and Φ κ ∈ I n (s, χ V ) is the weight κ standard section, that is, the unique standard section whose restriction to K G is the character det(k) κ . The derivative of the Whittaker function is taken with respect to s. Let x ∈ V n and assume that T = Q(x) is invertible. To prove Theorem 5.2 we employ the recursive formula (5.5) for the star product. It implies that for the main term of the local height function. The second summand on the right hand side of (5.8) vanishes when Q(x 1 ) ≤ 0, in which case D x 1 is empty. When Q(x 1 ) > 0, this quantity is a local height function in genus n − 1 for the quadratic space V 1 = x ⊥ 1 ⊂ V of signature (m − 1, 2). The divisor D x 1 is naturally isomorphic to the Grassmannian of V 1 . Let be the orthogonal projection and putx = (pr(x 2 ), . . . , pr(x n )) ∈ V n−1 1 . If we write T in block form as in (4.5) then the moment matrix ofx is Lemma 5.3. Assume the above notation. If z ∈ D x 1 , we have Here the height function on the right hand side is for the tuplex ∈ V n−1 1 .
Proof. If z ∈ D x 1 and y ∈ V , then R(y, z) = R(pr(y), z). Hence the assertion is a direct consequence of the definition of ξ 0 (y, z).
The following result gives a formula for the main term of the local height function.
The proof of Theorem 5.4 will be given in the next three subsections.

By Theorem 5.4, we obtain
If we use Theorem 5.2 in genus n − 1 to compute the last term on the right hand side, we get the assertion.

5.2.
The main term of the local height. In this subsection we assume again that n = m + 1. We give a first formula for the main term of the local height in terms of a certain Whittaker function. We begin by rewriting the Green function ξ(x, z) defined in (5.1) in terms of the Gaussian (5.14) Lemma 5.5. If x ∈ V and z ∈ D, we have Proof. The statement follows from the integral representation − Ei(−z) = ∞ 1 e −zt t dt by inserting the definitions of ξ 0 (x, z) and R(x, z).
By our assumption on m, the Schwartz form ϕ n−1 KM is a top degree differential form on D. We write it as ϕ n−1 KM (y, z) = ϕ n−1, * KM (y, z) · Ω n−1 . For x ∈ V n , we define a Schwartz function 1 by ψ * 11 (x, z) = ϕ G (x 1 , z) · ϕ n−1, * KM ((x 2 , . . . , x n ), z). (5.15) Proposition 5.6. Let x ∈ V n , put T = Q(x), and write T = T 1 T 12 t T 12 T 2 as in (4.5). The main term of the local height function is given by where B n,∞ denotes the constant in Proposition 3.1 and d(a) is given by (4.11).
Proof. Using Lemma 5.5 and (5.15), we see By the local Siegel Weil formula, Proposition 3.1, we have Inserting this, we obtain the assertion.

Some Lie algebra computations.
In this subsection, we temporarily drop the assumption that n = m + 1. We compute the Whittaker function W T (1, s, λ(ψ * 11 )) more explicitly. We begin by recalling from [BFK1,Section 5] some facts about the Lie algebra of G. Let g = k + p + + p − be the Harish-Chandra decomposition of g = Lie(G) ⊗ R C. Let S = Sym n (R). Then there are isomorphisms The group K G acts on g by the adjoint representation, Ad(k)g = kgk −1 , and on S C by k.X = kX t k for k ∈ K G . For the isomorphism (5.16) we have Ad(k)p + (X) = p + (k.X), (5.17) is invariant under the action of K G , and therefore p * ± ∼ = p ∓ as K G -modules. Let (e α ) be a basis of S, and write (e ∨ α ) for the dual basis with respect to the trace form. Then (p − (e ∨ α )) is a basis of p − , and we write (η ′ α ) for the dual basis of p * − . We identify p * − with S C by the map (5.19) Recall that the Lie algebra gl n (C) ∼ = Mat n (C) is isomorphic to k via the map Let E jk ∈ Mat n (C) be the elementary matrix having the entry 1 at the position (j, k) and all other entries 0. Then the matrices for 1 ≤ j, k ≤ n, form a basis of k.
We denote by C(ℓ) the K G -module given by the action of K G on C by multiplication with det(k) ℓ . Recall that the space of differential forms A p,q (H n ) on H n can be described by the isomorphism Here, the operator corresponding to∂ on A p,q (H n ) is given by (5.22) where η ′ α acts on ∧ · (p * ) by exterior multiplication. The following result, which describes the action of K G on ψ 11 , is taken from the unpublished manuscript [BFK2]. We thank Jens Funke and Steve Kudla for allowing to include it here.
which satisfies ψ(0) = Ω n−1 · 1 n and Proof. To prove this result we use the Fock model realization of the Weil representation as described in the appendix of [FM1]. Let F = F (C (m+2)×n ) be the space of polynomial functions on V n C ∼ = C (m+2)×n . As in [FM1] we denote the variables by z αj , z µj , where α = 1, . . . , m, µ = m + 1, m + 2, and j = 1, . . . , n. The Lie algebra g × so(V ) C acts on F via the Weil representation.
Let so(V ) C = k H ⊕ p H be the Cartan decomposition as in Section 3.1. Let X αµ be the standard basis of p H and denote by ω αµ the corresponding dual basis of p * H . The Kudla-Millson Schwartz forms can be viewed as elements of The Gaussian ϕ G,(j) corresponds to the constant polynomial 1 for every j.
We define the Schwartz form ψ = (ψ jk ) in the Fock model by putting (l) and This has the desired diagonal components. Using the intertwining operator between the Schrödinger and the Fock model of the Weil representation, it is easily checked that ψ jk (0) = 0 for j = k. On the other hand, by (3.4), we have ψ jj (0) = Ω n−1 , and therefore ψ(0) = Ω n−1 · 1 n . To verify the transformation law under K G , we compute the action of the Lie algebra k under the Weil representation. Recall that the basis element Y jk defined in (5.21) acts by In fact, since the element Y jk corresponds to 1 2i (w ′ k • w ′′ j ) in the notation of [FM1], this claim follows from [FM1,Lemma A.1]. Now a direct computation shows This implies that the ψ jk generate an irreducible representation of K G , which has ψ nn as a highest weight vector, and which is isomorphic to det κ ⊗ Sym 2 (C n ) ∨ . Hence, we obtain the claimed transformation law.
The intertwining operator λ : (5.23) and write Ψ(g, s) for the corresponding extension to a standard section.
This corollary characterizes Ψ uniquely. We now use the action of p − in the induced representation to find a different expression for Ψ. Proof. We first show that r(D)Φ κ (g, s) has the same K G -type as Ψ. Via the isomorphism (5.19), the operator D induces an operator It satisfies Ad(k)D = k −1 .D for k ∈ K G , where the action on the left hand side is on the first factor of the tensor product and the action on the right hand side on the second factor. In fact, we have (see also [BFK1,Lemma 5.1]). But this implies, again using the identification (5.19), that r(D)Φ κ has the transformation law In other works, it has the same K G -type as Ψ.
Since the different K G types in I(s, χ V ) have multiplicity one, there exists a constant c(s) such that r(D)Φ κ (g, s) = c(s)Ψ(g, s).
Proof. The first equality is a direct consequence of Proposition 5.9. It implies the second equality, since the Whittaker integral is an intertwining map of (g, K)-modules.
Proposition 5.11. For a ∈ GL n (R) we have Proof. For the proof we put X = e 11 and split p − (X) as in (5.25). We compute the action of the three terms individually. We have Next, we compute, using the action of K G , Finally, we notice that Putting the terms together, we obtain Corollary 5.12. Assume that n = m + 1 and det(T ) = 0. Write T in block form as in (4.5), and recall the definition (4.11) of d(a). For a ∈ R >0 we have W T (m(d(a)), 0, λ(ψ * 11 )) = 2 2πT 1 a 2 − κ 2 + a 2 ∂ ∂a W ′ T (m(d(a)), 0, Φ κ ).
Since n = m + 1, we have ρ n = κ. Moreover, because of the signature of V , the matrix T is not positive definite. Hence, according to Proposition 4.3, the Whittaker function on the right hand side vanishes at s = 0. This implies the assertion.
5.4. The main term of the local height revisited. Here we combine the results of the previous two subsections with the asymptotic properties of Whittaker functions derived in Section 4.2.
Proof of Theorem 5.4. Recall that n = m + 1, x ∈ V n , and that T = Q(x) is invertible.
We have to show that whereT 2 is defined by (5.10). According to Proposition 5.6 we know that Inserting (2.2) and the formula of Corollary 5.12, we obtain Noticing that d(a)), 0, Φ κ )e 2πQ(x 1 a) a −ρn .
We now employ Corollary 4.11 to evaluate the limit on the right hand side. We obtain Hence the claim follows from (3.5).
5.5. An alternative proof of Proposition 3.1. Here we use Corollary 5.10 and Proposition 5.1 to give an alternative way of computing the constant B n,∞ appearing in Proposition 3.1. Assume that n = m + 1.
In particular, we have B n,∞ /B n−1,∞ = i n Γ(ρn) (2π) ρn , and B 1,∞ = 1 πi , and B 2, 6. The local arithmetic Siegel-Weil formula at an odd prime p In this section we assume that p = 2 is a prime. Let W = W (F p ) be the Witt ring ofF p and K = W Q be the fraction field of W , which is the completion of the maximal unramified extension of Q p . Let σ be the Frobenius of W (such that its reduction toF p is the Frobenius x → x p ).
Let L be a unimodular quadratic lattice over Z p of rank n + 1 and put V = L Qp . Let C(L) be the Clifford algebra of L, and let D(L) = Hom(C(L), Z p ) be its dual. We writẽ H = GSpin(L) for the general Spin group over Z p , and notice thatH(Z p ) ⊂ C(L) × acts on C(L) via left multiplication and and thus acts on D L . Let ι be the main involution on C(V ) which fixes V point-wise. If δ ∈ C(V ) × with δ ι = −δ, then ψ δ (x, y) = tr(xδy ι ) defines a non-degenerate symplectic form on C(V ). We will require that δ ∈ C(L) and δδ ι ∈ Z × p , which implies that C(L) is unimodular under this symplectic form. This induces an embedding (6.1) i = i δ :H → GSp(C(L)).
It is also known thatH is 'cut out' by a family tensors (s α ), s α ∈ C(L) ⊗ , in the sense that for any Z p algebra R we havẽ 6.1. The local unramified Shimura datum and the Rapoport-Zink space associated toH. Here we set up some notation for the rest of this section. We recall the construction of an unramified local Shimura datum forH due to Howard and Pappas, and the associated Rapoport-Zink space. We refer to [HP] for details. Choose a Z p -basis e = {e 1 , . . . , e n+1 } of L with Gram matrix (6.2) ((e i , e j )) = diag(I n−2 , ǫ L , 0 1 1 0 ), Then (H, [b], {µ}, C(L)) is the local unramified Shimura datum constructed by Howard and Pappas in [HP,Section 4] forH. Here {µ} is the conjugacy class of the cocharacter µ underH(K), and [b] is the σ-conjugacy class of the basic element b, i.e., the set of elements h σ bh −1 with h ∈H(K). Associated to b there are two isocrystals A direct calculation shows that L has a Z p -basis e ′ = {e ′ 1 , . . . , e ′ n+1 } with Gram matrix (6.5) ((e ′ i , e ′ j )) = diag(I n−2 , ǫ L , p, −pu) where u ∈ Z × p with (p, u) = −1, and −uǫ L = ǫ L . We can actually take e ′ i = e i for i ≤ n − 2. In particular, V = L ⊗ Zp Q p is a quadratic space over Q p with the same dimension and the same determinant, but with opposite Hasse invariant as V .
According to [HP,Lemma 2.2.5], there is a unique p-divisible group Moreover, the Hodge filtration on D(X 0 )(F p ) is induced by µF p (up to conjugation). The symplectic form ψ δ induces a principal polarization λ 0 on X 0 . Let RZ(X 0 , λ 0 ) be the Rapoport-Zink space associated to GSp(C(L), ψ δ ), see [RZ] and [HP,Section 2.3]. It is a smooth formal scheme over Spf(W ) representing the moduli problem over Nilp W of triples (X, λ, ρ)/S, where S is a formal scheme over W on which p is Zariski locally nilpotent, (X, λ) is a p-divisible group with principal polarization λ, and ρ is a quasi-isogeny ρ : X 0 ×F pS X × SS ,S = S × WFp , which respects polarization up to a scalar, in the sense that Zariski locally onS, we have RZ = RZ(H, [b], {µ}, C(L)) be the GSpin Rapoport-Zink space constructed in [HP,Section 4]. This space comes with a closed immersion RZ → RZ(X 0 , λ 0 ), and by restricting the universal object one obtains a is a universal triple (X univ , λ univ , ρ univ ) over RZ. The universal quasi-isogeny preserves the polarization only up to scalar, which induces a decomposition of RZ as a union of open and closed formal subschemes where RZ (l) ⊂ RZ is cut out by the condition ord p c(ρ univ ) = l ∈ Z. According to [HP,Section 7] (see also Section 7.3 here), RZ can be used to uniformize the supersingular locus at p of some Shimura variety associated with (H, D).
Notice that V K acts on C(V ) K via right multiplication, which induces an action on the isocristal D K . This gives an embedding , we obtain an embedding V ⊂ End 0 (X 0 ). We call V the special endomorphism space of X 0 following [HP].
LetH be the algebraic group GSpin(V). ThenH(Q p ) = {h ∈H(K) | hb = bσ(h)} acts by automorphisms on D K , giving rise to a quasi-action on X 0 . This quasi-action has the property c(hρ) = µH(h)c(ρ), where µH is the spin character ofH. So h ∈H(Q p ) induces an isomorphism RZ (l) ∼ = RZ (l+ordp µH(h)) . In particular, we have According to [She,Corollary 7.8], RZ = p Z \ RZ is exactly the Rapoport-Zink space of H associated to the basic local unramified Shimura datum induced from the datum (H, [b], {µ}, C(L)). Finally, let J ⊂ V be an integral Z p -submodule of rank 1 ≤ r ≤ n. We define the special cycle Z(J), following Soylu [So], as the formal subscheme of RZ cut out by the condition Here, for an S-point α : S → RZ, X = α * (X univ ) and ρ = α * (ρ univ ) are the pull-backs of the universal objects. If J has a ordered Z p -basis x = (x 1 , . . . , x r ) ∈ V r , we also denote Z(J) = Z(x). The moment matrix T = Q(x) = 1 2 ((x i , x j )) in Sym r (Q p ) is determined by J up to Z p -equivalence. Soylu gave an explicit formula of the dimension of the reduced scheme Z(J) red underlying Z(J) in terms of T and L, see [So,Section 4.2].
The purpose of this section is to prove a local arithmetic Siegel-Weil formula for Z(J). We show that when Z(J) is 0-dimensional, the local height of each point P ∈ Z(J) depends only on T , not on the choice of the point P , and is equal to the central derivative of some local Whittaker function (Theorem 6.9).
6.2. Dual vertex lattices and decomposition of the Rapoport-Zink space.
Moreover, S Λ = S + Λ ∪ S − Λ has two connected components, both of which are smooth and projective of dimension t Λ /2 − 1 For a dual vertex lattice Λ of V, let RZ Λ be the closed formal subscheme of RZ defined by the condition ρ • Λ • ρ −1 ⊂ End(X). The following theorem summarizes some of the basic properties of RZ Λ and RZ. Assertions (1), (3), (4) are due to Howard and Pappas (see [HP], Proposition 5.1.2, Section 6.5, Remark 6.5.7). The second assertion is due to Li and Zhu [LZ,Theorem 4.2.11].
Theorem 6.1. The following are true.
with α i ∈ Z × p and 0 ≤ a 1 ≤ a 2 ≤ · · · ≤ a n+1 ≤ 1. The condition t Λ = dim Fp Λ ′ /Λ = t implies a 1 = · · · = a n−t+1 = 0, and a n−t+2 = · · · = a n+1 = 1. So we can change the basis to make (6.9) true. Since L is unimodular, V has Hasse invariant 1, and hence V has Hasse invariant −1, i.e., In particular, E = Q p ( (−1) r β) is the unique unramified quadratic field extension of Q p , and β is uniquely determined up to a square by this condition. On the other hand, det V = det V gives αβ = det L mod (Z × p ) 2 , which then determines α uniquely up to a square. 6.3. Special cycles and local heights. Recall the definition of the special cycle Z(J) at the end of Section 6.1. It is not hard to see [So,Section 4.2 The following theorem is part of [So,Theorems 4.13,4.16 and Proposition 4.15 ]. (Recall our convention that dim V = n + 1 and notice that our 2T is Soylu's T .) Theorem 6.3. (Soylu) Let Z(J) red be the reduced scheme of Z(J) and assume that J = J(x 1 , . . . , x n ) ⊂ V has rank n and is integral. Assume that T = Q(x) is Z p -equivalent to diag(T 1 , T 2 ) where T 1 is unimodular of rank r = r(T ) (which is also the rank of T (mod p) overF p ), and T 2 ∈ p Sym n−r (Z p ). Then Z(J) red is 0-dimensional if and only if one of the following conditions holds: (1) r(T ) = n − 1, n − 2.
In such a case, Proof. We give a sketch of the proof in this special case to give a rough idea what is involved in the general theorems of Soylu [So,Section 4]. Choose a Z p -basis e = {e 1 , . . . , e n } of J with 1 2 (e i , e j ) = diag(T 1 , T 2 ), and let M 1 be the submodule of J generated by e 1 , . . . , e r , which is unimodular. To have J ֒→ Λ, one has to have t Λ ≤ n − r + 1. In the case r = n − 1, n − 2, one has t Λ = 2. and RZ Λ = RZ red Λ is reduced of dimension 0. So In the case r(T ) = n − 3, one might have t Λ = 2 or 4. If t Λ = 4, then (as M 1 is unimodular) Therefore, if det(2T 1 ) = det L, we cannot embed J into a dual vertex lattice Λ with t Λ = 4, and thus Z(J) is 0-dimensional and reduced as argued above.
When det(2T 1 ) = det L, Soylu proved that there is indeed some embedding J ⊂ Λ with t Λ = 4. We refer to [So,Section 4] for the details.
Let M 1 be a unimodular quadratic Z p -lattice of rank r < n − 2, and assume that there are isometric embeddings M 1 ⊂ L and M 1 ⊂ Λ, where Λ ⊂ V is a dual vertex lattice. Write (6.10) Notice that, choosing proper bases of M 1 and L, the data b and µ defined in (6.3) and (6.4) still make sense for the unimodular lattice L 2 , so we have a local unramified Shimura datum (H(r), [b], {µ}, C(L 2 )) and its associated Rapoport-Zink space RZ(r). HereH(r) = GSpin(L 2 ). Moreover, one can easily check that V 2 = L 2 b•σ K is a direct summand of V, and Λ 2 is a dual vertex lattice of V 2 . The embdding L 2 ⊂ L induces a closed immersion (6.11) i(r) : RZ(r) ֒→ RZ .
The following proposition is a direct consequence of [LZ,Lemma 3.1.1].
Proposition 6.4. (Li-Zhu) Let the notation be as above and assume r ≤ n − 3.
(2) Assume that J = M 1 ⊕ J 2 is a Z p -submodule of Λ. Then i(r)Z RZ(r) Proof. Assume that {x 1 , . . . , x r } is a basis of M 1 with Gram matrix diag(α 1 , . . . , α r ) and α i ∈ Z × p . Applying [LZ,Lemma 3.1.1] r-times, we obtain the above proposition. Notice that their lemma still holds with the same proof when the norm of x n is a unit in Z p (not necessarily equal to 1).
From now on, we assume that Z(J) is 0-dimensional. For P ∈ Z(J), its local height index is defined to be (6.12) ht p (P ) = the length of the formal complete local ring O Z(J),P .
By Theorem 6.3, we have r(T ) ≥ n − 3. There is a decomposition with M 1 unimodular of rank n − 3. Furthermore we can choose bases of M 1 and J 2 so that the Gram matrix of J becomes 2T with T = diag(T 1 , T 2 ) where 2T 1 is the Gram matrix of M 1 and T 2 = diag(α 1 p a 1 , α 2 p a 2 , α 3 p a 3 ) is the matrix of J 2 with α i ∈ Z × p and 0 ≤ a 1 ≤ a 2 ≤ a 3 . We can always embed M 1 into L. Assume that Z(J)(F p ) is not empty. Then there is an embedding M 1 ⊂ J ⊂ Λ for some dual vertex lattice Λ.
The local height ht p (P * ) has been studied in [KRap1] (the case a 1 = 0 actually follows from [KRY2] with n − 3 replaced by n − 2). Assume the decompositions (6.10). Notice that L 2 is unimodular of rank 4. There are two cases: either (i) det L 2 = 1 and is the unique unramified quadratic field extension of Q p , i.e. (p, u) = −1. In the second case, L 2 is Z p -equivalent to Z 4 p with the quadratic form Q(x) = x 1 x 2 +x 2 3 −ux 2 4 , or more conceptionally where α ′ is the Galois conjugate of α. The second case only occurs when r(T ) ≥ n − 2, i.e., a 1 = 0. Indeed, if a 1 > 0, i.e., r(T ) = n − 3, then we would have det(2T 1 ) = det L = det M 1 det L 2 = det(2T 1 ) det L 2 , which implies 1 = det L 2 = u, a contradiction. The condition a 1 = 0 is exactly the condition given in [KRap1,Theorem 2] for Z RZ(n−3) (J 2 )(F p ) to be finite. So in both cases, RZ(n − 3) is associated to the supersingular locus at p of the Hilbert modular surface over a real quadratic field F with p split or inert in F , F p = Q p × Q p or E. In [KRap1], Kudla and Rapoport considered twisted Hilbert modular surfaces to avoid issues with the boundary. But their localization at p, considered in [KRap1,, is for our p the same as for a regular Hilbert modular surface, and hence their local results apply. We restate it here as the following theorem for convenience. In case (ii), a 1 = 0, it is [KRap1,Proposition 6.2]. In case (i), it is [GK,Proposition 5.4], restated in [KRap1,Proposition 11.2] with a minor mistake (it should not assume a 1 = 0 in this case). Proposition 6.6. (Kudla-Rapoport) Let the notation and hypotheses be as above, in particular Z RZ(n−3) (J 2 ) is 0-dimensional, and let P * ∈ Z RZ(n−3) (J 2 ). Recall that T 2 is Z pequivalent to diag(α 1 p a 1 , α 2 p a 2 , α 3 p a 3 ) with 0 ≤ a 1 ≤ a 2 ≤ a 3 , and α i ∈ Z × p . Then ht p (P * ) = ν p (T 2 ), where ν p (T 2 ) is given as follows: (1) When a 2 ≡ a 1 (mod 2), ν p (T 2 ) is equal to (2) When a 2 ≡ a 1 (mod 2), ν p (T 2 ) is equal to

Local Whittaker functions and the local arithmetic Siegel-Weil formula.
Let ψ = ψ p be the 'canonical' unramified additive character of Q p used in this paper. Let L be an integral quadratic lattice over Z p of rank l, and let χ L = ((−1) l(l−1) 2 det L, ·) p be the associated quadratic character. For every integer r ≥ 0 we also consider the lattice L (r) = L ⊕ H r , where H = Z 2 p is the standard hyperbolic plane with the quadratic form Q(x, y) = xy. We temporarily allow L to be non-unimodular.
Let T ∈ Sym n (Z p ) be non-singular with n ≤ l. Then according to [Ku2,Appendix] and [Ya], there is a local density polynomial α p (X, T, L) of X such that for every integer r ≥ 0, one has α p (p −r , T, L) = Here γ(L) = γ(L ⊗ Zp Q p ) is the local Weil index. We also recall [Ya, Section 2] that α p (p −r , T, L) is the local representation density α p (M T , L (r) ) = β p (M T , L (r) ) studied in Kitaoka's book [Ki1,Section 5.6]. Here M T = Z n p is the quadratic lattice associated to T , i.e., with the quadratic form Q(x) = t xT x. For a unimodular lattice L of rank l, define (6.13) δ L = 0 if l ≡ 1 (mod 2), χ L (p) if l ≡ 0 (mod 2).
(1) Assume that L = L 1 ⊕ L 0 is an integral lattice over Z p such that Q(x) ∈ pZ p for every x ∈ L 0 . Then α p (X, T, L) = α p (X, T, L 1 ).
(2) Assume that L is Z p -unimodular of rank l ≥ n. Then Here L − denotes the lattice L with the rescaled quadratic form Q − (x) = −Q(x).
Proof. LetL = L/pL with the F p -valued quadratic formQ(x) = Q(x) mod p for an integral quadratic Z p -lattice L. Replacing L by L (r) , we may assume X = 1 in the proof. For (1), write l and l i for the rank of L and L i respectively with l = l 0 + l 1 . Notice that L 0 is a zero quadratic space of dimension l 0 . Every isometry fromM T toL splits into the sum of an isometry fromM T toL 1 and a homomorphism fromM T toL 0 . So [Ki1,p. 99,exercise] gives For (2) where χ(M ) for an unimodular quadratic Z p -latticeM is defined as follows. When l = dimM is odd, χ(M ) = 0. When l is even, χ(M ) is ±1 depending on whetherM is equivalent to a direct sum of hyperbolic planes or not. Assume l = 2r + 2 is even. Since M is unimodular, M is equivalent to H r ⊕ M 0 with M 0 = Z 2 p with Q(x, y) = x 2 − ǫy 2 for some ǫ ∈ Z × p . Then χ(M ) = 1 if and only ifM 0 is a hyperbolic plane, which is the same as saying that ǫ is a square in F p , i.e., (ǫ, p) = 1. On the other hand, it is easy to check that So χ(M ) = χ M (p) in this case. This proves (2).
Proposition 6.8. Assume that T is Z p -equivalent to diag(T 1 , T 2 ) with T 1 being unimodular of rank n − 3 and T 2 = diag(α 1 p a 1 , α 2 p a 2 , α 3 p a 3 ) with α i ∈ Z × p and 0 ≤ a 1 ≤ a 2 ≤ a 3 . Let L be a unimodular lattice of rank n+1. Let M 1 be the unimodular quadratic lattice with Gram matrix 2T 1 , and fix an embedding M 1 ֒→ L, which results in a decomposition L = M 1 ⊕ L 2 . Then W T,p (1, 0, λ(ϕ L )) = 0 if and only if W T 2 ,p (1, 0, λ(ϕ L 2 )) = 0. In such a case, we have Here T u and T u 2 denote any unimodular symmetric matrices over Z p of order n and 3, respectively, and ν p (T 2 ) is given in Proposition 6.6.
This proves the first identity and also the claim about the vanishing at s = 0. Assume W T,p (1, 0, λ(ϕ L )) = 0. We have by [KRap1,Propostions 11.5 and 7.2], . On the other hand, Lemma 6.7 gives . Now the second identity is clear.
Combining Propositions 6.6 and 6.8 and Corollary 6.5, we obtain the following local arithmetic Siegel-Weil formula.
Theorem 6.9. Let L be a unimodular quadratic Z p -lattice of rank n + 1 with p = 2, and let RZ be the Rapoport-Zink space as in Section 6.1. Let T ∈ Sym n (Z p ) be of rank n and let J ⊂ V be a Z p -sublattice of rank n which has a basis with Gram matrix 2T . Assume Z(J) is 0-dimensional and let P ∈ Z(J). Then where T u is a unimodular matrix in Sym n (Z p ) (i.e., det T u ∈ Z × p ).

Arithmetic Siegel-Weil formulas
In this section, we will prove the arithmetic Siegel-Weil formulas as stated in Theorem 1.2 and Remark 1.3. Throughout, let V be a quadratic space over Q of signature (m, 2), and let H = SO(V ). 7.1. Vanishing of coefficients of Eisenstein series. Let n = m+1 and let C = ⊗ p≤∞ C p be the incoherent quadratic space over A defined in the introduction. Recall the G Aequivariant map λ = ⊗λ p : S(C n ) → I(0, χ V ), λ(φ)(g) = ω(g)φ(0).
For simplicity, we also write λ(φ) for the associated standard section in I(s, χ V ). Let φ C ∞ (x) = e −π tr(x,x) ∈ S(C n ∞ ), then λ ∞ (φ ∞ ) = Φ κ ∈ I(s, χ V ) is the standard section of weight κ = m+2 2 . Recall that for a standard factorizable section Φ = Φ p ∈ I(s, χ V ), the Eisenstein series has a meromorphic continuation to the whole complex s-plane and is holomorphic at s = 0. It has a Fourier expansion of the form E(g, s, Φ) = T ∈Sym n (Q) E T (g, s, Φ).
When T ∈ Sym n (Q) is non-singular, the T -th Fourier coefficient factorizes, into product of local Whittaker functions, see (2.1). For every φ ∈ S(V (A f ) n ) = S(C n A f ), we define the Siegel-Eisenstein series of weight κ on the Siegel upper half plane H n as where we write g τ = n(u)m(a) ∈ G R with u = ℜ(τ ) ∈ Sym n (R) and a ∈ GL n (R) such that a t a = v as usual. In particular, we have g τ (i1 n ) = τ . We could choose for a the positive symmetric square root of v but we do not have to. The Eisenstein series vanishes automatically at s = 0 due to the incoherence. The arithmetic Siegel-Weil formula, envisioned by Kudla, aims to give arithmetic meaning to its central derivative at s = 0. From now on, assume T = Sym n (Q) is non-singular, and let (7.2) Diff(C, T ) = {p ≤ ∞ : C p does not represent T } be Kudla's Diff set defined in the introduction. Then Diff(C, T ) is a finite set of odd order, and ∞ ∈ Diff(C, T ) if and only if T is not positive definite. Moreover, when p ∈ Diff(C, T ), then W T,p (g p , 0, λ p (φ p )) = 0. So (7.3) ord s=0 E T (g, s, λ(φ) ⊗ Φ κ ) ≥ | Diff(C, T )| for every φ ∈ S(V (A f ) n ).
7.2. The arithmetic Siegel-Weil formula at infinity. Here we prove Theorem 1.2 (2) of the introduction. We begin by recalling the global setup.
For a compact open subgroup K ⊂ H(A f ) we consider the Shimura variety X K whose associated complex space is It is a quasi-projective variety of dimension m, which has a canonical model over Q.
Given x = (x 1 , . . . , x n ) ∈ V (Q) n with Q(x) = 1 2 (x, x) = 1 2 ((x i , x j )) > 0, let H x be the stabilizer of x in H. gives rise to a cycle Z(h, x) in X K of codimension n. More generally, given a positive definite T ∈ Sym n (Q) and any K-invariant Schwartz function ϕ ∈ S(V n (A f )), Kudla [Ku3] defines a weighted cycle as follows: If there exists an x ∈ V n (Q) with Q(x) = T , put If there is no such x, set Z(T, ϕ) = 0. These weighted cycles behaves well under pull-back (for varying K). Moreover, if T ∈ Sym n (Q) is regular but not positive definite, we put Z(T, ϕ) = 0.
If T ∈ Sym n (Q) is regular, we define a Green current for the cycle Z(T, ϕ) by where z ∈ D, h ∈ H(A f ), and a t a = v = ℑ(τ ). The pair Z(T, ϕ, v) = Z(T, ϕ), G(T, ϕ, v) ∈ Ch n C (X K ) defines an arithmetic cycle, which depends on v. For the rest of this section we assume that n = m + 1. In this case, the cycles Z(T, ϕ) are all trivial (in the generic fiber) for signature reasons. However, for indefinite T , the arithmetic cycles Z(T, ϕ, v) typically have a non-trivial current part. We are interested in their archimedian arithmetic degree deg ∞ Z(T, ϕ, v) = 1 2 X K (C) G(T, ϕ, v).
We are now ready to prove Theorem 1.2 (2) of the introduction, which we restate here in a version which also gives an explicit value for the constant of proportionality.
Theorem 7.1. Assume that T ∈ Sym n (Q) is of signature (n − j, j) with j > 0 and that ϕ ∈ S(V (A f ) n ) is K-invariant. Then the arithmetic Siegel-Weil formula holds for T , i.e., deg ∞ Z(T, ϕ, v) · q T = C n,∞ · E ′ T (τ, 0, λ(ϕ) ⊗ Φ κ ), where the constant C n,∞ is given as follows. Let L ⊂ V be an integral lattice, and let d L h = p<∞ d Lp h be the associated Haar measure on H(A f ), and C(L) = p<∞ C(L p ) be the associated constant given in Proposition 2.3 (with respect to the unramified additive character ψ f of A f ). Then C n,∞ = −B n,∞ C(L) vol(K, d L h) .
Proof. The archimedian arithmetic degree is given by deg ∞ Z(T, ϕ, v) = 1 2 X K (C) G(T, ϕ, v) = 1 2 H(Q)\D×H(A f )/K x∈V (Q) n Q(x)=T ϕ(h −1 f x) · ξ n 0 (xa, z) dh f . and a morphism of Shimura varieties over Q from X K to the Siegel Shimura variety determined by the symplectic space (C(V ), ψ δ ) and a suitable compact open subgroup. The integral model of the Siegel Shimura variety induces then an integral model X = X K of X K [Ki], [AGHM,Section 4]. Kisin showed that X is smooth over Z (p) , if the compact open subgroup K p ⊂ H(A p f ) is sufficiently small. By pulling back the universal abelian scheme, we obtain a polarized abelian scheme (A KS , λ KS , η KS ) with level structure over X , the Kuga-Satake abelian scheme. It is equipped with a right C(L)-action.
Given a Z (p) -scheme S and an S-point α : S → X , we obtain a triple A α = (A, λ, η) = α * (A KS , λ KS , η KS ) by pulling back the Kuga-Satake scheme. In particular, η is aK p -level structure η : sending V A p f (theétale realization of the motive associated to the representation ofH on V ) onto V ⊗ A p f . Let V (A α ) ⊂ End C(L) (A) (p) be the space of special endomorphisms of A α defined in [So,Definition 3.3].
Given T ∈ Sym n (Q) with det T = 0, the special cycle Z(T ) → X is defined as the stack over X with functor of points Z(T )(S) = {(α, x) : α ∈ X (S), x = (x 1 , . . . , x n ) ∈ V (A α ) n , Q(x) = T, η • x j • η −1 ∈L (p) }, whereL (p) = l =p L l , andL = l L l . In this subsection we drop the Schwartz function ϕ from the notation of Z(T, ϕ), since we only consider it here for the characteristic function ofL n .
Soylu showed in [So,Proposition 3.7] that the image of the forgetful map Z(T ) → X sends Z(T )(F p ) into the supersingular locus X ss ⊂ X (F p ). According to [HP,Proposition 7.2.3], there exists an α 0 ∈ X ss such that the p-divisible group (X 0 , λ 0 ) associated to A α 0 is equal the p-divisible group (X 0 , λ 0 ) considered in Section 6.1. According to [HP,Theorem 7.2.4] or [She,Theorem 1.2], there is an isomorphism of formal schemes where ( X W ) /Xss is the completion of X W along the supersingular locus X ss , and RZ = p Z \ RZ. The above discussion implies that for every (α, x) ∈ Z(T )(F p ) the space of special endomorphisms satisfies where V is the neighboring quadratic space over Q associated with C at p.
Proposition 7.2. Let L be a fixed lattice of V such that L p is a dual vertex lattice in V p of type 2 as in Section 6.2 andL q ∼ =L q for q = p. Let ϕ L = char(L n ). Let T ∈ Sym n (Q) where C n,p = −B n,∞ C(L) vol(K, d L h) .
In particular, C n,p = C n,∞ .
Hence we obtain the claimed formula.
Remark 7.4. In this subsection we have assumed for convenience that ϕ is the characteristic function ofL, and that there is a compact open subgroupK ⊂H(A f ) containingẐ × and mapping onto K. Both can be relaxed. First, we can naturally modify the definition Z(T ) in [So] to include Z(T, ϕ) for all ϕ = ϕ p ϕ p ∈ S(V (A f ) n ) K with ϕ p = char(L n p ). The proof of Proposition 7.2 goes through without any change. As already mentioned, the assumption on K is always fulfilled if there exists an even lattice M ⊂ V which is stabilized by K and such that K acts trivially on M ′ /M. In other words, every 'sufficiently small' compact open subgroups K satisfes the condition. Finally, we indicate how the results can be modified to hold for general K. Take a compact open subgroupK 1 ofH(A f ) containingẐ × such that its image K 1 in H(A f ) is contained in K. Then there is a natural projection X K 1 → X K and an analogous projection of Rapoport-Zink spaces. The p-adic uniformization identity (7.4) still holds according to [She,Theorem 1.2]. For ϕ ∈ S(V (A f ) n ) K , the special cycle Z K 1 (T, ϕ) is K-invariant and descends to a special cycle Z K (T, ϕ) on X K .
Here the extra 1 2 makes sense as we do the integral over the whole symmetric domain D instead of its connected component D + , while at a finite prime p, we did it at each individual point (connected component). This reinterpretation is different from the previous ones used in [KRY2], [KRap1], [KRap2], and [HY] among others.