Cycles on Siegel threefolds and derivatives of Eisenstein series

We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p > 2, which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider algebraic cycles on this model which are characterized by the existence of certain special endomorphisms, and their intersections. We characterize that part of the intersection which consists of isolated points in characteristic p only. Furthermore, we relate the (naive) intersection multiplicities of the cycles at isolated points to special values of derivatives of certain Eisenstein series on the metaplectic group in 8 variables. © 2000 Editions scientifiques et medicales Elsevier SAS RESUME. Nous etudions Fespace de modules de Siegel de genre 2 et, pour un nombre premier p > 2, un modele p-entier de cet espace qui parametre les varietes abeliennes principalement polarisees de dimension 2 munies d'une structure de niveau. On considere des cycles algebriques sur ce modele definis par 1'existence d'endomorphismes speciaux, ainsi que leurs intersections. On caracterise la partie de ces intersections qui est constituee de points isoles de caracteristique p. On donne une formule qui relie la multiplicite d'intersection (naive) de ces cycles en les points isoles aux valeurs speciales de derivees de certaines series d'Eisenstein sur Ie groupe metaplectique en 8 variables. © 2000 Editions scientifiques et medicales Elsevier SAS


Introduction
The classical Siegel-Weil formula relates a special value of a Siegel-Eisenstein series, an analytic object, to the representation numbers of quadratic forms, essentially diophantine quantities. Recent work has revealed that analogous relations should exist between the special values of derivatives of such series and quantities in arithmetical algebraic geometry, e.g., heights. One such relation involving Shimura curves was proved by one of us in [19]. In that paper, it was established that the nonsingular Fourier coefficients of the derivative at 0 of certain Siegel-Eisenstein series of weight 3/2 on the metaplectic group in four variables 1 are closely related to the value of the height pairing of a pair of arithmetic cycles on a Shimura curve.
It is a hope, already expressed in [19], that a similar relation holds in general between the derivative at 0 of certain incoherent Siegel-Eisenstein series on the metaplectic group in 2n variables and the height pairing of suitable arithmetic cycles on Shimura varieties associated to orthogonal groups of signature (n -1,2). This would constitute an arithmetic analogue of the result of the first author [ 18] which relates the value at 1 /2 of certain coherent Siegel-Eisenstein series with the intersection pairing on suitable classical cycles on these Shimura varieties. As a basic first step in the incoherent case, it can be shown that (at least for nonsingular Fourier coefficients) both sides of the identity to be proved can be written as a sum of terms enumerated by the places of Q. One can then hope to prove identities between individual corresponding terms one place at a time.
This paper is the first of a pair in which we generalize some of the results of [19] to higher dimensions in the case of finite primes of good reduction.
A first difficulty in the general program is that models over the integers of the Shimura varieties associated to orthogonal groups are not well understood. For low values of n there are, however, exceptional isomorphisms which relate the groups in question to symplectic groups, and the Shimura varieties associated to them have integral models which one can investigate. In the present paper we are concerned with the exceptional isomorphism which relates the orthogonal group of signature (3,2) with the symplectic group in 4 variables. In the companion paper [21] we are concerned with the Shimura variety associated to an orthogonal group of signature (2,2) which is related to certain Hilbert-Blumenthal surfaces.
Let us now be more specific about the contents of this paper. Let B be an indefinite quaternion algebra over Q. Let C = M^ (B) and put (0. 1) V={xeC; x f =x,tT O where x ^ x' =^x L is the involution on C induced by the main involution on B. Then (V, q), with q defined by x 2 = q(x) • 1, is a quadratic space of signature (3,2) and the group G = G Spin(y) of V can be identified with a twisted form of the group of symplectic similitudes in 4 variables. Let V be the space of oriented negative 2-planes in V(R) and let K be a compact open subgroup of G(Ay). Then, the Shimura variety Sh(G,P)j<, whose complex points are given by The exceptional isomorphism of G = GSpin(V) with a form of GSp4 plays a fundamental role throughout the paper. In particular, we use it to construct a good integral model of Sh(G,V)K-More precisely, we fix a prime p > 2 such that B is unramified at p and take K of the form K = K p ' Kp, where K p c G(A^) is sufficiently small and where Kp is the natural maximal compact open subgroup of G(Qp). Then we use the modular interpretation of Sh(G,P)j< to construct a smooth model M over SpecZ(p), as a parameter space of certain abelian varieties with additional structure.
Algebraic cycles on Sh(G,P)j< were defined analytically in [18] as follows. For x e V^ let q(x) = ^((xi.Xj)) G Sym^(Q) be the matrix of inner products of the components of x for the symmetric bilinear form ( , ) associated to q. Assume that d = q(x) is positive-definite (hence n < 3), and let Vx be the subspace of oriented negative 2-planes orthogonal to all entries of x. Let Gx be the pointwise stabilizer of x. Then Sh(Ga;, Vx) is a sub-Shimura variety of Sh(G, V), and thus defines a cycle of codimension n in Sh(G,Z>)j<. These cycles are a special case of the totally geodesic cycles in locally symmetric spaces studied in [20] and elsewhere. A slight generalization of the previous construction yields a cycle Z{d,uj;K) of Sh(G,P)j< which is associated to any positive definite d e Sym^(Q) and any X-invariant compact open subset uj of V^Af^.
The next step is to give a modular definition of these cycles. First, for one of the abelian varieties parametrized by M, we define the notion of a special endomorphism (Definition 2.1). SERIE  The space of such endomorphisms is a finitely generated free Z(p)-module equipped with a quadratic form q. The cycle Z{d, uj\ K) (= Z(d, uj) if K is fixed) is then obtained by imposing an n-tuplej of special endomorphisms such that ^(j) = d, and satisfying an additional compatibility with respect to uj. If uj satisfies an integrality condition at p, this definition can be used to extend the cycle Z(d^ uj) to a cycle Z(d, uj) for the integral model M. of the Shimura variety. Here, by a cycle on M., we mean a scheme which maps by a finite unramified morphism to M.. At this point we meet a very important problem: in contrast to M., the cycles Z(d,cj) will no longer be smooth, in general. In fact, they often are not flat over Z(p) and may even have embedded components. Our justification for our choice of this integral extension of the classical cycles is that their definition is very simple, has a nice inductive structure with respect to intersection, and that we are able to prove something about them. Before stating these results, we note that, while the arithmetic cycles Z(d^) can be defined for any d e Sym^(Q), any n, they are nonempty only when d is positive semidefinite and with coefficients in Z(p).
We Here uj = uj\ x -• • x uj^' The summand on the right corresponding to T is the set of points ^ where T^ •= r. This decomposition illustrates the inductive nature of the special cycles mentioned above.
The decomposition (0.4) bears some formal similarity to the partitioning into isogeny classes that occurs in the approach of Langlands-Kottwitz to the calculation of the zeta function of a Shimura variety. In that approach the stable conjugacy class of the Frobenius endomorphism is the most basic invariant of an isogeny class. In our context this role is played by the fundamental matrix. One of our discoveries is that the fundamental matrix and more specifically its divisibility by p governs the intersection behaviour of the special cycles. In any case, Z(T^) = 0 if ordp det(T) = 0. Furthermore, if ^ <E Z(T, uj) with det(T) ^ 0, i.e., T = T^ is positive definite, then the point $ lies in characteristic p and is not the specialization of a point of Z in characteristic 0. In this case, the connected component Z(T,UJ) of Z containing ^ consists entirely of supersingular points of M.. Contrary to what one might expect, however, the condition det(T^) -=^ 0 is not sufficient to ensure that ^ is an isolated point of intersection. One of our main results is the characterization of when this is the case. 698 S.S. KUDLA AND M. RAPOPORT ate these lines. It turns out that the more divisible T^ is by p, the more components there will be. A more thorough analysis of the set of irreducible components can be found in [21]. We point out that this phenomenon of excess intersection does not occur in the case of Shimura curves at a place of good reduction [19], but it does occur at a place of bad reduction [22].
With the previous notation let us put where each isolated intersection point ^ appears with multiplicity e(^) = lg0^, the length of the local ring of Z at ^.
We next come to the relation with Eisenstein series, for which we refer the reader to Section 8 or the first part of [19] for more details. Let TV be a symplectic space over Q of dimension 8 and let For T in the sum, the diagonal blocks are di,..., dr\ T is represented by ^(A^), but not by V(Qp). Moreover, T represents 1 over Zp. On the right in (0.9), we are summing over certain Fourier coefficients of the derivative at 0 of the Eisenstein series for Mp^. Our second main result is the following identity (Corollary 9.4). where c=^vo\{SO(V / (R))).
Unexplained notation may be found in the body of the text. The identity is proved by unravelling both sides of (0.10), where, for the right side, we use the decomposition (0.4) and 4^^ SERIE -TOME 33 - 2000 -N° 5   CYCLES ON SIEGEL THREEFOLDS AND EISENSTEIN SERIES   699 Theorem 0.1. The identity then reduces to the statement that, for T e Sym4(Z(p))>o such that T is not represented by V(Qp) and where T represents 1 over Zp, we have (all) Here, in the first factor on the left, there appears a quotient of the derivative at 0 of a certain Whittaker function for the quadratic space V(Qp) by the value at 0 of a Whittaker function for a twist V^Qp), and, in the second factor, a Fourier coefficient of a theta integral. In fact, the second factor can also be identified with an orbital integral. It turns out that the first factor equals the multiplicity e(^) of any point ^ C Z(T, uj) (which is constant), while the second factor is equal to the number of points in Z(T, a;). For the multiplicity e($), the calculation can be reduced to a problem on one-dimensional formal groups of height 2 which has been solved by Gross and Keating [7]. For the calculation of the Whittaker functions we use the results of Kitaoka [14] on local representation densities. It should be pointed out that we are using here the length of the local ring Oz^ as the multiplicity of a point ^, whereas the sophisticated definition would also involve Tor-terms. It is a fundamental question whether these correction terms vanish. This question we have to leave open.
In summary, we may say that Theorem 0.2 is proved by explicitly computing both sides of (0.10) and comparing them. It would of course be highly desirable to find a more direct connection between the analytic side and the algebro-geometric side of this identity.
We now give an overview of the structure of this paper. In Section 1, we introduce the Shimura variety and formulate the moduli problem solved by M. Our special cycles are introduced in Section 2. We define the fundamental matrix in Section 3 and isolate there the part of Z lying purely in characteristic p. It is clear from the above description that to proceed further we need a thorough understanding of the supersingular locus of M. XspecZ( SpecFp. This is essentially due to Moret-Bailly [23] and Oort [24]. In Section 4, we give a presentation of their results in terms of Dieudonne theory, better suited for our needs. A similar presentation was independently given by Kaiser [11] for a different purpose. The heart of the paper is Section 5. In it we determine the space of special endomorphisms of certain Dieudonne modules and deduce the characterization of isolated intersection points (Theorems 5.11, 5.12 and 5.14). Here again the exceptional isomorphism plays a vital role. In Section 6, we explain the reduction of the calculation of e(^) to the result of Gross and Keating, and, in Section 7, we explain how to count the number of isolated points. Section 8 is a review of the Fourier coefficients of Siegel Eisenstein Series. In Section 9, we bring everything together and prove the identity (0.10) above. In Section 10, we review some results of Kitaoka and show how they can be used to prove the formulas on Whittaker functions needed in Section 9. Finally, there is an appendix containing some facts on Clifford algebras in our special situation.
In conclusion we wish to thank A. Genestier for very useful discussions on our special cycles which helped us to correct some misconceptions we had about them. We also thank Th. Zink and Ch. Kaiser for helpful remarks, and the referee for his comments. We thank the NSF and the DFG for their support. S.K. would like to express his appreciation for the hospitality of the Univ. Wuppertal and the Univ. of Cologne during January 1995 and May and June of 1997 respectively. Finally, M.R. is very grateful to the Mathematics Department of the University of Maryland for inviting him and making his stay in Washington a memorable pleasure.

The Shimura variety
In this section, we review the construction of the Siegel 3-folds associated to indefinite quaternion algebras over Q, and the corresponding moduli problem. The use of the Clifford algebra is modeled on [28]. We refer to the appendix for some facts on those Clifford algebras that will be relevant for our purposes.
Let B be an indefinite quaternion algebra over Q, let C = M^(B), with involution x' = t x L , and let We define a quadratic form q on V by setting be the Clifford algebra of the quadratic space (V, q). Since, for x e V C (7, x 2 = q(x), there is a natural algebra homomorphism C(V) -> C extending the inclusion of V into C. The restriction of this map to the even Clifford algebra C~^(V) induces an isomorphism The group G acts on V C C by conjugation and this action yields an exact sequence where Z is the center of G. Let V be the space of oriented negative 2-planes in V(R). This space has two connected components and the group G(R) acts transitively on it, via its action on V(R). For an oriented 2plane z C P, let z\, z^ e z be a properly oriented basis such that the restriction of the quadratic form q from V(R) to z has matrix -la for the basis ^i, z^. Let jz = z\z^ G C(R). Viewing jz as the image of the element z^ C C(y(R)), the Clifford algebra of V(R), and recalling the commutative diagram of Section A.3 of Appendix A, we see that j^ = -jz and that j 2 = -z^zj = -1. Hence, jzj'z = 1 and so, jz € G(R). There is an isomorphism of algebras over M, where C+ (z) is the even Clifford algebra of the real 2-plane z. The composition of this map with the map induces a morphism, defined over R, hz : § -> G, where S = Rc/R^m, as usual. Note that z (i) = jz' The space V can thus be viewed as the space of conjugacy classes of such maps under the action of the group G(M). The data (G, P) or (G, h^) defines a Shimura variety Sh(G, V) [2,3], whose canonical model is defined over Q. Note that V is isomorphic to two copies of the Siegel space of genus 2, and, if B = M2(Q), Sh(G,P) is just the Siegel modular variety of genus 2.
Since G satisfies the Hasse principle, the Shimura variety represents a certain moduli problem over (Sch /Q), [17]. To define this we must introduce more notation. 4°   where N°(c) is the reduced norm on C. (iii) A is a Q-class of polarizations on A which induce the involution * on C: Ao^oA-1^^* ).
(iv) 77 is a 7^-class of isomorphisms a For the precise meaning of the datum (iv) we refer to [17, p. 390]. In particular, if S = Speck is the spectrum of a field, the ^-class rj is supposed to be stable under the action of the Galois group G8i\(k/k) where k is the algebraic closure used to form the Tate module of A. Note that the abelian scheme A will have relative dimension 8 over S. Proof.-For the representability, see [17]. We prove the last assertion in detail, since the conventions involved will be used later.
For T C B x , as above, let Proof. -For the first assertion: For the second, write z == gzo for g € G(R), so that Since x ^-> re* is a positive involution, this gives the claim. D Let V^ be the connected component of V containing ZQ and V~ the connected component of V not containing ZQ. Then, for any z G 2^, we obtain a (principally) polarized abelian variety over C, (1.22) A,=(^(R)j^£7(Z),±(, )) with dimA^ = 8 and with an action, given by left multiplication, Thus F \ ^+ parametrizes such principally polarized abelian varieties, up to isomorphism. More generally, to (z, g) e V x G(Af), we associate the collection (A, L, A, fj) defined by: where Az is taken up to isogeny; • A is the Q-class of polarizations determined by ( , ); • 77 is the I^-class containing the isomorphism: Note that, if 7 € G(Q) and k e K, then (7^, ^/gk) defines a collection isomorphic to that defined Passage in the other direction is similar. For example, in the isogeny class A and for 77 G rj, there is an abelian variety B, unique up to prime to p isogeny, such that r]p (Tp (B)) = U^.
The above proposition tells us that, when K = K p • Kp, as above, then M.KP provides us with a smooth model of Sh(G, V)K over Zcp). From now on, we will use the same notation for both moduli problems, if this does not cause confusion.

Special cycles
In this section we give a modular definition of the special cycles in Sh(G,'P), which were defined analytically in [18]. We then explain the relation between the two definitions.
Recall that the quadratic form on the space V C C = M^(B) was defined by x 2 = q(x) • 12. Let be the corresponding bilinear form, so that q( This defines a quadratic map q: V 71 -> Sym^. Fix a positive integer n. For d G Sym^(Q) a symmetric rational matrix, let be the corresponding hyperboloid. The group G acts diagonally on V n and preserves Qd-Cycles in Sh(G,P) were defined analytically in [18] as follows. For x G J?d(Q), let {x) C V be the Q-subspace spanned by the components of x, and let Vx = {x} 1 -be its orthogonal complement. Let T>x denote the space of oriented negative 2-planes in Va;(M), and let Gx be the pointwise stabilizer of (x) in G. Note that Gx ^ GSpin(T4), and that Vx C V. Moreover, for z € T>x -> the homomorphism hz factors through Gx (H^) • Thus there is a natural morphism of Shimura varieties, rational over Q, (2.4) , Sh(O^)--Sh(G,P).
If the space (x) is not positive-definite, then Vx = 0. If (x) is positive-definite of dimension r then d is positive semi-definite of rank r, sig(Vc) == (3 -r, 2) and Vx has codimension r in V.
Hence the previous construction is only interesting when d is positive semi-definite and even only when d is positive definite with n ^ 3. We introduce the following definition, which will play a key role throughout the paper. Here * denotes the Rosati involution of A. Also note that End 0 (A, i) is a finite-dimensional semisimple Q-algebra, so that the reduced trace appearing here makes sense. Indeed, this is well known when S is the spectrum of a field. The case when S is irreducible follows by reduction to its generic point, and the general case follows by considering the irreducible components of S.
Proof. -Again we may reduce first to the case where S is irreducible and then to the case when S is the spectrum of a field. However, for rj C rjletx =rj"{j) G Endc(U(Af)) = C{Af). Under the last identification the adjoint involution * with respect to (, ) corresponds to the involution / on G(A) (cf. (1.11)). Hence x lies in V{Af) and the assertion follows   Proof. -The first two assertions are obvious by (1.11). To prove the last assertion let z\, z^ € z be a properly oriented basis such that the restriction of the quadratic form q to z has matrix -1î n terms of this basis. Let v € ^(IR) with (v, Zi) =ca,i= 1, 2. Then  Proof. -We may assume that S is the spectrum of a field. The assertion follows from the positivity of the Rosati involution, since We next give a modular definition of the cycles introduced above. We take here the point of view that a cycle is given by a finite unramified morphism into the ambient scheme. Let  The condition (2.15) asserts that 77* (j) = r(a;) for some x^uj. If this is the case, then and k~lxk e c^. Thus the condition (2.15) depends only on rj.
To interpret condition (2.16) we may assume S to be connected. Let ( , ) be the bilinear form on the space of special endomorphisms of (A, L, A, rj) associated to the quadratic form q of Lemma 2.2. Then q(]) = \ (UiJj))ij ^ Sym^(Q) is defined as in (2.2). The condition (2.16) requires that g(j) = d.  Proof. -The first statement follows easily from the second. Let us assume that K is neat. The relative representability of the forgetful morphism by a morphism of finite type follows in a standard way from Grothendieck's theory of Hilbert schemes since MK may be considered as a moduli scheme of polarized abelian varieties with additional structure. To verify the valuative criterion of properness for the morphism (2.19), we have to check that an endomorphism between the generic fibers of abelian schemes over the spectrum of a discrete valuation ring extends uniquely. This follows from the Neron property of abelian schemes. Since the matrix d gives the squares j^ of the special endomorphisms, the morphism is quasi-finite and hence finite. The unramifiedness follows from the rigidity theorem for abelian varieties.
The last statement is to be interpreted as an equality between the image of (2.5) and Z(d, a;) (C), and follows easily from Lemma 2.3 above. D We now assume that p\ 2D{B) and that K = K p • Kp with K p neat, as in Proposition 1.3, and we formulate a p-integral version of the previous moduli problem.
Before doing this let us point out that for a point $ = (A, L, A, ^p) e MKP (S) of the p-integral version of our moduli problem with values in a connected scheme S we may transpose the concepts above. Hence we introduce the Z(p)-algebra The latter is a Z(p)-module with a Z(p)-valued positive definite quadratic form. The elements of V^ will again be called the special endomorphisms of (A, L, A, ^p).
Let now again d C Sym^(Q). Let ^p C y(A^) n be a J^-invariant open compact subset. Then a point of the corresponding moduli problem Z(d,ujP) on a Z(p)-scheme S is an isomorphism class of 5-tuples (A,^A,7f;j) where (A,^A,7f) is an object of MKP^S) and where j C (End(A, b) (g) Z(p)) 7 ' 1 is an n-tuple of special endomorphisms which satisfies (2.16) above and, in addition, These conditions are to be interpreted in the same way as (2.15), (2.16) above.
To clarify the relation between the p-integral version Z(d, uj?) and the previous Z(d, a;), let ). Note that it may well happen that Z(d, ujP) is non-empty but where both sides of the equality in Proposition 2.6 are empty. In fact, we will later consider cases in which d e Syn^ (Z(p)) is positive definite so that Z(d, uj) = 0 and when Z(d^) ^ 0.
From now on, since we will be interested in the arithmetic situation, we will simplify our notation by denoting uj what is denoted by uJ P above, i.e., (2.25) ^Cl^)" is a KP -invariant open compact subset.

The intersection problem
We continue to fix p \ 2D(B) and a neat open compact subgroup KP C G^A^) as at the end of Section 2. Then M. = MKP is a regular noetherian scheme of dimension 4. We wish to consider the intersection of the cycles introduced in a modular way in the previous section. Let us set up our problem in a more precise way.
We fix integers n\,..., rir with 1 ^ ni ^ 4 and with 77,1 + • • • + Ur = 4. For each z, we choose dz C Sym^ (Q) positive definite, and a J^-invariant open compact subset c^ C ^(A^)^. Let be the fiber product of the corresponding special cycles. By what has been said in Section 2, since the codimensions of the generic fibres of our special cycles add up to the arithmetic dimension of MKP. one might expect that Z consists of finitely 710 S.S. KUDLA AND M. RAPOPORT many points of characteristic p. We will see that this is in fact quite false, but we will be able to determine that part of Z which lies purely in characteristic p and also determine the isolated points of Z.
Let ^ be a point of Z, with corresponding point (A^, ^ A, ^p) c M.. We denote by C^ and (V^, q^) the Z(p)-algebra and the quadratic Z(p)-module associated to (A^, A, if) (cf. (2.20)). The projections Z ->• Z{di,uji} define HI -tuples of special endomorphisms where ( , )^ is the bilinear form associated to q^. Here, as always, p ^ 2. The matrix Tî s called the fundamental matrix associated to the intersection point ^ of the special cycles Z(di,cc;i),..., Z(dr,^r) ' We note that the blocks on the diagonal of T^ are di,..., dr. By the results of Section 2, the function ^ i-^ 7^ is constant on each connected component of Z.
Therefore, for T e Sym4(Z(p)) we may introduce

3.4) == union of the connected components of Z consisting of the points ^ with T^ = T.
We note here the hereditary nature of our construction, given by alid provided that the blocks on the diagonal of T are di,..., dr. We may therefore write We shall see that the fundamental matrix governs the intersection behaviour of our special cycles. We first note the following result.  Here_7TAo denotes as usual the Frobenius endomorphism. If dimAo ^ 2 and AQ remains simple over Fp, then F is a CM field. Suppose that dimAo = 2, so that [F : Q] = 2 or 4. The second case is excluded, since then E = F is commutative. In the first case, E is a division quaternion algebra over F ramified only at places over p. Thus p splits in F and invv(E) = inVv(E) = \ for v | p. But the embedding B c -^ E yields an isomorphism B 0Q F ^ E. This is possible only ifp | -D(B) and F splits B at all other primes. If dim A = 4, then [F : Q] = 2, 4 or 8, and the last case is again excluded since E = F. In the case [F : Q] = 4, F is a quaternion algebra over F, ramified only at primes lying overp, and B 0Q F ^ E. This cannot occur if p \ D{B). Finally, if [F : Q] = 2, then p splits in F and F is a division algebra over F of dimension 16 with invariants ^ and | at the primes over p. There is no homomorphism from a quaternion algebra B (g)Q F into such an algebra. D Returning to A, and assuming that p \ D(B), we see that A cannot be simple and that any simple factor of A of dimension 1 or 2 must occur with multiplicity at least 2. Thus we have various possibilities for A, up to isogeny: More precisely, p splits in F and F Ĥ p (g)Q F, where Hp is the quaternion algebra over Q ramified at oo and p. Let B' be the quaternion algebra over Q whose invariants agree with those of B except at oo and p. Then  For the last identification we are using the proposition in Section A.4 of Appendix A. Indeed, by A.5 the Rosati involution on End°(A^) ^ Mg(Hp) is of main type. Since the Rosati involution induces via restriction to M2(B) the given involution ofneben type, its restriction to M2(B') is of main type by the proposition of A.4.
Note that dim V^ ^ 3, with the exception of the supersingular case (3.8.vi). As a consequence, we have the following: 4 SpecZ(p) with support in the special fibre.

with corresponding special cycle Z(T, io). Ifdet(T) ^ 0, then the point set underlying Z(T^ uj) maps to the supersingular locus of M XspecZ(p) SpecFp. In particular, Z(T,UJ) is proper over
Proof. -Indeed the previous results imply that this is true for closed points. D Having answered these very crude questions on the intersection behaviour of our special cycles, we are led to ask more precise questions. Again for i = 1,... ,r let di e Sym^.(Q) be positive-definite with HI + • • • + rir = 4 and let uji C ^(Ap^ with corresponding cycles Z(di,C(;i),.. .,Z{dr^r}' We then ask: (a) under which conditions do the cycles Z{d^, 0:1),..., Z(dr, uJr} intersect properly? More precisely, can one parametrize the isolated points of Z = Z{d\, uj\) x ^ • • • x ^ Z(dr, u^r} and calculate at such an isolated point y, (3)(4)(5)(6)(7)(8)(9) e(y)=\g^(0^y)7 X^Oz.^'-^Ozr) (cf. [22,27]). An important question to answer is when the derived tensor product here can be replaced by an ordinary tensor product, i.e., by Oz' In the case when Y is an isolated point this would mean that the length in (3.9) is in fact the intersection number of Z\,..., Zr at y. In particular one may ask, when does the intersection number along Y depend only on T with Y C ZT^ Related to this question is the problem of the singularities of the schemes Z(d,c<;): under which conditions are they Cohen-Macaulay, or even locally complete intersections? In general they are neither [21]. Our next task will be to investigate the structure of the supersingular locus M.^ C M. xspecZ(p) SpecFp.

Structure of the supersingular locus
As mentioned in the introduction, the results of this section are a presentation of results of Moret-Bailly [23] and Oort [24]. A similar presentation was independently given by Kaiser [11].
We put F = Fp, and let W = W(¥) be the ring of Witt vectors of F and JC = W ^ Qp its quotient field. Also write W[F, V] for the Cartier ring of F.
Throughout this section, we assume that p \ D{B), and we fix an isomorphism Oc <^z ^p M 4(Zp). Suppose that ^ = (A, L, A, 7f) e .M^F), and let A(p) be the p-divisible (formal) group of A. The action of Oc 0z Zp ^ M^Zp) on A(p) then induces a decomposition A(p) ^ Ao(p) 4 , where Ao(p) is a p-divisible formal group of dimension 2 and height 4. Let LQ be the (contravariant) Dieudonne module of Ao(p) and let C = LQ ^w K-be the associated isocrystal. This does not depend on the choice of $ e M ss (¥), up to isomorphism.
More precisely, we fix a base point <^o = (Ao, ^, Ao, rf^) e .M^F) and let £=LQ ^>w K be the isocrystal associated to it. The isocrystal C has a polarization ( , ), is isoclinic with slopeand has dim^c C = 4. Then F is a-linear, V = pF~1 is a~1 -linear, and For TV-lattices L, L' in £, we define the (generalized) index \L': L\ as If L is special, then [L 1 -: L\ e Z is divisible by 4. We can replace L by a • L, for a e K^ to obtain a lattice with L= L-L or L= pL^. In this case we call L standard. We note that, if L is an admissible lattice, then, since C is isoclinic of slope 1/2, we have We define a set of lattices as follows: If L e X then FL e X. This follows from (FT^ = V-1 • L^ (c/ (4.1)). The conditions in our moduli problem imply that the lattice L c C associated to ^ e M ss (¥) and an isogeny between ^ and ^o actually lies in X. Note that each admissible lattice is the Dieudonne module of a ^-divisible formal group of dimension 2 and height 4 over F.
Recall from (3.8.vi) that End°(A^, i)°P =: C' ^ M^B'), where B' is the definite quaternion algebra over Q with the same local invariants as B at all primes £ ^ p. As before, let V = [x € M2(B'); x' = x and tr(a;) = 0}. Let Note that the action of G'(Qp) on A^ (p) up to isogeny passes to C. In fact, The action of G'(Qp) preserves the set of lattices X. Fix an isomorphism ^(A^) ^ B^A^) and, hence, an isomorphism G(A^) ^ G'(A^). Then, the usual analysis identifies G'(Q) with the group of self-isogenies of ^o and yields an isomorphism (4.6) .
We will now describe the lattices in X in more detail.
Since a(L) = dimp Hom^y^y] (L, F), we see that a(L) is the a-number [24] of the ^-divisible group Ao(p) associated to L, i.e., Such lattices will be called superspecial.
In addition to the superspecial lattices, the following type of lattice will play a key role in the description of the structure of X. i.e., VL = FL, as claimed. Similarly one sees that if Z € X, then FL e X.
Starting with a distinguished lattice, we can scale it to obtain a distinguished lattice Z with either (4.10) Z^zZ^p-^Z or Z^Z^zp^Z-S with all indices equal to 2. We will call distinguished lattices scaled in this way standard. We note that if, in the identity defining a distinguished lattice L, the order of c is odd, then L may be scaled^ to be standard in the sense of the first alternative of (4.10) above. If the order of c is even, then L can be scaled to be standard in the sense of the second alternative of (4.10), and hence, FL can be scaled to be standard in the sense of the first alternative of (4.10). For any L 6 X and for any F Proof. -First, since FL = VL, we have Hence L is admissible. Next, we have The above proof infract shows the following. Suppose that L e X with FL =p•L ± . Then L^ =pL(£). lfFL ± =pL, then L(£) 1

-= L(£).
Thus to any distinguished L we have associated a projective line P(L/FL) and a family of admissible special lattices parametrized by the F-points of this projective line. These projective lines have a natural ¥p2 -structure which we now describe.
For any TV-lattice L in £, we have Then LQ is a Zp2 -module and If Lj^ X_is distinguished, then L i^ preserved by the a 2 -linear endomorphismp" 1^2 , and we have L^ LQ 0z^ W. Moreover, FL is also preserved by p~lF 2 , and (FL)o = F(£o). Thus, the two-dimensional F-vector space L/FL has a natural Fp2-structure: We may then view any line £ as an element of P(Lo/FLo)(¥). We denote by P~ the projective line P(Zo/FZo) over ¥p2.   On the other hand, F 2 !^ = F 2 L + pL. Let S = L/pL, and let / and v be the cr-linear respectively a~1 -linear endomorphisms of S induced by F and V. Since FV = VF = p, we have fv = vf = 0 and so ker(/) = im(v) and ker(^) = im(/) are two-dimensional subspaces of S. However, for any L^X there is some j ^ 2 with F 3 L C pL and hence / is nilpotent. If f 2 = 0, then F 2 L = pL since both lattices have index 4 in L and this would imply a(L) = 2, contrary to our assumption. Therefore, since im(/) is two-dimensional we must have that im(/ 2 ) is one-dimensional and im(/ 2 ) = im(/) n im(v). Hence where all inclusions are of index 1. It follows that FL e X and hence also L e X. The "distinguished curves" cross at the superspecial points. To describe this, it will be useful to have a normal form for superspecial lattices.    Comparing, we see that fi must be a unit and that /^C T = -1 modp, as claimed. It is easy to check that the case in which a is not a unit yields no solutions. The assertion (ii) is trivial. D  We would finally like to compute the stabilizers in G'(Qp) of the superspecial and distinguished lattices.
Let B' be as above, and, identifying Qp2 with a subfield of By, write B' = Qp2 + IIQp2 for an element II e B'y x with n 2 = p and such that Ha = a^II, for a G Qp2. Let Co be the fixed set for the automorphism p" 1^2 of £, and let II operate on Co by F. By construction, II 2 = p, and so Co is naturally a left vector space over By of dimension 2. The polarization on £ induces a Qp2-bilinear symplectic form on Co, which still satisfies    n z^j^zj-\ _n Thus, in classical language (c/ [29,8]), the superspecial lattices correspond to local components of the principal genus of quaternion Hermitian lattices, while the distinguished lattices correspond to local components of a non-principal genus of such lattices.
In less classical language we may describe our results in terms of the Bruhat-Tits building of the adjoint group G^ over Qp (comp. [11]). The building B(G^ Qp) is a tree and may be identified with the fixed points In these terms the stabilizer K^ of a distinguished lattice Z e X is a maximal compact subgroup of the first kind of G^Qp), and the stabilizer K 88 of a superspecial lattice L G XQ is a maximal compact subgroup of the second kind of G^Qp).
Remark 4.19. -We return, for a moment, to the global situation, and recall that X is the set of distinguished lattices in C. As observed in Remark 4.10, our calculations "show" that the supersingular locus M^ is a union of rational curves and that the irreducible components are in bijection with the set where K^ is the stabilizer in G'(Qp) of a fixed distinguished lattice Z e X. These curves cross, p + 1 at a time, at the superspecial points, and there are p 2 + 1 such crossing points on each component. The set of all crossing points is in bijection with the set where K^ is the stabilizer in G^Qp) of a fixed superspecial lattice L e XQ.
We finally observe two consequences of our description of M^. Here, we have fixed a maximal order R in B, and for £ \ D(B\ we fix an isomorphism Ms (B^) M 4(Q^) such that M^R^) ^ M^). Then let K^ = G(Q^) H IVL^). Thus, for ^ | D^ (respectively ^ | D^), KH is the stabilizer ofaHermitian 0^-lattice of principal (respectively nonprincipal) type, and, for i\D(B\ K^ is a hyperspecial maximal compact subgroup of G(Q^). Note that, in contrast to the general assumptions above, K is not neat. Still, for a fixed prime p\ D{B\ one can consider the coarse moduli space M.K (the quotient by a finite group of one of the schemes considered above) and its points over F. Let B^ denote the definite quaternion algebra with D(B^) = D(B)p. Then, by (4.25), the components of the supersingular locus in the fiber of M.K at p correspond to the classes of maximal Hermitian lattices in the genus of type (D-i^pD^) for B^. An explicit formula for this number H(D^^pD^) was found by Hashimoto and Ibukiyama [9]. In the case D(B) = 1, so that B = M2(Q), the abelian varieties parameterized by M-K^) have the form A ^ A^, where Ao is a principally polarized abelian surface. Thus, in this case, M.K ^ AS,!, and the description of the supersingular locus reduces to some of the information given by Katsura and Oort [12], Theorem 5.7, and Ibukiyama, Katsura and Oort [ 10]. In particular, the number of irreducible components of the supersingular locus is H (1, p).
As another example, fix a square free positive integer D and distinct primes p\ and p2 relatively prime to D. Consider indefinite quaternion algebras B\ and B^ over Q with discriminants D{B^) = Dp^ and D(B^) = Dp^. Let G\ and G^ be the associated groups, via (1.3). As in (4.5), let G[ be the twist of Gi at p^ and let G^ be the twist of C?2 at ^i. These groups are both associated to the definite quaternion algebra Bf^ ^ B^^, and are isomorphic. Here we have written (Mf x FpJ 8 -8 -for the supersingular locus of the fiber over p^ of M\\ where *i = ss, for example. These results are in the spirit of those ofRibet [25,26], who considers components and their crossing points for the fibers of Shimura curves and modular curves at primes of bad reduction.

Endomorphism algebras and points of proper intersection
In this section, we consider the points of intersection of the special cycles in the supersingular locus, using the information obtained in Section 4 about the structure of this locus. In particular, in the decomposition of (3.6), we fix a matrix T and we obtain a criterion, in terms of T, for Z(T,u) to consist of isolated points. We also show that, even when det(T) 7^ 0, there can be components of the supersingular locus in the image of Z(T, uj) in A^8 8 .
We retain the notation of Sections 2-4, and we begin by obtaining information about the endomorphism rings of various types of admissible lattices.
For an admissible lattice L, let OL = End^(L, F) be the Zp-algebra of TV-linear endomorphisms of L which commute with F. Note that Endw(L,F) is an order in the Qp-algebra EndK;(A F) = Cp = C' 0Q Qp ^ M^Bp). Also, observe that Endw(L, F) = Endw(F^L, F) for any j G Z. If L = c' L^ is special, we have Thus, to determine OL for L € XQ we may assume L = L ± . By Lemma 4.18, we immediately have the following.

_ LEMMA 5.1. -For any super special lattice L e XQ (respectively any distinguished lattice L € X) Endw(L, F) (respectively Endw(L, F)) is a maximal order in Cy
In either case, this order is isomorphic to M'z(0'\ where 0' = Zp2 + IIZp2, as in Section 4. The map M-^(0') -> M^W^) given by reduction modulo n can be described as follows. Consider the case of L e XQ. As in Section 4, let LQ be the fixed points ofp^F 2 on L. Then define is defined analogously. Note that L/FL ^ Lo/FLo (g)p 3 F, and that the endomorphism a induced on L/FL by a e End^v(Z, F) is red^a) (g) 1.
Next, suppose that L e X \ XQ, and let L be the unique distinguished lattice associated to L by Proposition 4.8. Recall that FL = FL + VL. In particular, for every element a e Endw(L, F), aFL C FL, so that aL C L, and there is a natural homomorphism which is injective, for some embedding ¥p4 ^ Ms (Fp2).
Proo/^ -As remarked above, the automorphism of L/FL = Lo/FLo 0F 3 F induced by p~lF' 2 is just 1 0 a 2 . Since, for any a e Endw{L,F), a commutes with this automorphism, a(£) C £ implies that Q^a 2^) ) C cr 2^) . Since a nonscalar endomorphism can have at most two eigenlines, a(£) C £ and a 4^) -^ £ implies that a == a • Is, for a <E Fp2. If a 4^) = ^ but a 2 (^) 7^ ^, and if a is not a scalar endomorphism, then £ and cr 2^) are the distinct eigenlines of a. Then ¥p2 [a] ^ Fp4, and any endomorphism f3, with f3 G Endyy (L, F) must lie in this subfield ofM2(Fp2). D Note that the lattices in (ii) of Lemma 5.2 are characterized intrinsically by the condition that F^L = p 2 L but F^L -^ pL. We let X^ be the set of lattices appearing in (ii) and X^ = X \ X(ii) \ Xo the set appearing in (i). Recall that OL = Endw(L, F) and 0^ = Endw(L, F). Then redL(0L) = TedL{Endw(L,F)) ^ M^(¥p2) ifLe Xo, Similarly, if FL = cL 1 -and x G 0^ then ;r*(£^) c Z^, i.e., x^FL) c FL, i.e., a;*(Z) c Z, since rr* commutes with F.
For L C X and for L € X, let (5.8) TVL-End^L^ny; and A^ = End^(£,F) n V;.
These are Zp-lattices in V? on which the quadratic form given by squaring, We now describe the reduction maps for distinguished and for superspecial lattices. We start with the case of distinguished lattices.  But the upper horizontal arrow is surjective since both algebras are generated by M. This proves that the lower horizontal arrow is surjective. By the statement at the beginning it is also injective which proves the equality sign at the south-west corner of the diagram in (ii). The rest of the Lemma follows from Lemma 5.3. D Next let us consider the case when L e XQ. We use somewhat similar notation: let HL == redL(A^) and m^ = redj^OM).
The same arguments yield: There is a commutative diagram: Our next task will be to show that the matrix T mod p in M^ (Fp) controls the size of m = mõ r rriL. Recall that, as in Lemma 5.5,
We now list the possibilities for m, which is the span of red(j'i),... ,red(j4), in the nonsuperspecial and the superspecial case separately. In cases (iii) [respectively (iv)], mo [respectively m] is a nondegenerate line, so that the quadratic form on it is isomorphic to either x 2 or ax 2 , with a e F^ \ F^' 2 , yielding a Clifford algebra Fp C Fp or Fp2. cpW.
Here we have used the fact that x^ = XQ and that x^ = 0, i.e., that x^{L) c FL. We conclude that FL C X and hence also L <E X. Next, we must show that every element of M preserves L or, equivalently, FL. In fact, we show that M • L c FL, so that red^(M) = 0. First, consider the reduction sequence This completes the proof of the lemma. D To finish the proof of (ii), we show that the distinguished lattice FL constructed in Lemma 5.13 is unique. Note that ker(^o) = im(^o). If L f = W • u + FL is another distinguished lattice, whose image V = L'' /FL is distinct from ker(;ro), then Since ^o commutes with I 7 ' and V, the eigenspaces Ei^and £"2 are preserved by F and V, hence FLi and -FI/2 are admissible. To see that FL\ and FL^ are distinguished, it suffices to see that FLi Cp' L^-, i.e., that Equivalently, we have to see that the eigenspaces E\ and E^ are isotropic with respect to the antihermitian form (5.11) on Lo/FLo. If v e ^, then XQV =£z-v and It follows that FLi and FL^^re distinguished.
The lattices Li = F~l(FLi) C X for i = 1 and 2 are the distinguished lattices appearing in the statement of (i). We have red^ (M) ^ 0 since XQ induces an automorphism of the eigenspace Ei. On the other hand any y G M with red^) = 0, i.e., with ?/(L) C -FL, also satisfies y{Li) c L cLi.lt follows that M C End^(Z^ F), hence (i), by the remark preceding Theorem 5.11. Now we consider case (ii). Let XQ e mo with x^ = e • 1, for e G F^ \ F^' 2 , and let ]/o be a generator of the radical r. Then XQVQ = -y^xo. Therefore Z/Q maps the eigenspace E\ of XQ in Lo/FLo to the eigenspace ^2 and the eigenspace E^ to Fi. Since ^2 = 0, but ZJQ ^ 0, precisely one of the two eigenspaces is annihilated by y^. The corresponding lattice is distinguished and yields as in case (i) the lattice L\ appearing in the statement of (ii). D

)). Then ^ is a point of proper intersection if and only if its fundamental matrix T =T^ is nonsingular and represents 1 over Zp. In this case $ is supersingular and superspeciaL
A topic we have not touched upon in the present paper is to describe the shape of the intersection of our cycles in the case of improper intersection, or, equivalently, to describe, for T e Sym4(Q)>o, the cycle Z(T^) when its dimension is positive. We refer to the companion paper [21] to the present one for more information on this topic.

Intersection multiplicities
In this section we consider the intersection multiplicity at a point of proper intersection. More precisely, we return to the setup of the third section, i.e., we fix a decomposition 4 = ni + • • • + nr, where n, > 1 for all z, elements ^ C Sym^ (Q)>o and ^ C l^A^ giving rise to special cycles Z(di,cc;i),...,Z(d^o^). We fix a point ^ e ^(^i,cc;i) x^ ••• x^ Z(d^c^) with det(T^) ^ 0 and where T = T^ represents 1 over Zp. Let <^ correspond to (A^A^Ji,.. .Jy.). Since det(T) ^-0 and since T represents 1 over Zp, the associated Dieudonne module L is superspecial and corresponds to a formal group A of dimension 2 and height 4, with a collection of endomorphisms j = (ji,... ,.74) spanning a Zp-submodule M of rank 4 in Endyy(L, F). By changing the trivialization of the rational Dieudonne module we may assume that L = L-L , i.e., that A is equipped with a principal quasi-polarization A^. By the theorem of Serre and Tate, the infinitesimal deformations of (A, L, X, ^p) correspond to those of (A, A^), i.e., (6.1) ^=Def(^A^).
Here M.^ denotes the formal completion of M at ^ and Def(A, A^) the formal deformation space of (A, A^) over Spf W. Similarly, for the special cycles one has, with obvious notation, The length e(^) of the local Artin ring appearing on the right was determined by Gross and Keating in Section 5 of [7]. Since we have assumed in all of the above that p -^ 2, we may as well continue to make this assumption, although Gross and Keating do not. Choose a basis ^i, ^2, ^3 for Mo such that q(u^ + u^2 + ^3^3) = dP^u^ + e^u^ + e^u^, with 0 ^ oi ^ a2 ^ 03, and £1, £2, ^3 ^ ^. Thus, over Zp, T is equivalent to the diagonal matrix diag^^^, e^, e^3). Recall that by Lemma 5.9, Z(T, uj) = 0 unless ordp det T 1 , i.e., 03 > 1. In addition, the matrix diag^^^R" 2 ,^^3) is represented by the norm form on the maximal order in the quaternion division algebra over Qp. This imposes additional restrictions on the a^, see Section 10. Proof. -The first statement follows from the formulas above. In this case, the cycles Z(di,uji) have to be irreducible and reduced locally at ^, and the intersection multiplicity in the sense of Serre, which is bounded by the length, is equal to 1. The rest follows from [5], Proposition 8.  Z(dr,uJr) at Ŵ e stress that this conjecture is reasonable only because M. is smooth over SpecZ(p). Indeed, Genestier [6] (comp. [22]) has shown that in the Drinfeld-Cherednik situation of bad reduction the analogues of the special cycles considered here may have embedded components. On the other hand, assume in our situation that Z(d^^) is an intersection of n, divisors in M. Then if ^ is an isolated intersection point of Z(di, c^i),..., Z(dr, c^r) it follows that each partial intersection Z(d^,c^J n • • • D Z(di^^) (1 ^ %i ^ • • • ^ ^ ^ r) is locally at ^ a complete intersection. Hence it also follows that the length e(^) is the intersection multiplicity of Z{d\, 0:1),..., Z(dr, u^r) at ^, and the above conjecture holds true. Comparing with the definitions in the companion paper to this one, we see that (A, a) is precisely one of the formal groups with Zp2-action considered there [21] and that the elements of M\ are special endomorphisms in the sense of that paper. In particular, the formal completion of Z(di,cji) D • • • D Z(dr,(^r) at $ coincides with the formal completion of the corresponding subvariety of the Hilbert-Blumenthal surface considered in [21].

The total contribution of isolated points
In this section we will consider the total contribution of the points of proper intersection of our special cycles. Using our previous results and a counting argument, we are able to give an explicit formula.
We return to the global situation of Sections 1 and 2 and fix data as follows. We assume as always that p \ 2D(B) and that K = K p • Kp where Kp is the standard maximal compact subgroup (see the end of Section 4), and where K 10 is neat. We then have the moduli scheme M. = M.KP which is smooth over SpecZ(p). As in Section 3, we fix ni,.. Here the sum runs over the points of proper intersection ^ in Z(d\, uj\) Xj^ • • -Xj^ Z(d^uJr)â nd e(^) denotes the length of the local ring at ^, as described in Section 6. Note that, if Conjecture 6.3 were known to hold, this is also the local intersection multiplicity at ^.
In the special case r = 1, we let di = T, and we have the cycle Z(T, c^), whose image in M lies in the supersingular locus .M 8 ®. Then Z(T, uj} is a collection of isolated points if and only if T represents 1 over Zp (Corollary 5.15). In this case we use the notation (7.2) {Z{T^)}^= ^ e(Q.

Z{T,UJ)
In general, by (3.6) and the analysis of the previous sections, we may write We will now give more explicit expressions for the above entities. For this it will suffice to give an expression for (7.2). But the results of Section 6 show that the intersection multiplicities e(^) in the sum of (7.2) only depend on T and even only on its Zp-equivalence class. As in Proposition 6.1, we denote this integer by ep(T) and thus may write (7.4) <Z(r^)>^e^(r). |Z(T^)(F) It remains to determine the cardinality of Z(T^ c<;)(F). As before, let B' be the definite quaternion algebra with discriminant D(B)p, let C' = M^B'), and let V = [x e C"; x' = x and tr(x) = 0}. Let Q be as in (4.5). Recall that we also have fixed an isomorphism G'(A^) c^ G(A^), and a base point <^o = (Ao,^,Ao,^) <E .M^F) such that the associated Dieudonne module Lo € X is superspecial, with stabilizer Kp in G^Qp). Then, under the parametrization (4.7), the set of superspecial points in A^F) corresponds to the double coset space We note that the condition (i) is equivalent to the assertion that the components of the 4-tuple g^y lie in (7.8) nZ^y^nEnd^^A^.
We let (pp be the characteristic function of V^Zp) 4 , let ^ = char(^) be the characteristic function of uj, and set ^ = ^ 0 ^. Then (^ e S^V^A^) 4 )^. Conditions (i) and (ii) can then be summarized as follows. The measure dg is induced by an arbitrary Haar measure on Z\Kf) \ G'{Kf) and the atomic measure on Z'(Q) \ G^Q). The coefficient Cp(T) is given by the formulas in Proposition 6.1. The identity of the theorem remains valid if T is nonsingular but not positive definite, since, in that case, T is not represented by V\ and hence both sides of the identity vanish.
Proof. -By Lemma 7.1, we see that (7.9) [Z(7>)(F)|= ^ ^(^y). In our main theorem (in Section 9), we will identify the right hand sides of the formulas of Theorem 7.2 and Corollary 7.3 as special values of derivatives of Fourier coefficients of certain Eisenstein series. In the next section we will explain more precisely the Eisenstein series in question.

Fourier coefficients of Siegel-Eisenstein series
In this section, we recall, from [19], the construction of certain incoherent Siegel-Eisenstein series and the structure of the Fourier coefficients of their derivative at s = 0, the center of symmetry. To be more precise, these Eisenstein series occur on the metaplectic cover of the symplectic group of rank 4 over Q, and have an odd functional equation. Their Fourier coefficients are parameterized by rational symmetric matrices T G Syn^Q). In [19], a formula was given for the derivative at s = 0 of such a coefficient, when det(T) 7^ 0.
We retain the notation of Section 1, and we refer to Sections 1-6 of [19] for more details. Thus B is an indefinite quaternion algebra over Q of discriminant D(B), C = M^{B), V is given by (1.1), and G is given by (1.3), etc. In particular, V is a five-dimensional quadratic space over Q with signature (3,2). Let \ = \y be the quadratic character of A^d^ attached to V: \(x) = (x, det(V))A, where ( , )A is the global Hilbert symbol. Note that ^oo(-l) = 1.
Let TV be a symplectic vector space of dimension 8 over Q, with a fixed symplectic basis ei,..., 64, e[,..., 64, and let H^ be the metaplectic extension of SP(WA), with Siegel parabolic PA. For s € C and for \ as above, let I^s^) be the global degenerate principal series representation of H^. As explained in [19], (2.9), the representation ^(O,^) has a direct sum decomposition into two types of irreducible representations. One of these types are the irreducible summands, like r^V), associated to five-dimensional quadratic spaces with character \y The other type are the irreducible summands associated to incoherent collections, in the sense of Section 2 of [19]. One such summand is Il4(C), associated to the incoherent collection C, defined as follows. For any finite prime t, C^ = V^, while Coo = V^, where V^ is the quadratic space over R of signature (5,0). There is a surjective map converges for Re{s) > 5/2, and its analytic continuation vanishes at the point 5=0, [19]. There is a Fourier expansion with respect to the unipotent radical of P. When ^(s) = (^ ^(s) is a factorizable section, and when det(T) -^ 0, there is a product formula where IVr,^^? 5, ^) is the local generalized Whittaker integral (cf. Section 4 of [19]). For fixed h, T, and ^, there is a finite set of places S such that, [19], Proposition 4.1, Thus, the only nonsingular T for which 2^(/i,0, ^) can be nonzero are those for which |Diff(T,C)| = 1. We will relate the value E^(h,0, ^) for Diff(T,C) = {p} to the numbers (^(T, uj))p in the previous section. Let us fix a finite prime p. We wish to give a formula for E^{h^ 0, ^) if T e Sym^Q) is nonsingular with Diff(r,C) = {p}. Let B' be the definite quaternion algebra over Q which is ramified at p and whose invariants coincide with those of B at all finite primes other than p. Let G'^lVhGBQ.andlet Recall, [31,19], that the local degenerate principal series representation ^4,^(0, \p) has a direct sum decomposition with irreducible factors Recall that the metaplectic group H^ acts on the space 5'(y(A) 4 ) via the Weil representation uj = uj^, defined using our fixed additive character '0 of A/Q. For g e G'(A) and h e H^, let (8.16) 0{g^)= ^ ^W}{g-1ŷ eV^Q) 4 be the theta function attached to (^/, and let for the Tamagawa measure dg on Z(A) \ G'(A), and where ev((//) denotes the projection of (// to the subspace of functions all of whose local components are even (cf. [19], (7.19)). Note that 0{g, h, (//) can be defined by the same formula for g <E 0(1^) (A), and that

The main theorem
In this section we assemble the results of previous sections and state our main results. We begin by further specializing the formula of Proposition 8. Fix the prime p with p \ 2D{B), and assume that ^p is the characteristic function of V(Zp) 4 .
Recall that ^p(s) is the standard section with ^p(O) = \p{^p). Also, let ^ be the characteristic function of the lattice y'(Zp) 4 , and let ^p(s) be the standard section with ^p(O) == \p{^p).
Recall that a nonsingular T G Syn^Qp) is represented by precisely one of the quadratic spaces V(Qp) and V^Qp), [19], Proposition 1.3.   The proof will be given in Section 10.
A subset uj C V^A^) 71 is said to be locally centrally symmetric if it is invariant under the action of the group ^2(A^). The characteristic function ^ e ^(^(A^) of such a set is locally even, as in (8.17), i.e., ^ = ev(^). The function ^ = ^ (g) ^ e ^(y^Aj)^) is then locally even as well, so that the expression (8.29) holds for the derivative of the Fourier coefficients of the associated Eisenstein series.
We can now state our main result. Note that, if r e Sym4(Z(p))>o does not represent 1, then Z(T,UJ) contains components of the supersingular locus (Corollary 5.15 and Theorems 5.12 and 5.14). In this case, we do not have a formula for the contribution of Z(T^ uj) to the intersection number.
In Theorem 9.2, the chosen gauge form ^ on 80(1^) = Z' \ Q determines the Haar measure on SC^y^R) used to compute vol(SO(y')(]R)). The corresponding gauge form on the inner twist SO(V) = Z\G determines the measure on Z^Af) \ G^Af) used to compute vol(pr(JC)). Note that the product vol(SO(y')(R)) vol(pr(^)) is independent of the choice of P roof of Theorem 9. . J logp • (p 2 + 1) (p -1). <Z(T, o;))p.
To finish the proof, we simply note the following relation between volumes. vol(pr(X))^vol(pr(^))_ vol(pr(^)) vol(pr(^)) ^ ' y^ / ' This finishes the proof of Theorem 9.2. D Proof of Lemma 9.3, following Kottwitz [16]. -We may replace G/Z and G'/Z' by their simply connected coverings G respectively G r and pr(Kp) and pr(^p) by their inverse images Kp respectively Kp. We use on G(Qp) respectively G^Qp) the Haar measure induced by a top differential form on the Zp-form of G respectively G' corresponding to an Iwahori subgroup TpCKp respectively Ip c Kp. These measures are compatible (cf. [16], p. 632). The volumes of Ip and Ip are related as follows. Choose as in [16] a maximal split torus S in G and a maximal torus S'i containing S which splits over an unramified extension. We also denote by 5i the canonical Zp-form of S'i. (cf. [19], (6.13)). This result gives an analogue of the results of [19].

Representation densities
In this section, we give the proof of Proposition 9.1, which is based on a formula of Kitaoka, [14], for representation densities. In this section, for x e Q^, \{x) = (x,p)p.
We begin by recalling the well-known relation between the values of the if T G Syn^Zp), since the factor 7p(Vp) in loc. cit. is 1 in our present case. Recall -see [14], Lemma 9 and the discussion on pp. 450-453, for example -that a? (Sr, T) is a rational function of X =p~r, i.e., there is a rational function As r(^0 such that (10.6) ap(Sr^T)=As,T{p~r).
We therefore have Of course, we would like to have analogous information about Wp (e, 0, ^p) and Wr,p (e, 0, p) for all T. At first, we simply restrict to the case where p \ T, so that we may assume that GO = 0, i.e., (10.10) T=dmg(£o,elp a \e2p a2 ,e3p a3 ).

10.15) ^"^l-J-
Then Sr is isomorphic to the split space H^r^ and Kitaoka gives an explicit formula for the representation density Op(H^rmT) for any ternary form T, [14]. His formulas, in the cases ai -as even and a\ -02 odd, are given as a sum of five double sums! These can be simplified to yield the following expressions:  and T is as in (10.13).
If eo is a square, then ap(S\£o)=l-p-\ [32]. On the other hand, 5" is just the norm form on the maximal order of the division quaternion algebra over Qp. The following result is due to Gross and Keating, [7], Proposition 6.10. For convenience, we give a proof. Proof. -Let B be the division quaternion algebra over Qp, and let R be its maximal order. Then, for a suitable Zp-basis, S" is the matrix for the quadratic form Q given by the reduced norm on R. Let  Thus, we may replace f by T' = diag(^i, e^p^-^, e^p 0 ' 3 -0 ' 1 ). Here e\ can be taken to be equal to either 1 or /3. Using reduction, we have Let (V, q) be a nondegenerate quadratic space of dimension 5 over a field F of characteristic not 2. Let C(V) be its Clifford algebra, with its 2-grading

C(V)=C^(V)eC~(V).
The Clifford involution c ^ c' of C(V) is the unique involution which acts by the identity map onVcC-(y).Thus where dimy± = 2 and V± are maximal isotropic subspaces of V. Let VQ e Vo be a basis vector with q{vo) = 1. We recall the Spin representation of G. We use the identifications of representations of C (V), c(y)/c(y)C(y-)>o = C(Y+ e Vo) =c(y+)(i+z;o)ec(y+)(i-^o).
As C(y) 4 "-modules the last two modules are isomorphic. Either one of them defines the Spin representation W of G. Its dimension is 4.
Fix an isomorphism A 2 V^= F and let In particulars/or g e G = G SpinfV),

{a(g)x,a(g)y)=v{g){x,y).
Here a(g) denotes the spin representation action ofg on W, and v. G -> F x , i/{g) = gg' is the restriction to G of the spinor norm on C(V).
Proof. -Choose a basis eo, ei, VQ, fo, f\ for V such that the matrix for the quadratic form is / 1 0\

A.3.
In this section F is again arbitrary, of characteristic not 2. Proof. -Recall that 6 e C~(V) is central in C(Y) and satisfies ^/ = 6. It is, thus, clear that x = 6v satisfies x' = x. On the other hand, x 2 = q(v)6 2 = a lies in F, the center of C^V). In addition, if re 7^ 0, then x cannot lie in the center of C'^^V), since, if it did, then v = 6~lx would lie in the center of C(Y), and this is not the case. If a = 0, so that x 2 = 0, the condition tr(x) = 0 is immediate. If x 2 = a ^ 0, choose u e V with q(u) -^ 0 but with (u, v) = 0, and set y = 6u. Then xy = -yx, and so, over an algebraic closure of -F, left multiplication by y gives an isomorphism between the =L^/a eigenspaces of x, and thus these spaces have the same dimension and tr(rr) = 0. This proves that 6V is contained in the space on the right hand side. The converse inclusion will be proved further down. D Similarly, applying the involution of nebentype on Mg(B), we have T^( x (g) y) = i^x (g) faV/i" 1 ) =i(^x (g) T y).
Every involution on E compatible with the isomorphism i: C\ 0 C^ -^ E is conjugate to one of these two by an element of the form g = i{g^ (g) g^), with t g L = ±g. Note that Proof.-Take r € B x such that r' = -r and r 2 < 0. Note that the condition on r 2 is automatic when B = H. Choose T] ^ B x such that ryr = -rr] and T^ = -rj. Then every element a; C B can be written uniquely in the form x = a + brj with a and & C R(r) ^ C. Then