Green Forms and the Local Archimedean Arithmetic Siegel-Weil Formula

GREEN FORMS AND THE LOCAL ARCHIMEDEAN ARITHMETIC SIEGEL-WEIL FORMULA

LUIS E. GARCIA, SIDDARTH SANKARAN

Abstract. The goal of this paper is to construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in any codimension, and to show that the resulting local archimedean height pairings are related to derivatives of Siegel Eisentein series. A conjecture put forward by S. S. Kudla relates these derivatives with arithmetic intersections of special cycles, and our results essentially settle the part of his conjecture involving local archimedean heights, for compact Shimura varieties attached to O(p, 2) and U(p, 1) (any p ≥ 1).

Contents 1. Introduction2 1.1. Shimura varieties and special cycles2 1.2. Kudla’s conjectures3 1.3. Main results4 1.4. Notation and conventions6 2. Green forms on hermitian symmetric domains7 2.1. Superconnections and characteristic forms of Koszul complexes7 2.2. Hermitian symmetric domains of orthogonal and unitary groups 13 2.3. Explicit formulas for O(p, 2) and U(p, 1) 16 2.4. Basic properties of the forms ϕ and ν 17 2.5. Behaviour under the Weil representation 20 2.6. Star products 23 3. Archimedean heights and derivatives of Whittaker functionals 26 3.1. Degenerate principal series of Mp2r(R) 26 3.2. Degenerate principal series of U(r, r) 29 3.3. Whittaker functionals 31 3.4. The integral of ξ 37 4. Green forms for special cycles on Shimura varieties 42 4.1. Orthogonal Shimura varieties 42 4.2. Unitary Shimura varieties 45 4.3. Star products on XK 46 4.4. The degenerate case 47 5. Archimedean heights and derivatives of Siegel Eisenstein series 53 5.1. Siegel Eisenstein series and the Siegel-Weil formula 53 5.2. Archimedean height pairings 56 References 58 1 2 LUIS E. GARCIA, SIDDARTH SANKARAN

1. Introduction The Arakelov theory of Shimura varieties has been intensively studied since, about twenty years ago, Kudla [23] launched a program relating their arithmetic Chow groups with derivatives of Eisenstein series and of Rankin-Selberg L-functions. Let us describe his proposal before stating our main results. For simplicity, in the introduction we focus on the case of orthogonal Shimura varieties; in the body of the paper we treat both the orthogonal and unitary cases. 1.1. Shimura varieties and special cycles. Let F be a totally real number ﬁeld of degree d with real embeddings σ1, . . . , σd : F → R and (V,Q) be a quadratic F -vector

space such that V = Vσ1 has signature (p, 2) and Vσi is positive deﬁnite for i ≥ 2. Let H = GSpin(V) and denote by D the hermitian symmetric domain associated with H(R). For every neat open compact subgroup K ⊂ H(Af ), the Shimura variety

(1.1) XK = H(Q)\D × H(Af )/K is a quasi-projective complex algebraic variety that carries a natural large family of algebraic cycles known as special cycles (see [22]). These exist in any codimension and are parametrized by pairs (T, ϕf ) consisting of a symmetric r-by-r matrix T r K with entries in F and an integer-valued Schwartz function ϕf ∈ S(V(Af ) ) ; the corresponding cycle has codimension rk(T ) and we denote it by rk(T ) (1.2) Z(T, ϕf ) ∈ CH (XK ).

A ﬁrst step in understanding special cycles is to consider their image [Z(T, ϕf )] ∈ 2rk(T ) ∨ H (XK ) under the cycle class map. Let L be the Hodge bundle on XK and L be its dual. In [24, 25, 26], Kudla and Millson show that the generating series d X ∨ r−rk(T ) T 2r T Y 2πitr(σi(T )τi) (1.3) [Z(T, ϕf )] ∪ (L ) q ∈ H (XK ) ⊗ C[[q]], q = e , T ≥0 i=1 1 is a holomorphic Hilbert-Siegel modular form of genus r and parallel weight 2 dim(V ). 2r For the proof they construct a theta series ΘKM(τ, ϕf ) ∈ A (XK ) that satisﬁes dΘKM(τ, ϕf ) = 0 and whose cohomology class is the generating series (1.3). The method shows that the cohomological properties of special cycles can be understood using the theta correspondence. For example, the cup product formula X ∨ r−rk(T ) (1.4) [Z(T1, ϕf,1)] ∪ [Z(T2, ϕf,2)] = [Z(T, ϕf,1 ⊗ ϕf,2)] ∪ (L ) T ≥0 T = T1 ∗ ∗ T2

for T1 and T2 positive deﬁnite follows easily. Kudla [22] also shows that, for compact XK , the degrees of special cycles with respect to L are given by X T 1 (1.5) degL(Z(T, ϕf ))q = Vol(XK , Ω) ·E(τ, λ(ϕf ), s0), s0 = 2 (p + 1 − r). T ≥0

Here Ω is the ﬁrst Chern form of the hermitian line bundle L and E(τ, λ(ϕf ), s) denotes 1 a Siegel Eisenstein series of weight 2 dim(V ) and level determined by ϕf , normalized GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 3 so that the center of the functional equation is at s = 0. The proof follows easily from the Siegel-Weil formula as generalized by Kudla-Rallis [27].

1.2. Kudla’s conjectures. Suppose now that XK has a regular model X over Spec( ) r Z and consider the arithmetic Chow groups CHc (X ) defined in [12], appropriately mod- ified if XK is not compact [9]. Here and in the following we will ignore many serious difficulties regarding the construction of good integral models of Shimura varieties and their special cycles, places of bad reduction and compactifications, as our focus is ˆ r on archimedean aspects. A class Z ∈ CHc (X ) is represented by a pair (Z, gZ ), where Z ∈ CHr(X ) is a Chow cycle whose restriction to X we denote by Z and r−1,r−1 (1.6) gZ ∈ D (X)/im∂ + im∂ is a Green current for Z (see (2.1)) with appropriate growth at infinity. These groups admit an intersection product r s r+s (1.7) · : CHc (X ) × CHc (X ) → CHc (X )Q. r s 0 0 If (Z, gZ ) ∈ CHc (X ) and (Z , gZ0 ) ∈ CHc (X ) and Z,Z intersect properly in X, their r+s 0 0 0 product is given by (Z, gZ ) · (Z , gZ0 ) = (Z·Z , gZ ∗ gZ0 ), where Z·Z ∈ CHc (X )Q is the usual intersection and gZ ∗ gZ0 is the star product c ∗ (1.8) gZ ∗ gZ0 = gZ ∧ δZ0 + [ωZ ] ∧ gZ0 , ωZ := dd gZ + δZ ∈ A (XK ). r r There is a map CHc (X ) → CH (XK ) given by pullback to XK , and we refer to any cycle mapping to Z as a lift of Z. Kudla conjectures that, for certain such models X , there is a family of lifts ˆ r (1.9) Z(T, y, ϕf ) ∈ CHc (X ) ∨ r−rk(T ) r of Z(T, ϕf ) · (L ) ∈ CH (X), depending on an auxiliary parameter y (a real symmetric positive definite r-by-r matrix), such that arithmetic analogues of (1.3)- (1.5) hold. More precisely, the generating series ˆ(r) X ˆ T (1.10) θ (τ, ϕf ) := Z(T, y1, ϕf )q , y1 = Im(τ1), T should be a (non-holomorphic) Hilbert-Siegel modular form of genus r and weight r 1 2 dim(V ), valued in CHc (X ). We also require that the product formula

ˆ(r) ˆ(s) ˆ(r+s) τ1 0 (1.11) θ (τ1, ϕf,1) · θ (τ2, ϕf,2) = θ 0 τ2 , ϕf,1 ⊗ ϕf,2 1 holds. Finally, assume that XK is compact. Writingω ˆ ∈ CHc (X ) for the natural lift p+1 of the Hodge bundle L and degd : CHc (X ) → R for the arithmetic degree map, we ask that

X ˆ p+1−r T d (1.12) deg(d Z(T, y1, ϕf ) · ωˆ )q = Vol(XK , Ω) E(τ, Φ, s) 1 ds s= (p+1−r) T 2 for a certain Eisenstein series E(τ, Φ, s) depending on ϕf . 4 LUIS E. GARCIA, SIDDARTH SANKARAN

For Shimura curves all this is proved in [30]. There are also analogous conjectures for unitary Shimura varieties (see [19, 20]). For recent progress, see [7,8] and the references therein.

1.3. Main results. Let XK be an arbitrary orthogonal or unitary (for a ﬁxed CM extension of F ) Shimura variety as described in Section4; throughout the paper we refer to the orthogonal group case as case 1 and to the unitary case as case 2. In the latter case special cycles on XK are parametrized by pairs (T, ϕf ), where T is a r K hermitian r-by-r matrix with entries in E and ϕf ∈ S(V(Af ) ) . To treat both cases simultaneously, we write Xr(F ) = Symr(F ) in case 1 and Xr(F ) = Herr(E) in case 2. The sought after arithmetic Chow cycles are of the form ˆ (1.13) Z(T, y, ϕf ) = (Z(T, ϕf ), g(T, y, ϕf )),

and in this paper we focus on the current g(T, y, ϕf ). Assume first that T is non- singular. Then the irreducible components of the special cycle Z(T, ϕf ) admit a complex uniformization by certain submanifolds Dv in D parametrized by vectors r v = (v1, . . . , vr) ∈ V such that (Q(vi, vj))i,j = 2T . Using results of Bismut, Gillet and Soulé[4,2,3] we define, for each such v, a natural Green form

∗ (1.14) g(v) ∈ D (D)

for Dv. We give a characterization of g(v) `ala Bott-Chern in Theorem 2.1, showing that it is the unique such form that decreases rapidly with the distance to Dv. Using this property we deﬁne our Green form

∗ (1.15) g(T, y, ϕf ) ∈ D (XK ) as an inﬁnite sum of the forms g(v). Our main results, Theorem 4.4 and Theorem 5.1, show that the currents g(T, y, ϕf ) have the properties required by the conjectural identities (1.11) and (1.12). Concern- ing (1.11) we prove the following result.

Theorem 4.4. Assume that r + s ≤ dim(XK ) + 1 and let T1 ∈ Xr(F ) and T2 ∈ Xs(F ) be non-singular. If ϕf,i = ⊗pϕp,i (i = 1, 2) and ϕp,1 ⊗ ϕp,2 is supported on the r+s regular locus of V(Fp) for some ﬁnite place p of F , then Z(T1, ϕf,1) and Z(T2, ϕf,2) intersect properly and

X y1 g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2) = g(T, ( y2 ) , ϕf,1 ⊗ ϕf,2) T = T1 ∗ ∗ T2

∗ in D (XK )/(im∂ + im∂).

In Section5, we introduce a Siegel Eisenstein series E(τ, λ(ϕf ), s) of genus r. For non-singular T and ϕf = ⊗v ∞ϕv, its Fourier coeﬃcient ET (τ, λ(ϕf ), s) is given by an - Q Euler product ET (τ, λ(ϕf ), s) = v WT,v(τ, λv(ϕv), s), and hence its derivative can be GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 5 written as a sum d X (1.16) E (τ, λ(ϕ ), s) = E 0 (τ, λ(ϕ ), s) ds T f T f v v over all places of F . Since the arithmetic degree map degd can also be written as a similar sum, the conjectural identity (1.12) suggests comparing the terms at each place v. For v = σ1 we obtain the following result.

Theorem 5.1. Assume that XK is compact and attached to a group of the form O(p, 2) or U(p, 1) and that r ≤ p + 1. Then

r Z (−1) ικ0 p+1−r T 0 g(T, y1, ϕf ) ∧ Ω q = ET (τ, λ(ϕf ), s0)σ1 2Vol(XK , Ω) [XK ] if T is not totally positive deﬁnite and

r Z (−1) ικ0 p+1−r T g(T, y1, ϕf ) ∧ Ω q 2Vol(XK , Ω) [XK ] ι = E 0 (τ, λ(ϕ ), s ) − E (τ, λ(ϕ ), s ) log det σ (T ) T f 0 σ1 T f 0 2 1 if T is totally positive deﬁnite.

(See (3.49) and (5.19) for the values of ι and κ0). This result generalizes those in [29, 10] (the case r = 1), the main result in [23](r = 2 for Shimura curves) and also Liu [33] (unitary Shimura varieties and r = p + 1). It seems worthwhile to point out that, in contrast with these works, the proof of Theorem 5.1 does not proceed by computing both sides explicitly and comparing them (using often difficult changes of variables). Instead, our proof is conceptual and relies on understanding: the reducibility and K-types of principal series representations of certain Lie groups; the correspondence of spherical harmonics under the theta correspondence as described by Howe; results of Shimura giving estimates for Whittaker coefficients. Now consider the case where T is possibly singular. Assuming again that the underlying group is O(p, 2) or U(p, 1), in Section 4.4 we define a current g(T, y, ϕf ) satisfying

c r−rk(T ) (1.17) dd g(T, y, ϕf ) + δZ(T,ϕf ) ∧ (−Ω) = ω(T, y, ϕf ),

P T where T ω(T, y, ϕf )q is the Fourier expansion of ΘKM(τ, ϕf ). The construction involves regularizing the integral used when T is non-singular (of course, in that case this current specializes to the one discussed above). For example, when T is identically zero, we show that

r−1 ∗ (1.18) g(0r, y, ϕf ) = − log det y · ϕf (0) · (−Ω) ∈ D (XK )/(im∂ + im∂). 6 LUIS E. GARCIA, SIDDARTH SANKARAN

τ 0 0 More generally, if T = 0r−r0 with τ a non-singular r -by-r matrix and ϕf = 0 00 ϕf ⊗ ϕf , we show that

00 h 0 0 r−r0 g(T, y, ϕf ) = ϕf (0) g(τ, y , ϕf ) ∧ (−Ω) (1.19) 00 r−r0−1i ∗ − log(det y )δ 0 ∧ (−Ω) ∈ D (X )/im∂ + im∂. Z(τ,ϕf ) K

Thus g(T, y, ϕf ) generalizes the currents considered in [30, §6.4, 6.5] for Shimura curves. In forthcoming work, we will show that a result similar to Theorem 5.1 holds for these currents attached to singular T .

1.4. Notation and conventions. Let k be a ﬁeld endowed with a (possibly trivial) k k involution a 7→ a. We write Symr( ) (resp. Herr( )) for the group of symmetric (resp. hermitian) r-by-r matrices with coeﬃcients in k under matrix addition. For a ∈ GLr(k) and b ∈ Herr(k), let a 0 1r b 1r (1.20) m(a) = t −1 , n(b) = , wr = . 0 a¯ 0 1r −1r

r For x = (x1, . . . , xr) ∈ k , we write

x1  . (1.21) d(x) =  ..  . xr

× We fix the following standard choice of additive character ψ = ψF : F → C when 2πix F is a local field of characteristic zero. If F = R we set ψ(x) = e ; if F = Qp −1 −2πi/p we choose ψ = ψQp so that ψ(p ) = e ; if F is a finite extension of Qv we set × ψF (x) = ψQv (trF/Qv (x)). If F is a global field, we write AF for the ideles of F and set × × × ψF : F \AF → C , where ψF = ⊗vψFv and the product runs over all places v of F . We denote the connected component of the identity of a Lie group G by G0. We denote by A∗(X) (resp. D∗(X)) the space of differential forms (resp. currents) ∗ k on a smooth manifold X. Given α ∈ A (X) = ⊕k≥0A (X), we write α[k] for its component of degree k. If α is closed, we write [α] ∈ H∗(X) for the cohomology class defined by α. If X is a complex manifold, we let dc = (4πi)−1(∂ − ∂), so that ddc = (−2πi)−1∂∂. ∗ k,k −k We denote by the operator on ⊕k≥0A (X) acting by multiplication by (−2πi) on Ak,k(X). The canonical orientation on X induces an inclusion Ap,q(X) ⊂ Dp,q(X) sending a differential form ω to the current given by integration against ω on X. We will sometimes denote this current by [ω] (and will be careful to avoid confusion with the notation for cohomology class described above) but often will still denote it by ω. If f(s) is a meromorphic function of a complex variable s, we write CT f(s) for the s=0 constant term of its Laurent expansion at s = 0. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 7

2. Green forms on hermitian symmetric domains Let X be a complex manifold and Z ⊂ X be a closed irreducible analytic subset c−1,c−1 of codimension c. A Green current for Z is a current gZ ∈ D (X) such that c (2.1) dd gZ + δZ = [ωZ ], where δZ denotes the current of integration on Z and ωZ is a smooth differential form on X. A Green form is a Green current given by a form that is locally integrable on X and smooth on X − Z (cf. [6, §1.1]). Here we will construct Green forms for certain complex submanifolds of the hermitian symmetric space D attached to O(p, 2) or U(p, q), where p, q > 0. Throughout the paper we will refer to the case involving O(p, 2) as case 1 and to the case involving U(p, q) as case 2. We would like to emphasize that our methods apply uniformly and all results in both cases are closely parallel. There is a natural hermitian holomorphic vector bundle E over D, and the submanifolds of D that we consider are the zero loci Z(s) of certain natural holomorphic sections s ∈ H0(Er)(r ≥ 1). In this setting, the results in [4,2,3] (reviewed in Sec- tion 2.1) can be applied to construct some natural differential forms on D, including a Green form gZ(s) for Z(s), as we show in Section 2.2. We give explicit examples in Section 2.3, showing that this construction recovers some differential forms considered in previous work on special cycles (cf. [23, 24]), and then (Sections 2.4 and 2.5) we establish the main properties of these forms. The final Section 2.6 considers star products and will be used in the proof of Theorem 4.4.

2.1. Superconnections and characteristic forms of Koszul complexes. In this section we review the construction of some characteristic differential forms attached to a pair (E, u), where E is a holomorphic hermitian vector bundle and u is a holomorphic section of its dual. Most of the results in this section are due to Bismut [4] and Bismut-Gillet-Soulé[2,3]. The exceptions are Theorem 2.1, which characterizes the forms ξ0 that we will use later to construct Green forms for special cycles on Shimura varieties, and the results in Section 2.6 leading to the rotation invariance of star products proved in Corollary 2.10. We will use Quillen’s formalism of superconnections and related notions of super- algebra. For more details, we refer the reader to [36,1]. We briefly recall that a super vector space V is just a complex Z/2Z-graded vector space; we write V = V0 ⊕V1 and refer to V0 and V1 as the even and odd part of V respectively. We write τ for the endo- deg(v) morphism of V determined by τ(v) = (−1) v. The supertrace trs : End(V ) → C is the linear form defined by

(2.2) trs(u) = tr(τu), a b where tr denotes the usual trace. Thus if u = ( c d ) with a ∈ End(V0), d ∈ End(V1), b ∈ Hom(V1,V0) and c ∈ Hom(V0,V1), then trs(u) = tr(a) − tr(d).

2.1.1. Let E be a holomorphic vector bundle on a complex manifold X and u ∈ H0(E∨) be a holomorphic section of its dual E∨. Let K(u) be the Koszul complex 8 LUIS E. GARCIA, SIDDARTH SANKARAN

of u: its underlying vector bundle is the exterior algebra ∧E and its diﬀerential u : ∧kE → ∧k−1E is deﬁned by

X i+1 (2.3) u(e1 ∧ · · · ∧ ek) = (−1) u(ei)e1 ∧ · · · ∧ eˆi ∧ · · · ∧ ek. 1≤i≤k The grading on K(u) is given by K(u)−k = ∧kE, so that K(u) is supported in non-positive degrees. We identify K(u) with the corresponding complex of sheaves of sections of ∧kE, and say that u is a regular section if the cohomology of K(u) vanishes in negative degrees. If u is regular with zero locus Z(u), then K(u) is quasi-isomorphic to OZ(u) (regarded as a complex supported in degree zero).

2.1.2. Assume now that E is endowed with a hermitian metric k · kE. This induces k j a hermitian metric on ∧E: for any x ∈ X, ∧ Ex is orthogonal to ∧ Ex if j 6= k, k and an orthonormal basis of ∧ Ex is given by all elements ei1 ∧ · · · ∧ eik , where 1 ≤ i1 < ··· < ik ≤ rkE and {e1, . . . , erkE} is an orthonormal basis of Ex. Let ∇ be the corresponding Chern connection on ∧E. We regard ∧E as a super vector bundle, with even part ∧evenE and odd part ∧oddE, and u as an odd endomorphism of ∧E. Let u∗ be the adjoint of u and deﬁne the superconnection √ ∗ (2.4) ∇u = ∇ + i 2π(u + u ) on ∧E. Then we have Quillen’s Chern form

2 0 0 ∇u M k,k (2.5) ϕ (u) = ϕ (E, k · kE, u) = trs(e ) ∈ A (X). k≥0 We recall some properties of ϕ0(u) established (in greater generality) by Quillen [36]. The form ϕ0(u) is closed and functorial: given a holomorphic map of complex manifolds f : X0 → X, consider the pullback bundle (f ∗E, f ∗k · k) and the pullback section f ∗u ∈ H0(f ∗E∨); then (2.6) ϕ0(f ∗u) = f ∗ϕ0(u).

∗ k,k −k Let be the operator on ⊕k≥0A (X) acting by multiplication by (−2πi) on Ak,k(X). Writing [ϕ0(u)∗] for the cohomology class of ϕ0(u)∗, we have (2.7) [ϕ0(u)∗] = ch(∧E) = ch(∧evenE) − ch(∧oddE).

0 ∗ 0 In particular, [ϕ (u) ] depends on E, but not on u. Thus the forms ϕ (tu) for t ∈ R>0 all belong to the same cohomology class, but as t → +∞ they concentrate on the zero 2 locus of u. More precisely, recall that ∇u is an even element of the (super)algebra ∗ (2.8) A(X, End(∧E)) := A (X)⊗ˆ C∞(X)Γ(End(∧E)). Given a relatively compact open subset U ⊂ X whose closure U is disjoint from Z(u)

and a non-negative integer k, consider an algebra seminorm k·kU,E,k on A(X, End(∧E)) measuring uniform convergence on U of partial derivatives of order at most k. Let ∨ ∨ k · kE∨ be the unique metric on E such that the isomorphism E ' E induced by GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 9 k · kE is an isometry. Quillen [36, §4] shows that there exist positive real numbers C, a such that

2 2 ∇tu −2πat (2.9) ke kU,E,k ≤ Ce , t ∈ R>0.

Moreover, for a one may take any number strictly less than minx∈U {ku(x)kE∨ }. Thus a similar bound holds for ϕ0(tu).

2.1.3. Instead of rescaling u 7→ tu, we may also rescale the metric k · kE on E. −1 ∗ Replacing k · kE with s k · kE preserves√ the Chern connection ∇ but rescales u to 2 ∗ 2 ∗ s u and hence replaces ∇u with ∇ + i 2π(u + s u ). More generally, given s, t ≥ 1, consider the superconnection √ 2 ∗ (2.10) ∇s,tu := ∇ + i 2π(tu + ts u ).

2 We will also need an estimate for e∇s,tu for large values of s, t. To obtain it, write 2 2 2 ∇s,tu = (∇s,tu)[0] + Rs,tu, where (∇s,tu)[0] has form-degree zero and Rs,tu has form- degree ≥ 1. Note that Rs,tu is nilpotent and that

2 2 ∗ 2 2 2 2 0 (2.11) (∇s,tu)[0] = −2π(tu + ts u ) = −2πs t kuk ⊗ id ∈ A (X) ⊗ End(∧E).

2 In particular, (∇s,tu)[0] and Rs,tu commute. Hence we have

2 2 e∇s,tu = e(∇s,tu)[0] eRs,tu (2.12) N −2πs2t2kuk2 X 1 k = e k! Rs,tu k=0

with N ≤ dimR X. Let k · kU,E,k be a seminorm as in 2.1.2 and a ∈ R>0 strictly less than minx∈U {ku(x)kE∨ }. Since Rs,tu is a polynomial in s, t, it follows from (2.12) 2 2 2 ∇s,tu −2πas t n that we can ﬁnd n ∈ N such that ke kU,E,k = O(e (st) ) for s, t ≥ 1. Since (st)n = O(eεs2t2 ) for any ε > 0, we conclude that for any positive a strictly less than

minx∈U {ku(x)kE∨ }, there is a positive constant C such that

2 2 2 ∇s,tu −2πas t (2.13) ke kU,E,k ≤ Ce , s, t ≥ 1.

0 −1 0 −1 Thus a similar bound holds for ϕ (E, s k · kE, tu); we say that ϕ (E, s k · kE, tu) decreases rapidly as st → +∞ in the C∞ topology of A∗(U).

d 0 1/2 2.1.4. It follows from (2.7) that the form dt ϕ (t u) (for t ∈ R>0) is exact, and one can ask for a construction of a functorial transgression of this form. Bismut, Gillet and Soul´e[2] construct such a transgression, and the resulting form is key to our results. To deﬁne it, let N ∈ End(∧E) be the number operator acting on ∧kE by multiplication by −k and set

2 0 ∇u M k,k (2.14) ν (u) = trs(Ne ) ∈ A (X). k≥0 10 LUIS E. GARCIA, SIDDARTH SANKARAN

Then ν0(u) is functorial with respect to holomorphic maps f : X0 → X and satisfies ([2, Thm. 1.15]) 1 d (2.15) − ∂∂ν0(t1/2u) = ϕ0(t1/2u), t > 0. t dt 2.1.5. Assume that the section u has no zeroes on X. Then the Koszul complex K(u) is acyclic and (2.7) shows that ϕ0(u) is exact. In this case one can define a characteristic form ξ0(u) giving a ∂∂-transgression of ϕ0(u). Namely, let Z +∞ dt (2.16) ξ0(u) = ν0(t1/2u) . 1 t The bound (2.9) implies that all partial derivatives of ν0(t1/2u) decrease rapidly as k,k t → +∞. In particular, the integral converges and defines a form in ⊕k≥0A (X); moreover, one can differentiate under the integral sign, so that by (2.15) we have (2.17) ∂∂ξ0(u) = ϕ0(u). The following theorem characterizes the forms ξ0(u). Note that its statement and proof are very similar to [39, Thm. IV.2] on Bott-Chern forms. The difference is that the form ξ0(u) transgresses Quillen’s Chern form ϕ0(u) instead of the Chern-Weil ∇2 form trs(e ), and that instead of requiring that our transgression vanishes for split complexes we impose a rapid decrease condition. For our purposes these conditions are natural, as we want to define Green forms for cycles on arithmetic quotients Γ\D by averaging natural forms on D over translates by γ ∈ Γ and the rapid decrease of ξ0(u) and ϕ0(u) as u grows guarantees the convergence of these series. Given a relatively compact open subset U of X with closure U and a positive integer ∗ k, we denote by k · kU,k any seminorm on A (X) measuring uniform convergence on U of partial derivatives of order at most k. Theorem 2.1. For a hermitian vector bundle (E, k·k) on a complex manifold X and a nowhere vanishing section u ∈ H0(E∨), define Z +∞ dt ξ0(u) = ξ0(E, k · k, u) = ν0(t1/2u) . 1 t 0 k,k Then ξ (u) ∈ ⊕k≥0A (X) and satisfies the following properties: 0 ∇2 (i) (Transgression) ∂∂ξ (u) = trs(e u ). (ii) (Functoriality) For any holomorphic map f : X0 → X, we have ξ0(f ∗u) = f ∗ξ0(u).

(iii) For any seminorm k · kU,k, there are positive constants a, C such that

0 −1 −2πa(st)2 kξ (E, s k · kE, tu)kU,k ≤ Ce for all s, t ≥ 1. Moreover, if (E, k · k, u) 7→ ξ˜0(u) is another assignment as above satisfying (i)-(iii), then ξ0(u) − ξ˜0(u) ∈ im∂ + im∂; GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 11

∗ more precisely, we can ﬁnd forms ηi(u) ∈ A (X) (i = 1, 2) satisfying 0 ˜0 ξ (u) − ξ (u) = ∂η1(u) + ∂η2(u), ∗ ∗ 0 ηi(f u) = f ηi(u) for any holomorphic map f : X → X, ∞ ∗ ηi(tu) decrease rapidly as t → +∞ in the C topology of A (X). Proof. As remarked before the statement, estimate (2.9) shows that one can diﬀeren- tiate under the integral sign in (2.16), hence (i) follows from (2.15). Property (ii) is clear.

To prove (iii), let a ∈ R with 0 < a < minx∈U {ku(x)kE∨ }. For s, t ≥ 1, estimate (2.13) gives Z +∞ √ dt0 kξ0(E, s−1k · k , tu)k ≤ kν0(E, s−1k · k , t t0u)k E U,k E U,k t0 (2.18) 1 Z +∞ ≤ C e−2πa(st)2t0 dt0 = C(2πa(st)2)−1e−2πa(st)2 1 for some positive constant C, and (iii) follows. For the uniqueness statement, the proof follows that of [39, Thm. IV.2]. Namely, given a pair (E, u) on X and a point x ∈ X, denote the hermitian metric on Ex by k · kx. Let π : X × C → X be the canonical projection and consider the hermitian vector bundle E˜ on X × C given by E˜ = π∗E, with metric 2 −1 (2.19) kg · k(x,z) := (1 + |z| ) k · kx, (x ∈ X, z ∈ C). Letu ˜ = π∗u ∈ H0(E˜∨). The form ξ˜0(˜u) − ξ0(˜u) ∈ A∗(X × C×) is then ∂∂-closed by (i). Writing the diﬀerentials on X × C as ∂ = ∂X + ∂z and ∂ = ∂X + ∂z, we ﬁnd that Z 0 = ∂∂(ξ˜0(˜u) − ξ0(˜u)) log |z|2 P1 Z ˜0 0 2 = ∂z∂z(ξ (˜u) − ξ (˜u)) log |z| P1 Z (2.20) ˜0 0 2 + (∂X ∂ − ∂X ∂z)(ξ (˜u) − ξ (˜u)) log |z| P1 = −2πi(ξ˜0(u) − ξ0(u)) Z ˜0 0 2 + (∂X ∂ − ∂X ∂z)(ξ (˜u) − ξ (˜u)) log |z| P1 2 where the last equation follows from the current identity ∂z∂z log |z| = −2πi(δ0 −δ∞) 1 0 −1 ˜0 −1 on P and the rapid decrease of ξ (E, s k · kE, u) and ξ (E, s k · kE, u) as s → +∞. We set Z −1 ˜0 0 2 η1(u) = (−2πi) ∂(ξ (˜u) − ξ (˜u)) log |z| , P1 (2.21) Z −1 0 ˜0 2 η2(u) = (−2πi) ∂z(ξ (˜u) − ξ (˜u)) log |z| . P1 12 LUIS E. GARCIA, SIDDARTH SANKARAN

The integrals converge to smooth forms on X by (iii), which also shows that the ∗ ∗ forms ηi(tu) decrease rapidly as t → +∞. We have ηi(f u) = f ηi(u) for i = 1, 2 and any holomorphic map f : X0 → X, and (2.20) can be rewritten as 0 ˜0 (2.22) ξ (u) − ξ (u) = ∂η1(u) + ∂η2(u). This finishes the proof. We remark that similar forms ϕ0 and ξ0 can be defined for arbitrary acyclic complexes of holomorphic hermitian vector bundles, and that the above theorem generalizes to this setting; we will not need this more general version. 2.1.6. We now go back to the general setting where we have a hermitian holomorphic vector bundle E on X and a section u ∈ H0(E∨), and now make the assumption that u is regular (see 2.1.1) and the zero locus Z(u) is smooth; thus K(u) is a resolution of OZ(u) and the codimension of Z(u) (if non-empty) in X equals rk(E). Define 0 ∗ g(u) = ξ (u)[2rk(E)−2] (2.23) Z +∞ i rk(E)−1 0 1/2 dt rk(E)−1,rk(E)−1 = 2π ν (t u)[2rk(E)−2] ∈ A (X). 1 t The following proposition is contained in [3]. It shows that g(u) is a Green form for Z(u). Proposition 2.2. (1) The integral (2.23) converges to a smooth differential form g(u) on X − Z(u). (2) The form g(u) is locally integrable on X. (3) As currents on X we have c 0 ∗ dd g(u) + δZ(u) = ϕ (u)[2rk(E)]. Proof. Part (1) has already been discussed in Section 2.1.5. Part (2) is shown to hold in the course of the proof of [3, Thm. 3.3]. Namely, recall that in that paper one considers an immersion i : M 0 → M of complex manifolds, a vector bundle η on M 0 and a complex (ξ, v) of holomorphic hermitian vector bundles on M that gives a resolution of i∗OM 0 (η). Assume that Z(u) is non-empty. We set 0 M = X, M = Z(u), η = OZ(u), and for the complex (ξ, v) we take the Koszul complex 0 1/2 (∧E, u); with these choices, the form ν (t u) agrees with the form αt defined in [3, (3.12)]. Let z0 ∈ Z(u) and chose local coordinates z1, . . . , zN (N = dim X) around z0 such that Z(u) is defined by the equations z1 = ··· = zrk(E) = 0 and let |y| = 2 2 1/2 (|z1| + ··· + |zrk(E)| ) . Locally around z0 we can write

2rk(E)−2 0 X 0 p (2.24) ν (u)[2rk(E)−2] = ν (u) , p=0 0 p 0 where ν (u) corresponds to the part of ν (u)[2rk(E)−2] whose degree in the variables dzi, dzi (1 ≤ i ≤ rk(E)) is p. Since the complex normal bundle to Z(u) in X is GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 13 canonically identiﬁed with E, the equations [3, (3.24), (3.26)] show that, for any p, Z +∞ du (2.25) |y|2rk(E)−1 ν0(t1/2u)p 1 u −(2rk(E)−1) is bounded in a neighborhood of z0; part (2) follows since |y| is locally integrable around z0. To prove (3), let β be a compactly supported form on X. Then Z Z a Z c 0 1/2 ∗ c dt g(u) ∧ dd β = lim ν (t u)[2rk(E)−2] ∧ dd β X a→+∞ 1 X t Z a Z c 0 1/2 ∗ dt = lim dd ν (t u)[2rk(E)−2] ∧ β a→+∞ 1 X t (2.26) Z a Z d 0 1/2 ∗ dt = lim (−t ϕ (t u)[2rk(E)]) ∧ β a→+∞ 1 X dt t Z Z 0 ∗ 0 1/2 ∗ = ϕ (u)[2rk(E)] ∧ β − lim ϕ (a u)[2rk(E)] ∧ β. X a→+∞ X Here the ﬁrst equality follows from dominated convergence since the integrals a du (2.27) |y|2rk(E)−1 ∫ ν0(t1/2u)p 1 u

for a ≥ 1 are uniformly bounded in a neighborhood of z0, as shown in the proof of [3, Thm. 3.3]. The third equality follows from (2.15). This establishes (3) since 0 1/2 ∗ ϕ (a u)[2rk(E)] approaches δZ(u) as a → +∞, as shown by [3, (1.14), (1.19)]. 2.2. Hermitian symmetric domains of orthogonal and unitary groups.

2.2.1. Let case 1 (2.28) k = R C case 2 and id case 1 (2.29) σ = complex conjugation case 2. Let m ≥ 2 be a positive integer and let V be a k-vector space of dimension m endowed with a non-degenerate σ-Hermitian bilinear form Q of signature (p, q). We assume that pq 6= 0 and in case 1 we further assume that q = 2. Let U(V ) := Aut(V,Q) be the isometry group of Q and let SO(p, 2)0 case 1 (2.30) G = U(V )0 ∼= U(p, q) case 2.

+ − + − We fix an orthogonal decomposition V = V ⊕V with dimk V = p and dimk V = q, and with V + and V − positive and negative definite respectively; in case 1 we also fix an orientation of V −. Let K be the centralizer in G of the isometry of V acting 14 LUIS E. GARCIA, SIDDARTH SANKARAN

as the identity on V + and as −1 on V −. Then K is a maximal compact subgroup of G, given by SO(V +) × SO(V −) case 1 (2.31) K = U(V +) × U(V −) case 2. Let Gr(q, V ) be the Grassmannian of q-dimensional subspaces of V ; in case 1 we assume furthermore that these subspaces are oriented, so that in this case Gr(q, V ) consists of two copies of the usual Grassmannian. Let

(2.32) D = {z ∈ Gr(q, V )| Q|z < 0} be the open subset of Gr(q, V ) consisting of negative definite subspaces. Then D has two connected components in case 1 and is connected in case 2. Let z0 ∈ D be the point corresponding to V − and D+ be the connected component of D containing + + z0. Then G acts transitively on D and the stabilizer of z0 is K; thus D ' G/K is the symmetric domain associated with G. In case 2 it is clear that D carries a U(V )-invariant complex structure; to see this in case 1, one can use the model (2.33) D '{[v] ∈ P(V (C))|Q(v, v) = 0,Q(v, v) < 0}. 0 The correspondence between both models sends z ∈ D to the line [ez +iez] ∈ P(V (C)), 0 0 0 where ez and ez form an oriented orthogonal basis of z satisfying Q(ez, ez) = Q(ez, ez). 2.2.2. Let E be the tautological bundle on D, whose fiber over z ∈ D is the subspace z ⊂ V . Thus E is a holomorphic line bundle in case 1 (for it corresponds to the pullback of OP(V (C))(−1) under the isomorphism (2.33)), and a holomorphic vector bundle of rank q in case 2. It carries a natural hermitian metric hE defined by −Q(vz, vz) case 1 (2.34) hE(vz) = −Q(vz, vz) case 2

for vz ∈ Ez = z. This metric is equivariant for the natural U(V )-equivariant structure on E. We denote by ∇E the corresponding Chern connection on E and by top ∗ i rkE 2 rkE,rkE (2.35) Ω = ΩE := c (E, ∇E) = 2π det(∇E) ∈ A (D) its Chern-Weil form of top degree.

We denote by ΩD the K¨ahlerform

(2.36) ΩD = ∂∂ log kD(z, z),

where kD is the Bergmann kernel function of D. As shown in [42, p. 219], the invariant i top top form − 2π ΩD agrees with the first Chern form c1(ΩX ) of the canonical bundle ΩX on any quotient X = Γ\D+ by a discrete torsion free subgroup Γ ⊂ G. When E has rank one, the canonical bundle on D is naturally isomorphic to E⊗p in case 1 (as an application of the adjunction formula shows) and to E⊗(p+1) in case 2, and so in both i cases − 2π ΩD is a positive integral multiple of ΩE. ∨ 0 An element v ∈ V defines a global holomorphic section sv of E : for vz ∈ Ez, we define 0 0 (2.37) sv(vz) = Q(vz, v). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 15

+ + Let Dv be the zero locus of sv on D and set Dv = Dv ∩ D . We have

(2.38) Dv = {z ∈ D|v ⊥ z}

and so Dv is non-empty only if Q(v, v) > 0 or v = 0. Assume that Q(v, v) > 0, ⊥ so that the orthogonal complement v of v has signature (p − 1, q). Writing Gv 0 ⊥ 0 + for the stabilizer of v in G, we ﬁnd that Gv ' Aut(v ,Q) acts transitively on Dv with stabilizers isomorphic to SO(p − 1) × SO(q) in case 1 and to U(p − 1) × U(q) + 0 in case 2. Thus Dv is the symmetric domain attached to Gv. We conclude that ∨ ∨ codimDDv = rkE and hence that sv is a regular section of E . In fact, this shows that sv is regular whenever v 6= 0, since in the remaining case we have Q(v, v) ≤ 0 and so sv does not vanish on D. r More generally, given a positive integer r and a vector v = (v1, . . . , vr) ∈ V , there ∨ r is a holomorphic section sv = (sv1 , . . . , svr ) of (E ) , with zero locus

(2.39) Dv := Z(sv) = ∩1≤i≤rDvi .

Note that Dv depends only on the span hv1, . . . , vri. It is non-empty if and only if hv1, . . . , vri is a a positive deﬁnite subpace of (V,Q) of positive dimension; in that + + case, its (complex) codimension in D is rk(E)·dimkhv1, . . . , vri. We set Dv = Dv ∩D .

2.2.3. We will now specialize the constructions in Section 2.1 to the setting of hermitian symmetric domains. Namely, recall the deﬁnitions of (V,Q), D, E and 0 ∨ sv ∈ H (E ) in 2.2.1 and 2.2.2. Let r be a positive integer and v = (v1, . . . , vr) be an r-tuple of vectors in V . We write K(v) := K(sv) for the Koszul complex associated with the section sv = r ∨ r (sv1 , . . . , svr ) of (E ) . On its underlying vector bundle ∧(E ), we consider the superconnection √ ∗ (2.40) ∇v := ∇sv = ∇ + i 2π(sv + sv), and we deﬁne forms

k 2 0 0 ∗ X i ∇v ϕ (v) := ϕ (sv) = 2π trs(e )[2k], k≥0 (2.41) k 2 0 0 ∗ X i ∇v ν (v) := ν (sv) = 2π trs(Ne )[2k], k≥0

where N is the number operator on ∧(Er) acting on ∧k(Er) by multiplication by −k.

r Deﬁnition 2.3. For v = (v1, . . . , vr) ∈ V , write Q(v, v) = Q(v1, v1)+···+Q(vr, vr) and deﬁne

ϕ(v) = e−πQ(v,v)ϕ0(v), ν(v) = e−πQ(v,v)ν0(v). 16 LUIS E. GARCIA, SIDDARTH SANKARAN

k,k r ∨ Thus ϕ(v) and ν(v) belong to ⊕k≥0A (D). If sv is a regular section of (E ) , define Z +∞ dt M ξ0(v) = ν0(t1/2v) ∈ Ak,k( − ), t D Dv 1 k≥0 0 g(v) = ξ (v)[2r·rk(E)−2], ξ(v) = e−πQ(v,v)ξ0(v). The forms ϕ(v) and ν(v) were already defined and studied in the setting of general period domains in [11]. As to the form g(v), note that the section sv is regular if dimkhv1, . . . , vri = r; in this case we say that v is non-degenerate. For non-degenerate v, Proposition 2.2 shows that g(v) is smooth on D − Dv, locally integrable on D and, as a current, satisfies c 0 (2.42) dd g(v) + δDv = ϕ (v)[2r·rk(E)],

so that g(v) is a Green form for Dv. 2.3. Explicit formulas for O(p, 2) and U(p, 1). Let us give some explicit formulas when the tautological bundle E is a line bundle. Thus V is a either a real vector space of signature (p, 2) (case 1) or a complex vector space of signature (p, 1) (case 2). There is a unique hermitian metric on E∨ making the isomorphism E ∼= E∨ induced by hE an isometry; we denote this metric by h and its Chern connection by ∇E∨ . For v ∈ V we have −π(Q(v,v)+2h(sv)) ∂h(sv) ∧ ∂h(sv) ϕ(v)[2] = e i − ΩE , (2.43) h(sv) −1 ∨ ∗ ϕ(v) = ϕ(v)[2] ∧ Td (E , ∇) , where 2 ! −∇ ∨ −1 ∨ 1 − e E (2.44) Td (E , ∇) = det 2 ∇E∨

∨ denotes the inverse Todd form of (E , ∇E∨ ). This is a special case of the Mathai- Quillen formula [34, Thm. 8.5]; see also [11, §3] for a proof in our setting, where it is also shown that the form ϕ(v)[2] coincides with the form ϕKM(v) deﬁned by Kudla and Millson in [24]. Let us now consider the form ν(v) for v ∈ V . Here the Koszul complex K(v) has sv just two terms: K(v) = (E −→OD), and the operator N acts by zero on OD and by −1 on E. For the component of degree zero of ν0(v) we obtain

2 0 ∇v ν (v)[0] = trs(Ne )[0] 2 (∇v)[0] = trs(Ne ) (2.45) −2πh(sv) = trs(Ne ) = e−2πh(sv). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 17

Given z ∈ D, let z⊥ be the orthogonal complement of z in V , so that V = z ⊕ z⊥; we ⊥ write vz and vz⊥ for the orthogonal projection of v ∈ V to z and z respectively. Let Qz be the (positive deﬁnite) Siegel majorant of Q deﬁned by

(2.46) Qz(v, v) = Q(vz⊥ , vz⊥ ) − Q(vz, vz).

Then we have Qz(v, v) = Q(v, v) + 2hz(sv) and we conclude that ν(v)[0] is just the Siegel gaussian:

SG −πQz(v,v) (2.47) ν(v)[0] = ϕ (v) := e . We also compute that, for v 6= 0, Z +∞ 0 0 1/2 dt ξ (v)[0] = ν (t v)[0] 1 t (2.48) Z +∞ dt = e−2πth(sv) 1 t = −Ei(−2πh(sv)), ∞ −zt dt 0 where Ei(−z) = − ∫1 e t denotes the exponential integral. Thus ξ (v)[0] coincides with the Green form defined in [23, (11.24)] in case 1 when p = 1. 2.4. Basic properties of the forms ϕ and ν. 2.4.1. We will first give formulas for the restriction of ϕ and ν to special cycles. Let + + w ∈ V with Q(w, w) > 0 and recall the group Gw and complex submanifold Dw ⊂ D ⊥ defined in 2.2.2. Then Gw is identified with the isometry group of (w ,Q), and we + 0 may identify Dw with the hermitian symmetric domain attached to Gw. We write 0 0 ϕDw (v ) and νDw (v ) for the forms on Dw given in Definition 2.3. Lemma 2.4. Let v, w ∈ V with Q(w, w) > 0 and write v = v0 + v00 with v0 ∈ w⊥ and v00 ∈ hwi. Then −πQ(v00,v00) 0 ν(v)|Dw = e νDw (v ), −πQ(v00,v00) 0 ϕ(v)|Dw = e ϕDw (v ). ⊥ Proof. Let Ew be the tautological bundle on Dw whose fiber over z ∈ Dw is z ⊂ w . The restriction of E to Dw is isometric to Ew. For any v ∈ V , this isometry induces ∼ 0 0 an isomorphism K(v)|Dw = K(v ), where K(v ) denotes the Koszul complex with 0 0 underlying vector bundle ∧Ew and differential sv . Thus ∇v|Dw = ∇v and the lemma follows. 2.4.2. The proof of the next proposition is a straightforward consequence of general properties of Koszul complexes and Chern forms. r Proposition 2.5. Let r ≥ 1 and v = (v1, . . . , vr) ∈ V . Then:

(a) ϕ(v) = ϕ(v1) ∧ ... ∧ ϕ(vr). (b) ϕ(v) is closed. (c) ϕ(v)[k] = 0 if k < 2r · rk(E). ∗ (d) For every g ∈ U(V ), we have g ϕ(gv1, . . . , gvr) = ϕ(v1, . . . , vr). 18 LUIS E. GARCIA, SIDDARTH SANKARAN

(e) ϕ(0) = ctop(E∨, ∇)∗ ∧ Td−1(E∨, ∇)∗ (here we assume r = 1). (f) Let O(r), case 1 h ∈ U(r), case 2.

Then ϕ((v1, . . . , vr) · h) = ϕ(v1, . . . , vr). Proof. Except for (c), which follows from (a) and the Mathai-Quillen formula [34, Thm. 8.5], all statements are proved in [11, Prop. 2.3 and (2.17)] in case 1, and the proof there extends without modiﬁcation to case 2. Consider now the form ν. We write

2 rk(F ) (2.49) c(F, ∇F ) = det(t∇F + 1rk(F )) = 1 + c1(F, ∇F )t + ... + crk(F )(F, ∇F )t

for the total Chern-Weil form of a vector bundle F with connection ∇F .

r Proposition 2.6. Let r ≥ 1 and v = (v1, . . . , vr) ∈ V . P (a) We have ν(v) = 1≤i≤r νi(v), where

νi(v) := ν(vi) ∧ ϕ(v1,..., vˆi, . . . , vr). (b) For t > 0: d ddcν0(t1/2v) = −t ϕ0(t1/2v). dt

(c) ν(v)[k] = 0 if k < 2r · rk(E) − 2. ∗ (d) For any g ∈ U(V ), we have g ν(gv1, . . . , gvr) = ν(v1, . . . , vr). (e) Assume that r = 1. Then

∨ ∗ ν(0)[2rk(E)−2] = crk(E)−1(E , ∇) .

(In particular, ν(0)[0] = 1 when E has rank one.) (f) Let O(r), case 1 h ∈ U(r), case 2.

Then ν((v1, . . . , vr) · h) = ν(v1, . . . , vr).

−πQ(v,v) ∇2 Proof. Recall that ν(v) = e trs(Ne v ), where N is the number operator on the Koszul complex K(v) ' ⊗1≤i≤rK(vi). Letting Ni be the number operator on K(v ), we can write N = N + ··· + N and ∇2 = ∇2 + ··· + ∇2 , where [N , ∇2 ] = i 1 r v v1 vr i vj [∇2 , ∇2 ] = 0 for i 6= j; this proves (a). vi vj Part (b) follows from (2.15). By part (a) and Proposition 2.5.(c), it suﬃces to prove (c) when r = 1. Then (c) is vacuously true if rk(E) = 1, and in general it follows from [5, (3.72),(3.35), Thm. 3.10]. For any v and any g ∈ U(V ), the U(V )-equivariant structure on E induces an isomorphism g∗K(gv) ' K(v) preserving the metric; this proves (d). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 19

Part (e) follows from the explicit computation (2.47) when E has rank one. In the general case, by taking u = 0 in [5, (3.35)] we ﬁnd that

∇2 ∗ d i 2 trs(Ne ) = − det ∇ ∨ − b1rk(E) [2rk(E)−2] db 2π E b=0 r−1 d i 2 = (−1) det ∇ + b1rk(E) (2.50) db 2π E b=0 r−1 ∗ = (−1) crk(E)−1(E, ∇) ∨ ∗ = crk(E)−1(E , ∇) (note that the number operator N in op. cit. has sign opposite from ours). To prove (f), note that h induces an isometry

i(h) (2.51) K(v1, . . . , vr) −−→ K((v1, . . . , vr) · h) −1 that commutes with N and such that ∇(v1,...,vr)·h = i(h) ∇(v1,...,vr)i(h). Thus (f) follows from the conjugation invariance of trs. The properties of ϕ(v) and ν(v) established in Propositions 2.5 and 2.6 also hold for ϕ0(v) and ν0(v). In contrast, the next result, showing that the forms ϕ and ν are rapidly decreasing as functions of v, really requires the additional factor e−πQ(v,v) to 0 hold (note for example that the restriction of ϕ (tv) to Dv is independent of t ∈ R). Let S(V r) be the Schwartz space of complex-valued functions on V r all whose derivatives are rapidly decreasing.

Lemma 2.7. For ﬁxed z ∈ D and r ≥ 1, we have r ∗ ϕ(·, z), ν(·, z) ∈ S(V ) ⊗ ∧Tz D. Proof. By Proposition 2.5.(a) and Proposition 2.6.(a), we may assume that r = 1. 1 Recall that the quadratic form Qz(v) = 2 Q(v, v) + hz(sv) on V is positive deﬁnite. 2 2 Write ∇v(z) = (∇v)[0](z) + S(v, z). By Duhamel’s formula ([39, p. 144]) we have (2.52) 2 e−πQ(v,v)e∇v(z) = e−2πQz(v) Z X k −2π(1−tk)Qz(v) −2π(tk−tk−1)Qz(v) −2πt1Qz(v) + (−1) e S(v, z)e ··· S(v, z)e dt1 ··· dtk. k k≥1 ∆

k k Here ∆ = {(t1, . . . , tk) ∈ R |0 ≤ t1 ≤ ... ≤ tk ≤ 1} is the k-simplex and the sum is ﬁnite since S(v, z) has positive degree. Let k · kU,k be an algebra seminorm as in 2.1.2 and let QU a positive deﬁnite quadratic form on V such that QU < Qz for all z ∈ U. Then

−2πQz(v) −2πQU (v) (2.53) ke kU,k < Ce , v ∈ V, for some positive constant C. Since S(v, z) grows linearly with v,(2.52) implies that

2 −πQ(v,v) ∇v(z) −2πQU (v) (2.54) ke e kU,k < Ce , v ∈ V, 20 LUIS E. GARCIA, SIDDARTH SANKARAN

(with diﬀerent C) and hence that the same bound holds for kφ(v)kU,k where φ(v) is any derivative of ϕ or ν in the v variable.

2.5. Behaviour under the Weil representation.

2.5.1. Let r be a positive integer. We write 0 and 1r for the identically zero and r identity r-by-r matrices respectively. Consider the vector space Wr := k endowed with the σ-skew-Hermitian form determined by 0 1 (2.55) r . −1r 0 Its isometry group is the symplectic group

0 1r t 0 1r (2.56) Sp2r(R) = g ∈ GL2r(R) g −1r 0 g = −1r 0 in case 1 and the quasi-split unitary group

0 1r t 0 1r (2.57) U(r, r) = g ∈ GL2r(C) g −1r 0 g = −1r 0 in case 2. We denote by Mp2r(R) the metaplectic double cover of Sp2r(R), and identify it (as a set) with Sp2r(R) × {±1} as in [37]. Deﬁne Mp ( ), case 1 (2.58) G0 = 2r R r U(r, r), case 2.

0 Let Nr and Mr be the subgroups of Gr given by

(2.59) Nr = {(n(b), 1)|b ∈ Symr(R)} (2.60) Mr = {(m(a), ) |a ∈ GLr(R), = ±1} in case 1 and by

Nr = {n(b)|b ∈ Herr} , (2.61) Mr = {m(a)|a ∈ GLr(C)} 0 0 in case 2. We also ﬁx a maximal compact subgroup Kr of Gr as follows. In case 1 0 we let Kr be the inverse image under the metaplectic cover of the standard maximal compact subgroup a −b (2.62) a + ib ∈ U(r) ∼= U(r) b a 0 0 0 of Sp2r(R). In case 2 we deﬁne Kr = Gr ∩U(2r). Thus in this case Kr ' U(r)×U(r); an explicit isomorphism is

' 0 U(r) × U(r) −→ Kr (2.63) 1 k1 + k2 −ik1 + ik2 (k1, k2) 7→ [k1, k2] := . 2 ik1 − ik2 k1 + k2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 21

× m 2.5.2. In case 2, we ﬁx a character χ = χV of C such that χ|R× = sgn(·) . Then 0 r Gr × U(V ) acts on S(V ) via the Weil representation ω , case 1 (2.64) ω = ψ ωψ,χ, case 2 (see [14, §1]). Here the action of U(V ) is in both cases given by (2.65) ω(g)φ(v) = φ(g−1v), g ∈ U(V ), φ ∈ S(V r). 0 To describe the action of Gr, we write (m(a), 1), for a ∈ GL ( ) in case 1, m(a) = r R m(a), for a ∈ GLr(C) in case 2, (n(b), 1), for b ∈ Sym ( ) in case 1, (2.66) n(b) = r R n(b), for b ∈ Herr in case 2, (wr, 1), case 1, wr = wr, case 2. k We write | · |k for the normalized absolute value on ; thus |z|R = |z| and |z|C = zz. Then we have (see [21] and also [16, §4.2] for explicit formulas for χψ and γV r ): m/2 ω(m(a))φ(v) = | det a|k φ(v · a) χ (det a) case 1 · ψ (2.67) χ(det a) case 2, ω(n(b))φ(v) = ψ(tr(bT (v)))φ(v), ˆ ω(wr)φ(v) = γV r φ(v). 1 ˆ Here for v = (v1, . . . , vr) we deﬁne T (v) = 2 (Q(vi, vj)) and φ(v) denotes the Fourier transform Z ˆ 1 (2.68) φ(v) = φ(w)ψ( 2 trC/Rtr(Q(v, w)))dw, V r where dw is the self-dual Haar measure on V r with respect to ψ.

0 2.5.3. For our purposes it is crucial to understand the action of Kr on the Schwartz forms ϕ and ν. This was done for the form ϕ in [24], where it is shown that ϕ 0 spans a one-dimensional representation of Kr. We will show that ν also generates an 0 irreducible representation of Kr. To describe it we next recall the parametrization of 0 irreducible representations of Kr by highest weights; we denote by πλ the (unique up to isomorphism) representation with highest weight λ. 0 Consider ﬁrst case 1. Then Kr is a double cover of U(r) and its irreducible representations are πλ with r 1 r (2.69) λ = (l1, . . . , lr) ∈ Z ∪ ( 2 + Z) , l1 ≥ · · · ≥ lr. k/2 0 For an integer k, we write det for the character of Kr whose square factors through U(r) and deﬁnes the k-th power of the usual determinant character det : U(r) → C×. 22 LUIS E. GARCIA, SIDDARTH SANKARAN

Kudla and Millson show that 0 0 m/2 0 0 (2.70) ω(k )ϕ = det(k ) ϕ, k ∈ Kr, 0 and so ϕ aﬀords the one-dimensional representation πl of Kr with highest weight m (2.71) l := 2 (1,..., 1). Now consider case 2. Using the isomorphism (2.63) we can write uniquely (up 0 to isomorphism) any irreducible representation of Kr as πλ1 πλ2 for a unique pair λ = (λ1, λ2) of dominant weights of U(r). Let k(χ) be the unique integer such that χ(z) = (z/|z|)k(χ), and note that k(χ) and m have the same parity. Kudla and Millson show that (m+k(χ))/2 (−m+k(χ))/2 (2.72) ω([k1, k2])ϕ = (det k1) (det k2) ϕ, k1, k2 ∈ U(r), 0 and so ϕ aﬀords the one-dimensional representation πl of Kr with highest weight m+k(χ) −m+k(χ) (2.73) l := ( 2 (1,..., 1), 2 (1,..., 1)). 0 We will now determine the Kr representation generated by the Schwartz form ν(v)[2r−2]. Recall that by Proposition 2.6 we can write X (2.74) ν(v) = νi(v), 1≤i≤r where

2 −πQ(vi,vi) ∇v (2.75) νi(v) = e trs(Nie i ) ∧ ϕ(v1,..., vˆi, . . . , vr).

Let r = 1r and, for 1 ≤ i < r, set   1i−1  0 1 (2.76) i =   ∈ SO(r).  1r−i−1  −1 0 Then

(2.77) νi(v) = ω(m(i))νr(v), 1 ≤ i ≤ r; 0 thus ν(v)[2r−2] belongs to the Kr representation generated by νr(v)[2r−2]. 0 Lemma 2.8. Let r ≥ 1 and assume that rk(E) = 1. Under the action of Kr via the

Weil representation, the form ν(v)[2r−2] generates an irreducible representation πλ0 with highest weight m m m λ0 := ( 2 ,..., 2 , 2 − 2) in case 1 and m+k(χ) m+k(χ) m+k(χ) −m+k(χ) −m+k(χ) −m+k(χ) λ0 := (( 2 ,..., 2 , 2 − 1), ( 2 ,..., 2 , 2 + 1))

in case 2. In both cases the form νr(v)[2r−2] is a highest weight vector in πλ0 . GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 23

Proof. Assume ﬁrst that r = 1. By (2.47), ν(z)[0] is the Siegel gaussian, which has weight

p−q m (2.78) λ0 = 2 = 2 − 2 in case 1 and

p−q+k(χ) q−p+k(χ) m+k(χ) −m+k(χ) (2.79) λ0 = ( 2 , 2 ) = ( 2 − 1, 2 + 1) in case 2. Now assume that r > 1. By Proposition 2.5.(c), we have

(2.80) νr(v)[2r−2] = ν(vr)[0] · ϕ(v1, . . . , vr−1)[2r−2]

and hence by (2.70), (2.72), (2.78) and (2.79), the form νr(v)[2r−2] has weight λ0. To show that νr(v)[2r − 2] is a highest weight vector, one needs to check that + ω(α)νr(v)[2r−2] = 0 for every compact positive root α ∈ ∆c (see (3.5) and (3.28)). This can be done by a direct computation using (2.67) and the explicit formulas (2.47) and (2.43) for ν[0] and ϕ, or alternatively as follows. Evaluating at z0,(2.47) and (2.43) show that

r ∗ K (2.81) νr(v, z0) ∈ (S(V ) ⊗ ∧p ) ;

r r −πQ (v,v) here S(V ) ⊂ S(V ) is the subspace spanned by functions of the form e z0 p(v), r where p(v) is a polynomial on V . For νr(v, z0), the degree of these polynomials (called the Howe degree) is 2r − 2. Now it follows from the formulas in [18, §III.6] 0 (see also [13, Prop. 4.2.1] in case 1) that the only Kr representation containing the r weight λ0 realized in S(V ) in Howe degree 2r−2 is πλ0 , and the statement follows.

2.6. Star products. Let k, l be positive integers with k + l ≤ dim(D) + 1. Given a k 0 non-degenerate k-tuple v = (v1, . . . , vk) ∈ V , we have defined a form ξ (v) such that 0 g(v) = ξ (v)[2k−2] is a Green form for the cycle Dv (see Definition 2.3 and (2.42)). 0 0 0 We now consider star products of these currents. Fix tuples v = (v1, . . . , vk) and 00 00 00 l v = (v1 , . . . , vl ) ∈ V such that the tuple 0 0 00 00 k+l (2.82) v = (v1, . . . , vk+l) := (v1, . . . , vk, v1 , . . . , vl ) ∈ V is non-degenerate (note that this implies that v0 and v00 are both non-degenerate). We define

0 00 0 0 0 00 2(k+l)rk(E)−2 (2.83) g(v ) ∗ g(v ) = g(v ) ∧ δDv00 + ϕ (v )[2rk(E)k] ∧ g(v ) ∈ D (D).

Theorem 2.9. There exist currents α(v0, v00) ∈ D(k+l)rk(E)−2,(k+l)rk(E)−1(D) and β(v0, v00) ∈ (k+l)rk(E)−1,(k+l)rk(E)−2 0 00 0 00 0 00 0 00 D (D) such that g∗α(v , v ) = α(gv , gv ), g∗β(v , v ) = β(gv , gv ) for g ∈ G and satisfying g(v0) ∗ g(v00) − g(v) = ∂α(v0, v00) + ∂β(v0, v00).

k+l Proof. We abbreviate r = rk(E). Let π : D × R>0 → D be the canonical projection k+l and denote by t1, . . . , tk+l the coordinates on R>0 . Consider the super vector bundle 24 LUIS E. GARCIA, SIDDARTH SANKARAN

∗ k+l k+l π ∧(E ) on D × R>0 , endowed with the superconnection ∇t1v1,...,tk+lvk+l . Consider 0 ∗ k+l the formν ˜ (v) ∈ A (D × R>0 ) deﬁned by (2.84) 0 X 0 1/2 0 1/2 [1/2 1/2 dti ν˜ (v) = ν (ti vi)[2r−2] ∧ ϕ (t1 v1,..., ti vi, . . . , tk+lvk+l)[2(k+l−1)r] ∧ . ti 1≤i≤k+l

k+l For a piecewise smooth path γ : I → R>0 (I ⊂ R a closed interval) and α ∈ ∗ k+l A (D × R>0 ), let Z ∗−1 (2.85) α ∈ A (D) γ be the form obtained by integrating (id × γ)∗α along the ﬁbers of the projection of D × I → D. Fix a real number M > 1 and consider the paths

γd,M (t) = (t, . . . , t), 0 00 γk,M (t) = (1,..., 1, t, . . . , t), γk,M (t) = (t, . . . , t,M,...,M), (2.86) | {z } | {z } k k 0 00 γk,M = γk,M t γk,M , with t ∈ [1,M]. Let

k+l (2.87) ∆k,M = {(t1, . . . , t1, t2, . . . , t2)|1 ≤ t1 ≤ t2 ≤ M} ⊂ R>0 , | {z } k oriented so that ∂∆k,M = γd,M − γk,M . By Proposition 2.6.(a), we have Z Z M 0 0 1/2 dt (2.88) ν˜ (v) = ν (t v)[2(k+l)r−2] ; γd,M 1 t this approaches g(v) as M → ∞. Next, note that

0 ∗ 0 0 1/2 00 0 0 dt (id × γk,M ) ν˜ (v) = ν (t v )[2lr−2] ∧ ϕ (v )[2kr] ∧ , (2.89) t dt (id × γ00 )∗ν˜0(v) = ν0(t1/2v0) ∧ ϕ0(M 1/2v00) ∧ . k,M [2kr−2] [2lr] t 0 1/2 00 By [4, Thm. 3.2], as M → ∞ we have ϕ (M v )[2lr] → δDv00 as currents, and hence Z (2.90) lim ν˜0(v) = g(v0) ∗ g(v00). M→∞ γk,M

k+l Let d = d1 + d2 be the differential on D × R>0 , where d1 = ∂ + ∂ is the differential p+1 on D and d2 is the differential on R>0 . Then we have Z Z Z 0 0 0 (2.91) ν˜ (v) − ν˜ (v) = d2ν˜ (v). γd,M γk,M ∆k,M GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 25

0 The restriction ofν ˜ (v) to D × ∆k,M is dt ν˜0(v)| = ν0(t1/2v0) ∧ ϕ0(t1/2v00) ∧ 1 D×∆k,M 1 [2kr−2] 2 [2lr] t (2.92) 1 0 1/2 0 0 1/2 00 dt2 + ϕ (t1 v )[2kr] ∧ ν (t2 v )[2lr−2] . t2 Applying Proposition 2.6.(b) we obtain 0 d2ν˜ (v)|D×∆k,M d 0 1/2 0 0 1/2 00 = t1 ϕ (t1 v )[2kr] ∧ ν (t2 v )[2lr−2] dt1 0 1/2 0 d 0 1/2 00 dt1dt2 −ν (t1 v )[2kr−2] ∧ t2 ϕ (t2 v )[2lr] ∧ (2.93) dt2 t1t2 −1 0 1/2 0 0 1/2 00 = (−2πi) −(∂∂ν (t1 v ))[2kr] ∧ ν (t2 v )[2lr−2]

0 1/2 0 0 1/2 00 dt1dt2 +ν (t1 v )[2kr−2] ∧ (∂∂ν (t2 v ))[2lr] ∧ t1t2 0 00 0 00 = ∂α(t1, t2, v , v ) + ∂β(t1, t2, v , v ), with dt dt α(t , t , v0, v00) = (−2πi)−1ν0(t1/2v0) ∧ ∂(ν0(t1/2v00) ) ∧ 1 2 , 1 2 1 [2kr−2] 2 [2lr−2] t t (2.94) 1 2 0 00 −1 0 1/2 0 0 1/2 00 dt1dt2 β(t1, t2, v , v ) = (−2πi) ∂(ν (t1 v )[2kr−2]) ∧ ν (t2 v )[2lr−2] ∧ . t1t2 We set Z 0 00 0 00 α(v , v ) = α(t1, t2, v , v ), ∆k,+∞ (2.95) Z 0 00 0 00 β(v , v ) = β(t1, t2, v , v ). ∆k,+∞

The estimate (2.9) shows that the integrals converge to smooth forms on D − (Dv0 ∪ Dv00 ). These forms are locally integrable on D by Proposition 2.2 (note that Dv0 ∩ 0 00 0 00 Dv00 = ∅ since v is non-degenerate) and so we can regard α(v , v ) and β(v , v ) as currents. The statement follows by taking the limit as M → +∞ in (2.91), and the equivariance property under g ∈ G follows from Proposition 2.6.(d). As a corollary, we obtain the following invariance property of star products. Corollary 2.10. Let h ∈ O(k+l) (case 1) or h ∈ U(k+l) (case 2) and v = (v0, v00) ∈ k+l 0 00 0 00 V be non-degenerate. Let vh, vh be deﬁned by v · h = (vh, vh) and set 0 00 0 00 0 00 0 00 0 00 0 00 α(h; v , v ) = α(v , v ) − α(vh, vh), β(h; v , v ) = β(v , v ) − β(vh, vh) Then 0 00 0 00 0 00 0 00 2(k+l)rk(E)−2 g(v ) ∗ g(v ) − g(vh) ∗ g(vh) = ∂α(h; v , v ) + ∂β(h; v , v ) ∈ D (D). 26 LUIS E. GARCIA, SIDDARTH SANKARAN

Proof. This is a consequence of the theorem and the invariance property ξ0(v · h) = 0 ξ (v), which follows from Proposition 2.6.(f). When G = SO(1, 2)0, this invariance property is one of the main results in [23], where it is shown to hold by a long explicit computation (see also [33] for a similar proof in arbitrary dimension in case 2 when q = 1 and k + l = p + 1). Our corollary, and its global counterpart, Corollary 5.2 below, generalizes this to G = SO(p, 2)0 (p ≥ 1) and gives a conceptual proof covering both cases.

3. Archimedean heights and derivatives of Whittaker functionals r Let r ≤ p + 1 and v = (v1, . . . , vr) ∈ V be non-degenerate (recall that by this we mean that v1, . . . , vr are linearly independent) and denote by StabGhv1, . . . , vri the pointwise stabilizer of hv1, . . . , vri in G. Let ξ(v) be the form given in Deﬁnition 2.3. We will next, assuming that rk(E) = 1, compute the integral Z (3.1) ξ(v) ∧ Ωp−r+1, + Γv\D

where Γv is a discrete subgroup of ﬁnite covolume in StabGhv1, . . . , vri, under some additional conditions ensuring that it converges. The result is stated in Theorem 3.10 and relates this integral to the derivative of a Whittaker functional deﬁned on a 0 degenerate principal series representation of Gr. Despite the length of this section, its proof is conceptually simple and follows easily from Lemma 2.8 determining the weight of ν(v)[2r−2], together with results in [28, 31] concerning reducibility of these representations and multiplicity one for their K-types (we review these results in Sections 3.1 and 3.2) and estimates of Shimura [38] for Whittaker functionals that we review in Section 3.3.

3.1. Degenerate principal series of Mp2r(R). In this section we ﬁx r ≥ 1 and let 0 0 0 G = Mp2r(R). We abbreviate N = Nr, M = Mr and K = Kr (see 2.5.1). 3.1.1. We denote by g0 the complexiﬁed Lie algebra of G0. We have the Harish- Chandra decomposition 0 0 (3.2) g = p+ ⊕ p− ⊕ k , with 0 X1 X2 t t k = X1 = −X1,X2 = X2 , −X2 X1 (3.3) 1 X iX t p+ = p+(X) = X = X , p− = p+. 2 iX −X 0 0 0 0 Let K the maximal compact subgroup of G with Lie(K )C = k ; it is the inverse image under the metaplectic cover of the standard maximal compact U(r) of Sp2r(R) r in (2.62). For x = (x1, . . . , xr) ∈ C , we write −id(x) (3.4) h(x) = id(x) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 27

0 r 0 and deﬁne ej(h(x)) = xj. Then h = {h(x)|x ∈ C } is a Cartan subalgebra of k , and + + + we choose the set of positive roots ∆ = ∆c t ∆nc given by + ∆c = {ei − ej|1 ≤ i < j ≤ r}, (3.5) + ∆nc = {ei + ej|1 ≤ i ≤ j ≤ r}, + + where ∆c and ∆nc denote the compact and non-compact roots respectively. 3.1.2. The group P = MN is a maximal parabolic subgroup of G0, the inverse image under the covering map of the standard Siegel parabolic of Sp2r(R). The group M has a character of order four given by i, if det a < 0, (3.6) χ(m(a), ) = · 1, if det a > 0.

For α ∈ Z/4Z and s ∈ C, consider the character α s α s (3.7) χ | · | : M → R, (m(a), ) 7→ χ(m(a), ) | det a| and the smooth induced representation α G0 α s (3.8) I (s) = IndP χ | · | with its C∞ topology, where the induction is normalized so that Iα(s) is unitary when Re(s) = 0. In concrete terms, Iα(s) consists of smooth functions Φ : G0 → C satisfying

0 α s+ρr 0 r+1 (3.9) Φ((m(a), )n(b)g ) = χ(m(a), ) | det a| Φ(g ), ρr := 2 , with the action of G0 defined by r(g0)Φ(x) = Φ(xg0). Note that, by the Cartan decomposition G0 = PK0, any such function is determined by its restriction to K0; in α α particular, given Φ(s0) ∈ I (s0), there is a unique family (Φ(s) ∈ I (s))s∈C such that α Φ(s)|K0 = Φ(s0)K0 for all s. Such a family is called a standard section of I (s). 3.1.3. We denote by χα the character of K0 whose differential restricted to h0 has α 0 α weight 2 (1,..., 1). We will use some of the results about the K -types of I (s) established by Kudla and Rallis. Namely, in [28] they show that the K0-types that 0 appear are the irreducible representations πλ of K with highest weight λ = (l1, . . . , lr) α −1 (here l1 ≥ · · · ≥ lr) such that πλ ⊗ (χ ) descends to an irreducible representation of U(r) and satisfies α (3.10) l ∈ + 2 , 1 ≤ i ≤ r. i 2 Z Moreover, these K0-types appear with multiplicity one in Iα(s) (op. cit., p.31). If Φλ(·, s) ∈ Iα(s) is a non-zero highest weight vector of weight λ, then Φλ(e, s) 6= 0 (op. cit., Prop. 1.1), hence from now on we normalize all such highest weight vectors so that Φλ(e, s) = 1. Note that the restriction of Φλ to K0 is independent of s. For α scalar weights λ = l(1,..., 1) with l ∈ 2 + 2Z, we have (3.11) Φl(k0, s) := Φl(1,...,1)(k0, s) = (det k0)l, k0 ∈ K0. 28 LUIS E. GARCIA, SIDDARTH SANKARAN

3.1.4. Suppose that X ∈ g0 is a highest weight vector for K0 of weight λ. Then XΦl(s) is a highest weight vector of weight λ + l and hence we have

(3.12) XΦl(s) = c(X, l, s)Φλ+l(s)

for some constant c(X, l, s) ∈ C. We will need to determine this constant explicitly for certain X and l. Namely, let

(3.13) ei = (0,..., 0, 1 , 0,..., 0) i−th

− − 0 and deﬁne Xi = p−(d(ei)). Then Xr is a highest weight vector for K of weight 0 −2er. Let ιr : Mp2(R) → G be the embedding deﬁned by    1r−1 0r−1 a b  a b  (3.14) , 7→   ,  . c d 0r−1 1r−1   c d

α − Then, for any Φ ∈ I (s), the value of Xr Φ(e, s) only depends on the pullback function ∗ 0 ιrΦ(s) : Mp2(R) → C. Recall that every element g of Mp2(R) can be written uniquely in the form

0 1/2 ˜ (3.15) g = (n(x)m(y ), 1)kθ,

where x ∈ R, y ∈ R>0 and θ ∈ R/4πZ, where we deﬁne ˜ (kθ, 1), if − π < θ ≤ π, cos θ sin θ (3.16) kθ = kθ = . (kθ, −1), if π < θ ≤ 3π, − sin θ cos θ

We think of x, y and θ as coordinates on Mp2(R). In terms of these coordinates, we have d i ∂ (3.17) X−Φ(e) = −2iy + ι∗Φ(e), r dτ 2 ∂θ r

α d 1 ∂ ∂ for any Φ ∈ I (s), where dτ = 2 ∂x + i ∂y . The following lemma is a straightforward computation using the explicit expression

1 r+1 s+ ∗ l 2 2 iθl (3.18) ιrΦ (x, y, θ, s) = y e .

α Lemma 3.1. Let l ∈ 2 + 2Z. Then

− l(1,...,1) 1 r+1 l(1,...,1)−2er Xr Φ (s) = 2 s + 2 − l Φ (s).

− 1 r+1 In other words, this shows that c(Xr , l, s) = 2 s + 2 − l . GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 29

3.1.5. We now return to the quadratic space V over R of signature (p, 2). Let α(V ) = p − 2 and deﬁne I(V, s) = Iα(V )(s), (3.19) p − r + 1 s = . 0 2 We assume that r ≤ p + 1, so that

(3.20) s0 ≥ 0. 0 r Consider the Weil representation ω = ωψ of G × O(V ) on S(V ) and let Rr(V ) be its maximal quotient on which (so(V ), O(V +) × O(V −)) acts trivially. Note that for any φ ∈ S(V r), the function (3.21) Φ(g0) = ω(g0)φ(0), g0 ∈ G0, r 0 belongs to I(V, s0); the resulting map φ 7→ Φ: S(V ) → I(V, s0) is G -intertwining and factors through Rr(V ). By [28], it deﬁnes an embedding

(3.22) Rr(V ) ,→ I(V, s0). 3.2. Degenerate principal series of U(r, r). In this section we ﬁx r ≥ 1 and let 0 0 0 G = U(r, r). We abbreviate N = Nr, M = Mr and K = Kr (see 2.5.1). Our setup follows [17].

0 0 0 3.2.1. We denote by g the complexiﬁed Lie algebra of G and let gss = {X ∈ g0|trX = 0}. We set 1 1 1 (3.23) r = √ r r ∈ U(2r) 2 i1r −i1r and deﬁne φ(g) = rgr−1 for g ∈ G0; note that φ−1(G0) is the isometry group of the 1r 0 Hermitian form determined by 0 −1r . Then we have 0 k1 0 (3.24) Kr = φ k1, k2 ∈ U(r) 0 k2 and the Harish-Chandra decomposition 0 0 (3.25) g = p+ ⊕ p− ⊕ k , 0 with g = M2r(C) and 0 0 X1 0 k = Lie(Kr)C = φ X1,X2 ∈ Mr(C) , 0 X2 0 X (3.26) p+ = p+(X) := φ X ∈ Mr(C) , 0 0 0 0 p− = p−(X) := φ X ∈ Mr(C) . X 0 0 0 0 Let kss = k ∩ gss and 2r (3.27) h = φ(d(x))|x = (x1, . . . , x2r) ∈ C , x1 + ... + x2r = 0 . 30 LUIS E. GARCIA, SIDDARTH SANKARAN

0 Then h is a Cartan subalgebra of kss. For 1 ≤ i ≤ 2r, the assignment φ(d(x)) 7→ xi deﬁnes a functional ei : h → C. We write ∆ for the set of roots of (gss, h) and ﬁx the + + + set of positive roots ∆ = ∆c t ∆nc given by + ∆c = {ei − ej|1 ≤ i < j ≤ r} ∪ {−ei + ej|r < i < j ≤ 2r}, (3.28) + ∆nc = {ei − ej|1 ≤ i ≤ r < j ≤ 2r}, + + where ∆c and ∆nc denote the compact and non-compact roots respectively.

3.2.2. The group P = MN is the Siegel parabolic of G0. For a character χ of C× and s ∈ , deﬁne a character χ| · |s : P → × by C C C s s (3.29) χ| · |C(m(a)n(b)) = χ(det a)| det a|C and let G0 s (3.30) I(χ, s) = IndP (χ| · |C) be the degenerate principal series representation of G0. Thus I(χ, s) is the space of smooth functions Φ : G0 → C satisfying (3.31) Φ(m(a)n(b)g0) = χ(det a)| det a|s+ρr Φ(g0), ρ := r , C r 2 and the action of G0 is via right translation: r(g0)Φ(x) = Φ(xg0). Here the induction is normalized so that the I(χ, s) is unitary when χ is a unitary character and s = 0. Note that, by the Cartan decomposition G0 = PK0, any such function is determined 0 by its restriction to K ; in particular, given Φ(s0) ∈ I(χ, s0), there is a unique family (Φ(s) ∈ I(χ, s))s∈C such that Φ(s)|K0 = Φ(s0)K0 for all s. Such a family is called a standard section of I(χ, s).

3.2.3. We will need some results on K0-types of I(χ, s) that were proved by Lee [31]. Namely, I(χ, s) is multiplicity free as a representation of K0. If Φ(λ1,λ2)(·, s) ∈ I(χ, s) (λ1,λ2) is a highest weight vector of weight (λ1, λ2), then Φ (e, s) 6= 0, hence from now on we normalize all such highest weight vectors so that Φ(λ1,λ2)(e, s) = 1. Note that the restriction of Φ(λ1,λ2) to K0 is independent of s. For scalar weights

(3.32) l = (l1(1,..., 1), l2(1,..., 1)), l1, l2 ∈ Z, we have (3.33) l (l1(1,...,1),l2(1,...,1)) l1 l2 Φ ([k1, k2], s) := Φ ([k1, k2], s) = (det k1) (det k2) , k1, k2 ∈ U(r).

− − 3.2.4. Let Xr ∈ p− be as in 3.1.4; that is, Xr = p−(d(er)) with er = (0,..., 0, 1) ∈ r − 0 C . Then Xr is a highest weight vector for K of weight (−er, er) and so

− l − l+(−er,er) (3.34) Xr Φ (s) = c(Xr , l, s)Φ (s) − for some constant c(Xr , l, s) ∈ C. To compute this constant, note that U(r, r) ∩ − GL2r(R) = Sp2r(R) and Xr ∈ sp2r,C. Moreover, if Φ ∈ I(χ, s), then the restriction α 0 0 × Φ|Sp2r(R) belongs to I (s ), where s = 2s + (r − 1)/2 and α = 0 if χ|R is trivial GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 31 and α = 2 otherwise (this follows directly from a comparison of (3.9) and (3.31)). If l = (l1(1,..., 1), l2(1,..., 1)) is a scalar weight, then

l (l1−l2)·(1,...,1) 0 α 0 (3.35) Φ |Sp2r(R)(s) = Φ (s ) ∈ I (s ) and so Lemma 3.1 and the above remarks show that the constant in (3.34) is given by l − l (3.36) c(X−, l, s) = s + ρ − 1 2 . r r 2 3.2.5. We now return to the m-dimensional hermitian space V over C of signature × (p, q) with pq 6= 0. From now on we fix a character χ = χV of C such that χ|R× = sgn(·)m and define I(V, s) = I(χ, s), (3.37) m − r s = . 0 2 We assume that r ≤ p + 1, so that q − 1 (3.38) s ≥ ≥ 0, 0 2 with equality only when q = 1 and r = p + 1. 0 r Consider the Weil representation ω = ωψ,χ of G ×U(V ) on S(V ) and let Rr(V ) be its maximal quotient on which (u(V ),K) acts trivially. Note that for any φ ∈ S(V r), the function (3.39) Φ(g0) = ω(g0)φ(0), g0 ∈ G0, r 0 belongs to I(V, s0); the resulting map ϕ 7→ Φ: S(V ) → I(V, s0) is G -intertwining and factors through Rr(V ). By [32, Thm. 4.1], it defines an embedding

(3.40) Rr(V ) ,→ I(V, s0). 3.3. Whittaker functionals. 3.3.1. Let Sym ( ), case 1 (3.41) T ∈ r R Herr, case 2 × be a non-singular matrix and let ψT : N → C the character deﬁned by ψT (n(b)) = ψ(tr(T b)). Let I(V, s) and s0 be as in (3.19) (resp. (3.37)) in case 1 (resp. in case 2). A continuous functional l : I(V, s0) → C is a called a Whittaker functional if it satisﬁes

(3.42) l(r(n(b))Φ) = ψT (n(b))l(Φ), n(b) ∈ N, Φ ∈ I(V, s0). Such a functional can be constructed as follows. Let dn(b) be the Haar measure on N 0 0 that is self-dual with respect to the pairing (b, b ) 7→ ψ(tr(bb )). Embed Φ ∈ I(V, s0) in a (unique) standard section (Φ(s))s∈C and deﬁne Z −1 (3.43) WT (Φ, s) = Φ(wr n(b), s)ψT (−n(b))dn(b) Nr 32 LUIS E. GARCIA, SIDDARTH SANKARAN

The integral converges for Re(s) 0 and admits holomorphic continuation to all s ([41]); its value at s0 deﬁnes a Whittaker functional WT (s0). It is shown in op. cit. 0 0 that WT (s0) spans the space of Whittaker functionals. For g ∈ Gr and Φ ∈ I(V, s0), deﬁne Z 0 −1 0 (3.44) WT (g , Φ, s) = Φ(wr n(b)g , s)ψT (−n(b))dn(b). Nr and

0 0 d 0 (3.45) W (g , Φ, s0) = WT (g , Φ, s) . T ds s=s0 3.3.2. Let Symr(R), case 1 (3.46) Xr = Herr, case 2 + and denote by Xr the open subset of Xr consisting of positive deﬁnite matrices. Let Φl(s) be the normalized highest weight vector of I(V, s) of scalar weight ( m 2 , case 1 (3.47) l = m+k(χ) −m+k(χ) 2 , 2 , case 2 0 l (see (3.11) and (3.33)). We will need some results on the value WT (g , Φ , s0) and its 0 0 l derivative WT (g , Φ , s0) established by Shimura in [38]. For s ∈ C, we set 1 (s − s + m), 1 (s − s ) case 1, (3.48) (α, β) = 2 0 2 0 (s − s0 + m, s − s0) case 2. t + Let m(a) ∈ Mr and y = a a ∈ Xr . Let (3.49) ι = [k : R], (r + 1)/2, case 1 (3.50) κ = r, case 2. As in [23, Lemma 9.2] and [33, §4A] one shows that

−r(κ−1)/2 l s+ρr (3.51) 2 WT (m(a), Φ , s) = | det a|k ξ(y, T ; α, β), where Z (3.52) ξ(y, T ; α, β) = e−2πitr(T x) det(x + iy)−α det(x − iy)−βdx Xr and dx denotes Lebesgue measure (see [38, (1.11), (1.25)]). The integral in (3.52) converges for Re(α) > κ − 1 and Re(β) > 2κ − 1. As in [38, (1.29)], we can rewrite this as follows. Let Γ0(s) := 1 and Z −tr(x) s−κ Γr(s) := e det(x) dx + Xr (3.53) r−1 Y = πιr(r−1)/4 Γ(s − ιj/2) . j=0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 33

Then (2π)rκ(−i)r(α−β) (3.54) ξ(y, T ; α, β) = r(κ−1) η(2y, πT ; α, β), 2 Γr(α)Γr(β) where Z (3.55) η(2y, T ; α, β) = e−2tr(xy) det(x + T )α−κ det (x − T )β−κ dx Qr(T ) + and Qr(T ) = {x ∈ Xr|x − T, x + T ∈ Xr }. t Lemma 3.2. Let a ∈ GLr(k) and y = a · a. Then l WT (m(a), Φ , s0) = 0 if T is not positive deﬁnite and (−2πi)ιrm/2 l ιs0 m/4 −2πtr(yT ) WT (m(a), Φ , s0) = r(κ−1)/2 det(T ) | det y|k e 2 Γr(ιm/2) if T is positive deﬁnite.

Proof. Assume that T has r+ positive and r− negative eigenvalues, so that r++r− = r. According to [38, Thm. 4.2], the function −1 −1 (3.56) Γr− (α − ιr+/2) Γr+ (β − ιr−/2) η(2y, πT ; α, β) is holomorphic at s = s0. Thus, by (3.51) and (3.54), the order of vanishing of 0 l WT (g , Φ , s) at s = s0 is bounded below by that of Γ (α − ιr /2)Γ (β − ιr /2) 1 (3.57) r− + r+ − = . Γr(α)Γr(β) Γr+ (α)Γr− (β)

By our hypothesis that s0 ≥ 0, the function Γr+ (α) is regular and non-vanishing at −1 s = s0. Moreover, Γr− (β) vanishes at s = s0 if r− > 0; this shows that WT vanishes at s = s0 when T is not positive deﬁnite. Now assume that T is positive deﬁnite. By [38, (4.3),(4.6.K),(4.35.K)], setting −rα −1 κ−α α+β−κ (3.58) ω(2y, πT ; α, β) = 2 Γr(β) det(2πyT ) det(2y) η(2y, πT ; α, β), we have −rκ −2πtr(yT ) (3.59) ω(2y, πT ; α(s0), β(s0)) = ω(2y, πT ; ιm/2, 0) = 2 e ,

and the expression for WT (s0) in this case follows by a direct computation.

3.3.3. For y > 0, T ∈ Xr invertible and s ∈ C, deﬁne 2πtr(yT ) 1/2 l − ιrm (3.60) fT (y, s) = e WT (m(y · 1r), Φ , s)y 4 . Note that by Lemma 3.2, we have (−2πi)ιrm/2 ιs0 (3.61) fT (y, s0) = r(κ−1)/2 det(T ) 2 Γr(ιm/2) 34 LUIS E. GARCIA, SIDDARTH SANKARAN

if T is positive deﬁnite and fT (y, s0) = 0 otherwise. We will need to understand the asymptotic behaviour of the derivative

0 d 2πtr(yT ) 0 1/2 l − ιrm (3.62) f (y, s ) := f (y, s)| = e W (m(y · 1 ), Φ , s )y 4 T 0 ds T s=s0 T r 0 as y → +∞. First we consider the case when T is not positive deﬁnite.

0 −ay Lemma 3.3. Assume that T is not positive deﬁnite. Then fT (y, s0) = O(e ) for some a > 0 as y → +∞.

Proof. Let r+ and r− be the number of positive and negative eigenvalues of T respectively. As in the proof of Lemma 3.2, the function Γ (α − ιr /2)Γ (β − ιr /2) 1 (3.63) r− + r+ − = . Γr(α)Γr(β) Γr+ (α)Γr− (β)

vanishes at s = s0 and the function ˜ −1 −1 (3.64) fT (y, s) = Γr− (α − ιr+/2) Γr+ (β − ιr−/2) η(2y, πT ; α, β) 0 l ˜ is regular at s = s0. It follows that WT (·, Φ , s0) is a constant multiple of fT (·, s0). By [38, Thm. 4.2], we have a bound of the form ˜ −2πyτ(T ) B (3.65) |fT (y, s0)| ≤ Ae y , where A and B are positive constants and τ(T ) is the sum of the absolute values of all eigenvalues of T ; this proves the statement since tr(T ) − τ(T ) is negative.

1 t 1 Assume now that T is positive deﬁnite. Write T = (T 2 ) · (T 2 ), for some matrix 1 T 2 ∈ GLr(k), and let

t 1 1 (3.66) g := 4π (T 2 ) · y · (T 2 ); then using [38, (1.29), (3.3), (3.6)] we may write

l m/4 β+ιs0 −2π tr(T y) (3.67) WT (m(a), Φ , s) = C(β) | det y|k (det T ) · e ω(g; α, β) where (−2πi)ιrm/2 πrβ (3.68) C(β) := r(κ−1)/2 2 Γr(β + ιm/2) is a function independent of T and y, and ω(g; α, β) is an entire function in (α, β) ∈ C2, initially deﬁned by the formula −1 β ω(g; α, β) = Γr(β) det(g) Z −tr(gx) α−κ β−κ (3.69) × e det(x + 1r) det(x) dx + Xr on the region Re(β) > κ − 1, where the integral is absolutely convergent; its analytic + continuation is proven in [38, Theorem 3.1]. We say that a function F : Xr → C + vanishes at inﬁnity if F (tg) → 0 as t → +∞, for any g ∈ Xr . Lemma 3.4. (1) [38, (3.15)] ω(g; ιm/2, 0) = 1. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 35

(2) The special value of the derivative ∂ ω(·; β + ιm/2, β) ∂β β=0 vanishes at infinity. Proof. We are interested in the value and derivative of ω(g) at the parameters (α, β) in (3.48), which is outside the range of absolute convergence of (3.69). However, Shimura’s proof of analytic continuation involves a useful recurrence: let ∆ denote the differential operator [38, (3.10.I-II)], which satisfies the identity (3.70) ∆ e−tr(g) = (−1)r e−tr(g). Set ( 0, if m/2 ∈ Z (3.71) := 1 2 , if m/2 ∈/ Z in case 1 and = 0 in case 2. Then (3.12) and (3.7) of [38] together imply that ω(g; β + ιm/2, β) = (−1)r(ιm/2−)etr(g) det(g)β · ∆ιm/2− e−tr(g) det(g)−βω(g; β + , β) =(−1)r(ιm/2−)etr(g) det(g)β (3.72) × ∆ιm/2− e−tr(g) det(g)−βω (g; κ − β, κ − β − ) . The final incarnation of ω in (3.72) is now in the range of convergence of (3.69); for convenience, rewrite (3.69) for these parameters as

(3.73) ωe(g, β) := ω (g; κ − β, κ − β − ) , which may also be expressed as

−1 −β (3.74) ωe(g, β) = Γr (κ − β − ) det(g) ζe(g, β) where Z (3.75) ζe(g, β) := det(g)κ− e−tr(gx) det(x + 1)−β det(x)−β−dx. + Xr Note that by [38, 1.16], Z −tr(gx) s−κ −s (3.76) e det(x) dx = Γr(s) det(g) , for Re(s) > κ − 1 + Xr so ˜ (3.77) ζ(g, 0) = Γr (κ − ) , and is in particular independent of g. This equation implies

(3.78) ωe(g, 0) = 1, and hence ω(g; ιm/2, 0) = 1 as well, via (3.72). 36 LUIS E. GARCIA, SIDDARTH SANKARAN

As for the derivative, begin by noting Z ∂ κ− −tr(gx) − ζe(g, β) = − det(g) e log(det x) det(x) dx β=0 + ∂β Xr (3.79) Z − det(g)κ− e−tr(gx) log(det(x + 1)) det(x)−dx. + Xr For the first integral, writing g = γ · tγ and applying the change of variables x 7→ tγ−1 · x · γ−1 yields Z − det(g)κ− e−tr(gx) log(det x) det(x)−dx + Xr Z = − e−tr(x) (log(det x) − log det(g)) det(x)−dx + (3.80) Xr Z = log det g · ζ˜(g, 0) − e−tr(x) (log(det x)) det(x)−dx + Xr 0 = log det g · Γr (κ − ) − Γr (κ − ) . Turning to the second integral, for convenience write Z I(g) := − det(g)κ− e−tr(gx) log(det(x + 1)) det(x)−dx + Xr (3.81) Z = − e−tr(x) log det tγ−1 · x · γ−1 + 1 det(x)−dx. + Xr A comparison with (3.69) reveals that the integral defining I(g) is absolutely con- t vergent for all g. For a fixed g0 = γo · γ0 and parameter t ≥ 1, the integrand appearing in I(tg0) can be bounded as follows: e−tr(x) log det t−1 tγ−1 · x · γ−1 + 1 det(x)− (3.82) 0 0 −tr(x) t −1 −1 − ≤ e log det γ0 · x · γ0 + 1 det(x) As the latter function is absolutely integrable in x, it follows immediately that I(g) vanishes at infinity. By differentiating under the integral, one can similarly show that all partial derivatives of I(g) also vanish at infinity. We have shown ζe0(g, 0) Γ0 1 (3.83) = log det g − + · I(g) ζe(g, 0) Γ Γ 0 0 where, for legibility, we have abbreviated Γ = Γr (κ − ) and Γ = Γr (κ − ). Now taking the logarithmic derivative in (3.74), and recalling ωe(g, 0) = 1, yields ∂ Γ0 ζe0(g, 0) ω(g, β) = − log det g + e β=0 (3.84) ∂β Γ ζe(g, 0) = Γ−1I(g). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 37

Finally, we obtain part (ii) of the lemma as follows: taking the logarithmic derivative of (3.72) at β = 0 (recall here that ω(g; ιm/2, 0) = 1) gives (3.85) ∂ ιm/2− −tr(g) −β ω(g; β + ιm/2, β) = log det g + dlog∆ e det(g) ω(g; β) ∂β β=0 e β=0 = log det g + (−1)r(ιm/2−)etr(g)∆ιm/2− e−tr(g) − log det g + Γ−1I(g) . Note that the partial derivatives of log det g vanish at infinity; since tr(g) is linear in the entries of g, repeated differentiation gives (3.86) (−1)r(ιm/2−)etr(g)∆ιm/2− e−tr(g) {− log det g} = − log det g + J(g) for some function J(g) vanishing at infinity. Similarly, (3.87) etr(g)∆ιm/2− e−tr(g)I(g) can be expressed as a linear combination of partial derivatives of I(g), and hence also vanishes at infinity. Putting everything together, we find ∂ (3.88) ω(g; l + β, β) = f(g) ∂β β=0 for some function f(g) vanishing at infinity, as required.

Lemma 3.5. Assume that T is positive deﬁnite and let a ∈ GLr(k). Then 0 l 0 WT (m(ta), Φ , s0) ι Γr(ιm/2) lim l = log det(πT ) − . t→∞ WT (m(ta), Φ , s0) 2 Γr(ιm/2) Proof. This follows from (3.67) and (3.68) by taking logarithmic derivatives with respect to s and applying Lemma 3.4. Combined with Lemma 3.2, this shows that 0 0 l 2πtr(T ) ι Γr(ιm/2) (3.89) lim fT (y, s0) = WT (e, Φ , s0)e log det(πT ) − y→+∞ 2 Γr(ιm/2) when T is positive deﬁnite.

3.4. The integral of ξ. For the rest of Section3 we assume that rk( E) = 1.

3.4.1. Let r ≥ 1 and i (1 ≤ i ≤ r) be as in (2.76), so that νi(v) = ω(m(i))νr(v). r Consider the Schwartz functionsν ˜i, ν˜ ∈ S(V ) deﬁned by p−r+1 p νi(v, z0) ∧ Ω (z0) =ν ˜i(v)Ω (z0), 1 ≤ i ≤ r, (3.90) ν˜ :=ν ˜1 + ..., +˜νr.

SG r−1 Since ϕ (0) = 1 and ϕ(0) = −Ω, we have νr(0)[2r−2] = (−Ω) and hence

0 r−1 λ0 0 0 0 (3.91) ω(g )˜νr(0) = (−1) Φ (g , s0), g ∈ Gr, 38 LUIS E. GARCIA, SIDDARTH SANKARAN

since both sides deﬁne highest weight vectors of weight λ0 (cf. Lemma 2.8) in I(V, s0) that agree for g0 = 1 and the K-types in this representation appear with multiplicity one. Thus, setting

˜ X λ0 (3.92) Φ(s) = r(m(i))Φ (s) ∈ I(V, s), 1≤i≤r we have 0 r−1 ˜ 0 (3.93) ω(g )˜ν(0) = (−1) Φ(g , s0). ˜ The following lemma relating the Whittaker functional WT (·, s0) evaluated at Φ 0 l with the derivative WT (Φ , s0) is the main ingredient in the proof of Theorem 3.10. ˜ Lemma 3.6. Let fT (y, s) and Φ be given by (3.60) and (3.92) respectively. Then, for y > 0,

2πtr(yT ) 1/2 − ιrm dy 2 d 0 e W (m(y · 1 ), Φ˜, s )y 4 = f (y, s )dy. T r 0 y ι dy T 0

Proof. Since fT is real analytic [38, Thm. 5.9], it suﬃces to show that, for y > 0 and s 6= s0, 2πtr(yT ) 1/2 ˜ − ιrm dy 2 d fT (y, s) − fT (y, s0) (3.94) e WT (m(y · 1r), Φ, s)y 4 = dy. y ι dy s − s0 This is a direct computation using (3.17). By Lemma 3.1 (in case 1) and (3.36) (in case 2), we can write 2 X (3.95) Φ(˜ s) = (s − s )−1 Ad(m( ))X−Φl(s). ι 0 i r 1≤i≤r l ˜ ˜ Let F (g) = WT (g, Φ , s) and F (g) = WT (g, Φ, s). Then we have (3.96) −1 ˜ 1/2 X − 1/2 ι2 (s − s0)F (m(y · 1r) = Ad(m(j))Xr F (m(y · 1r)) 1≤j≤r X d i d = (−2iy + )F (m(y1/2 · 1 )) j dτ 2 dθ r 1≤j≤r j j X d d i d = (−iy + y + )F (m(y1/2 · 1 )) j dx j dy 2 dθ r 1≤j≤r j j j ! ! X d d ιrm = −iy + y − F (m(y1/2 · 1 )) j dx j dy 4 r 1≤j≤r j j ! ! X d d ιrm = −iy + y − F (m(y1/2 · 1 )) j dx dy 4 r 1≤j≤r j d ιrm = 2πtr(yT ) + y − F (m(y1/2 · 1 )) dy 4 r GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 39 and hence (3.97)

−1 2πtr(yT ) 1/2 − ιrm dy ι2 (s − s )e W (m(y · 1 ), Φ˜, s)y 4 0 T r y 2πtr(yT ) d ιrm 1/2 − ιrm dy = e 2πtr(yT ) + y − F (m(y · 1 )) y 4 dy 4 r y

2πtr(yT ) 1/2 − ιrm dy 2πtr(yT ) d 1/2 − ιrm = e 2πtr(yT )F (m(y · 1 ))y 4 + e F (m(y · 1 ))y 4 dy r y dy r

d 2πtr(yT ) 1/2 − ιrm 2πtr(yT ) d 1/2 − ιrm = e F (m(y · 1 ))y 4 dy + e F (m(y · 1 ))y 4 dy dy r dy r

d 2πtr(yT ) 1/2 − ιrm = e F (m(y · 1 ))y 4 dy dy r d = (f (y, s) − f (y, s )) dy, dy T T 0

where the last equality holds since fT (y, s0) is independent of y by (3.61).

3.4.2. For T ∈ Xr with det T 6= 0, let r (3.98) ΩT (V ) = {(v1, . . . , vr) ∈ V |(Q(vi, vj))i,j = 2T }.

Thus ΩT (V ) 6= ∅ if and only if (V,Q) represents T , and in this case U(V ) acts transitively on ΩT (V ); assuming this, let dµ(v) be an U(V )-invariant measure on r ΩT (V ) and consider the functional S(V ) → C defined by Z (3.99) φ 7→ φ(v)dµ(v). ΩT (V ) This functional is obviously U(V )-invariant and non-zero and hence defines a Whit- SW taker functional on Rr(V ). We denote by dµ(v) the unique U(V )-invariant measure r on ΩT (V ) such that for any φ ∈ S(V ): Z SW −1 (3.100) φ(v)dµ(v) = γV r · WT (e, Φ, s0), where Φ(g) := ω(g)φ(0). ΩT (V ) (2) (2) We denote by dg the invariant measure on U(V ) defined as dg = dlpdrk, where + − + − drk is the unique Haar measure on O(V )×O(V ) (resp. U(V )×U(V )) with total volume one in case 1 (resp. case 2), and dlp is the left Haar measure on P induced p + ∼ by the invariant volume form Ω on D = G/K.

Lemma 3.7. Let λ0 be as in Lemma 2.8, T ∈ Xr with non-zero determinant and v = (v1, . . . , vr) ∈ ΩT (V ). Let Gv ⊂ U(V ) be the pointwise stabilizer of hv1, . . . , vri 0 0 0 and Γv ⊂ Gv be a torsion free subgroup of ﬁnite covolume. Then, for any g ∈ Gr, we have Z 0 p−r+1 ι 0 ˜ ω(g )ν(v) ∧ Ω = − CT,Γv WT (g , Φ, s0), + 2 Γv\D 40 LUIS E. GARCIA, SIDDARTH SANKARAN ˜ where Φ is as in (3.92) and CT,Γv is the non-zero constant given by

(2) 2 r dg /dgv −1 C = (−1) Vol(Γ \G , dg ) γ r . T,Γv ι v v v dµ(v)SW V

(2) Here dgv is an arbitrary Haar measure on Gv and dg /dgv denotes the quotient ∼ (2) measure on ΩT (V ) = U(V )/Gv induced by dg and dgv.

0 Proof. The hypothesis that det T 6= 0 implies that Gv is reductive, hence the invariant (2) measure dg /dgv exists. By (3.90) and Proposition 2.6.(d), we haveν ˜(gv) =ν ˜(v) for any g ∈ U(V ) stabilizing V −. We compute (3.101) Z Z ω(g0)ν(v) ∧ Ωp−r+1 = ω(g0)˜ν(g−1v)dg(2) + Γv\D Γv\U(V ) Z (2) 0 −1 dg = Vol(Γv\Gv, dgv) ω(g )˜ν(g v) Gv\U(V ) dgv (2) Z dg /dgv 0 SW = Vol(Γv\Gv, dgv) SW ω(g )˜ν(v)dµ(v) dµ(v) ΩT (V ) (2) r−1 dg /dgv −1 0 = (−1) Vol(Γ \G , dg ) γ r W (g , Φ˜, s ), v v v dµ(v)SW V T 0

where the last equality follows from (3.100) and (3.93).

3.4.3. We will now apply the results obtained in this section to compute the integral (3.1) under our standing assumption that rk(E) = 1. Rather than aiming for the most general result, we restrict to cocompact arithmetic subgroups Γ and integral vectors v (see below); this suﬃces for the application to compact Shimura varieties in Section5. Fix a lattice L ⊂ V and a torsion free cocompact arithmetic subgroup Γ ⊂ G + stabilizing L, and let Γv = StabΓhv1, . . . , vri and XΓ = Γ\D . r Lemma 3.8. Assume that v ∈ L is non-degenerate and let Z(v)Γ be the image of + the natural map Γv\Dv → XΓ. (a) The sum X −1 g(v)Γ = g(γ v)

γ∈Γv\Γ

converges absolutely to a smooth form on XΓ − Z(v)Γ that is locally integrable on XΓ. (b) As currents on XΓ we have Z M X 0 1/2 −1 dt M→+∞ ν (t γ v)[2r−2] −−−−−→ g(v)Γ. 1 t γ∈Γv\Γ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 41

+ Proof. Let us first show that g(v)Γ converges as claimed. Let z ∈ D and pick a relatively compact neighborhood U of z. Recall that there is a positive definite form Qz defined by (2.46) that varies continuously with z. Write Γv = S1 t S2 (disjoint union), with

0 0 0 (3.102) S1 = {v ∈ Γv| min Qz(v , v ) < 1}; z∈U

r then S1 is finite since Γv ⊂ L . We can split the sum defining g(v)Γ accordingly; + the sum over S1 converges to a locally integrable form on D that is smooth outside + 0 ∪v ∈S1 Dv0 by Proposition 2.2. The sum over S2 converges to a smooth form on U by the estimate (2.9) and the standard argument of convergence of theta series. By Proposition 2.6.(d), the current defined by g(v)Γ is invariant under Γ, and this shows 0 M 0 1/2 0 dt 0 (a). For part (b), note that for each v ∈ S1 we have ∫1 ν (t v )[2r−2] t → g(v ) as M → +∞ (as currents on D+) by dominated convergence, as remarked in the proof of Proposition 2.2. For the sum over S2 we apply the bound (2.54), and (b) follows.

Lemma 3.9. Assume that v ∈ Lr is non-degenerate. Then the integral (3.1) converges and

Z Z M Z dt ξ0(v) ∧ Ωp−r+1 = lim ν0(t1/2v) ∧ Ωp−r+1 . + M→+∞ + t Γv\D 1 Γv\D

p−r+1 Proof. By Lemma 3.8.(a), the integral (3.1) equals ∫Γ\D+ g(v)Γ ∧ Ω and so it converges since XΓ is compact. Applying Lemma 3.8.(b) and unfolding the sum and the integral proves the claim.

r Theorem 3.10. Let T ∈ Xr with non-zero determinant and v ∈ ΩT (V ) ∩ L . Let

CT,Γv be as in Lemma 3.7 and l be the weight in (3.47). Recall that we assume that rk(E) = 1.

(a) If T is not positive deﬁnite (so that Dv = ∅), then

Z p−r+1 0 l ξ(v) ∧ Ω = CT,Γv WT (e, Φ , s0). + Γv\D

(b) If T is positive deﬁnite (so that Dv 6= ∅), then

Z p−r+1 0 l ξ(v) ∧ Ω = CT,Γv WT (e, Φ , s0) + Γv\D 0 l ι Γr(ιm/2) − CT,Γv WT (e, Φ , s0) log det(πT ) − . 2 Γr(ιm/2) 42 LUIS E. GARCIA, SIDDARTH SANKARAN

Proof. The integral converges by Lemma 3.9. We compute (3.103) Z Z ∞ Z dy ξ0(v) ∧ Ωp−r+1 = ν0(y1/2v) ∧ Ωp−r+1 + + y Γv\D 1 Γv\D Z ∞ Z dy = e2πtr(yT ) ν(y1/2v) ∧ Ωp−r+1 + y 1 Γv\D Z ∞ Z ιrm 2πtr(yT ) 1/2 p−r+1 − dy = e ω(m(y · 1r))ν(v) ∧ Ω y 4 + y 1 Γv\D Z ∞ ιrm ι − dy 2πtr(yT ) 1/2 ˜ 4 = − CT,Γv e WT (m(y · 1r), Φ, s0)y , 2 1 y where we have used Lemma 3.9 and Lemma 3.7 for the first and last equality respectively. Applying Lemma 3.6, we conclude that Z 0 p−r+1 0 0 (3.104) ξ (v) ∧ Ω = CT,Γv (fT (1, s0) − lim fT (y, s0)). + y→∞ Γv\D If T is not positive definite, then the limit in the above expression vanishes by part (a) of Lemma 3.3; this proves (a). If T is positive definite, then (b) follows from (3.89).

4. Green forms for special cycles on Shimura varieties We now shift focus from the hermitian symmetric domain D to its quotients Γ\D by arithmetic subgroups. Namely, we consider the special cycles Z(T, ϕf ) on orthogonal and unitary Shimura varieties introduced by Kudla in [22]. We review their definition and apply our results to construct Green forms g(T, y, ϕf ) for those special cycles Z(T, ϕf ) parametrized by a non-singular matrix T . Section 4.1 deals with the orthogonal case and Section 4.2 with the unitary one. In Section 4.3 we prove our main result concerning star products of the Green forms g(T, y, ϕf ). Motivated by the conjectures in Section 1.2, we would like to define currents g(T, y, ϕf ) also when T is singular. We will achieve this in Section 4.4 by regularizing 0 the integral defining ξ in Definition 2.3. In Example 4.8 we compute g(T, y, ϕf ) in a special case and show how, specialized to Shimura curves over Q, this construction recovers the current in [30, §6.4], introduced there in an ad hoc way to force agreement with a certain Eisenstein series. Throughout, we fix a totally real number field F with real embeddings σ1, . . . , σd.

4.1. Orthogonal Shimura varieties.

4.1.1. Let us brieﬂy recall the deﬁnition and basic properties of orthogonal Shimura varieties attached to quadratic spaces over F ; see [22] for more detail. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 43

Let p ≥ 1 and V be a quadratic space over F of dimension p+2, with corresponding bilinear form Q(·, ·). Assume that ( (p, 2) if i = 1 (4.1) signature(Vσi ) = (p + 2, 0) if i 6= 1,

where we abbreviate Vσi := V ⊗F,σi R for the real quadratic spaces at each place.

Attached to Vσ1 is the symmetric space D(Vσ1 ) deﬁned in (2.33); we set

(4.2) D(V) = D(Vσ1 ) = {[v] ∈ P(Vσ1 (C))|Q(v, v) = 0,Q(v, v)i < 0}.

Here Q(·, ·) is the C−bilinear extension of the bilinear form on Vσ1 . Let

(4.3) H := ResF/Q GSpin(V). Then

(4.4) H(R) ' GSpin(Vσ1 ) × · · · × GSpin(Vσd ) acts transitively on D(V) via the ﬁrst factor.

Deﬁnition 4.1. For a compact open subgroup K ⊂ H(Af ), consider the Shimura variety XK := H(Q)\D(V) × H(Af )/K 1 If K is suﬃciently small (neat), then XK is a complex quasi-projective algebraic 2 variety. If V is moreover anisotropic, then XK is projective.

The space XK may be written in a perhaps more familiar fashion: fix a connected component D+ ⊂ D(V) and let H(R)+ denote its stabilizer in H(R)+. Setting + + H(Q) = H(R) ∩ H(Q), there exist finitely many elements h1, . . . ht ∈ H(Af ) such that a + (4.5) H(Af ) = H(Q) hjK; j then we may write a + a (4.6) XK ' Γj D =: Xj j j as a disjoint union of quotients of D+ by discrete subgroups + −1 (4.7) Γj = H(Q) ∩ hjKhj . + We regard the quotients XK and Γj\D as orbifolds. In particular, for a Γj- invariant differential form η of top degree on D+, we define Z Z (4.8) η = [Γ : Γ0]−1 η, + 0 + [Γj \D ] Γ \D

1 In fact, the theory of canonical models implies that XK admits a model deﬁned over F , but we will not require this fact in the sequel. 2 Our assumptions on the signature of V imply that this is always the case when F 6= Q. 44 LUIS E. GARCIA, SIDDARTH SANKARAN

0 where Γ ⊂ Γj is a (equivalently, any) neat subgroup of ﬁnite index, and set ∫[X ] = P R K + . j [Γj \D ] 4.1.2. Recall that in 2.2.2 we have deﬁned the tautological line bundle E over D(V) r r and a global section sx of E for any x ∈ Vσ1 whose zero locus Z(sx) we denote by r Dx. Given a rational vector v = (v1, . . . , vr) ∈ V , we set

(4.9) Dv = Z(sσ1(v)) + + and Dv = Dv ∩ D . Let Hv(Q) be the pointwise stabilizer of hv1, . . . , vri in H(Q). + Given a component Xj = Γj\D ⊂ XK associated to hj as in (4.6), let Γj(v) = Γj ∩ Hv(Q); then the natural map + + (4.10) Γj(v)\Dv → Γj\D

defines an (algebraic) cycle on XK that we denote by c(v, hj,K). We require a final bit of notation: the moment matrix attached to v ∈ Vr is defined to be 1 (4.11) T (v) := 2 Q(vi, vj) i,j ∈ Symr(F ).

Given T ∈ Symr(F ), we let r (4.12) ΩT (V) = {v ∈ V |T (v) = T }.

Definition 4.2 ([22]). Fix a non-singular matrix T ∈ Symr(F ), and let ϕf ∈ r K S(V(Af ) ) be a K-invariant Schwartz function. Define the weighted special cycle ` Z(T, ϕf ,K) on XK = Xj by X X −1 Z(T, ϕf ,K) = ϕf (hj v) c(v, hj,K). j v∈ΩT (V) mod Γj Note that this definition is independent of all choices. If K0 ⊂ K is an open subgroup of finite index and π : XK0 → XK is the natural covering map, then ∗ 0 π Z(T, ϕf ,K) = Z(T, ϕ, K ); as a consequence, when K is understood we will denote this cycle simply by Z(T, ϕf ). See [22, §5] for a proof of this and further properties of these cycles.

4.1.3. Let T ∈ Symr(F ) with det T 6= 0. Any collection of vectors v = (v1, . . . , vr) ∈ Vr, with T (v) = T , is necessarily linearly independent. For such v, we have deﬁned in 2.2.3 a form g(σ1(v)) satisfying

c 0 (4.13) dd g(σ1(v)) + δDv = ϕ (σ1(v))[2r].

We introduce Green forms that depend on an auxiliary parameter y ∈ Symr(R)>0 t as follows: ﬁx some element α ∈ GLn(R) such that y = α · α and, for a Schwartz r K function ϕf ∈ S(V(Af ) ) , deﬁne a form gj(T, y, ϕf ) on D by setting X −1 (4.14) gj(T, y, ϕf ) = ϕf (hj v) g(σ1(v) · α). v∈ΩT (V) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 45

The convergence of this sum on the complement of Z(T, ϕf ) follows from Lemma 3.8.(a) and the fact that the number of orbits of Γj on Supp(ϕf ) ∩ ΩT (V) is ﬁnite. Note that the deﬁnition is independent of the choice of α, by Proposition 2.6.(f). Moreover, the form gj(T, y, ϕf ) is invariant under the action of Γj by Proposition 2.6.(d), and so it descends to a form on the connected component Xj that, abusing notation, we also denote by gj(T, y, ϕf ). Finally, let g(T, y, ϕf ) denote the form on XK whose restriction to Xj is gj(T, y, ϕf ). Essentially by construction, it is a Green’s form for the cycle Z(T, ϕf ); more precisely, let X −1 0 (4.15) ω(T, y, ϕf ) = ϕf (h v) ϕ (σ1(v) · α)[2r],

v∈ΩT (V) 0 viewed as a diﬀerential form on X. Let Hr be the symmetric space of Gr (see (2.58)); for τ = x + iy ∈ Hr, we set 0 1/2 0 (4.16) gτ = n(x)m(y ) ∈ Pr.

Then ω(T, y, ϕf ) is the T ’th coeﬃcient of the theta function

X T ΘKM(τ; ϕf ) = ω(T, y1, ϕf ) q , (4.17) d d T Y 2πitr(τiσi(T )) where τ = (xi + iyi)1≤i≤d ∈ Hr , q = e , i=1 considered (in much greater generality) by Kudla and Millson [26], who showed that ω(T, y1, ϕf ) is a Poincar´edual form to the special cycle Z(T, ϕf ); hence ω(T, y1, ϕf )−

δZ(T,ϕf ) is exact. Applying the identity (4.13) of currents on D, summing over v with T (v) = T and descending to the Shimura variety XK yields the equation c (4.18) dd g(T, y, ϕf ) + δZ(T,ϕf ) = ω(T, y, ϕf )

of currents on XK .

4.2. Unitary Shimura varieties. The results in the previous section carry over, essentially verbatim, to the unitary case. To describe the setup, suppose that E is a CM extension of the totally real ﬁeld F , and that V is a Hermitian space over E, with Hermitian form Q(·, ·).

For each real embedding σi : F → R, the space Vσi = V ⊗σi,F R is a complex Hermitian space; we assume ( (p, q), if i = 1 (4.19) signature Vσi = (p + q, 0), if i = 2, . . . d. for some integers p, q > 0. Let

D(V) = D(Vσ1 ) := {z ⊂ Vσ1 negative-deﬁnite subspace, dimC z = q} , 46 LUIS E. GARCIA, SIDDARTH SANKARAN

(cf. (2.32)) be the symmetric space attached to the real points of the unitary group

(4.20) H := ResF/Q GU(V).

Just as in Section 4.1, a suﬃciently small compact open subgroup K ⊂ H(Af ) deter- mines a Shimura variety

(4.21) XK := H(Q)\D × H(Af )/K which is quasi-projective, and projective when V is anisotropic; choosing representa- tives h1, . . . , ht for the double coset space H(Q)\H(Af )/K gives a decomposition a a −1 (4.22) X ' Γj\D =: Xj, where Γj := H(Q) ∩ hjKhj . j j

As above we can deﬁne E, sσ1(v), Dv and a special cycle c(v, hj,K) for any vector r r K v ∈ V . Given a K-invariant Schwartz function ϕf ∈ S(V(Af ) ) , and a non-singular matrix T ∈ Herr(E), deﬁne the special cycle

X X −1 (4.23) Z(T, ϕf ) = ϕf (hj v) c(v, hj,K) j v∈ΩT (V) mod Γj exactly as in Deﬁnition 4.2, which is either empty or has components of codimension rq. A Green form g(T, y, ϕf ) for Z(T, ϕf ), depending on an auxiliary parameter y ∈ (Herr)>0, can be deﬁned by asking that its restriction to Xj be given by the t expression (4.14), where α ∈ GLr(C) is such that y = α · α. As before, g(T, y, ϕf ) is independent of the choice of α and descends to a current on XK satisfying (4.18).

4.3. Star products on XK . We now consider both cases and write Symr(F ), case 1 (4.24) Xr(F ) = Herr(E), case 2. Given a place v of F , deﬁne the regular locus

r r (4.25) V(Fv)reg = {v = (v1, . . . , vr) ∈ V(Fv) | dimhv1, . . . , vri = r} . r r Note that V(Fv)reg is open in V(Fv) .

Lemma 4.3. Let r1, r2 be positive integers such that r = r1 + r2 ≤ dim(XK ) + 1. ri For i = 1, 2, let Ti ∈ Xri (F ) be non-singular and let ϕf,i = ⊗pϕp,i ∈ S(V(Af ) ). r Assume that ϕp,1 ⊗ ϕp,2 has support contained in the regular locus V(Fp)reg for some ﬁnite place p of F . Then Z(T1, ϕf,1) and Z(T2, ϕf,2) intersect properly and

g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2) = lim g(T1, y1, ϕf,1)≤M ∧ ω(T2, My2, ϕf,2) M→+∞

+ ω(T1, y1, ϕf,1) ∧ g(T2, y2, ϕf,2)≤M .

Proof. Let vi ∈ ΩTi (V) such that ϕf,i(vi) 6= 0. As remarked in the proof of Propo- sition 2.2, we have g(σ (v )) → g(σ (v )) and ϕ0(M 1/2σ (v )) → δ as 1 i ≤M 1 i 1 2 [2r2rk(E)] Dv2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 47 currents on D, as M → +∞. By our hypothesis on ϕf , the cycles Dv1 and Dv2 intersect transversely, hence the product g(σ (v )) ∧ δ exists and we have 1 1 Dv2 (4.26) g(σ (v )) ∧ δ + ϕ0(σ (v )) ∧ g(σ (v )) 1 1 Dv2 1 1 [2r1rk(E)] 1 2 0 1/2 0 = lim g(σ1(v1))≤M ∧ ϕ (M σ1(v2))[2r rk(E)] + ϕ (σ1(v1))[2r rk(E)] ∧ g(σ1(v2))≤M M→∞ 2 1 (see the discussion after equation (2.48) in [5]). The claim now follows by the same argument as the proof of Lemma 3.8. Theorem 4.4. With the notation and hypotheses of the Lemma 4.3, we have

X y1 g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2) = g(T, ( y2 ) , ϕf,1 ⊗ ϕf,2)

T ∈Xr(F ) T = T1 ∗ ∗ T2 ˜ ∗ ∗ in D (XK ) := D (XK )/(im∂ + im∂). Proof. Lemma 4.3 shows that the left hand side is the limit as M → +∞ of the expression X 0 1/2 ϕf,1(v1)ϕf,2(v2)·(g(σ1(v1) · α1)≤M ϕ (M σ1(v2) · α2)[2r2rk(E)] v ∈Ω ( ) (4.27) 1 T1 V v2∈ΩT2 (V) 0 + ϕ (σ1(v1) · α1)[2r1rk(E)]g(σ1(v2) · α2)≤M ) With the notation of the proof of Theorem 2.9, this expression equals Z X 0 (4.28) ϕf,1(v1)ϕf,2(v2) ν˜ (σ1(v1) · α1, σ1(v2) · α2) γk,M v1∈ΩT1 (V) v2∈ΩT2 (V) and so, up to exact forms ∂α and ∂β, it agrees with the same expression with ∫ γk,M replaced by ∫ , which equals g(T, ( y1 ) , ϕ ⊗ϕ ) . This finishes the proof. γd,M y2 f,1 f,2 ≤M 4.4. The degenerate case. We now turn to the case where det T = 0. Throughout this section we assume that V is anisotropic and that rk(E) = 1. r First, for a tuple v ∈ Vσ1 and a complex parameter s, let r √ 0 X 0 0 √ √ dti (4.29) νe (v; s)[2r−2] := ν ( tivi)[0] ∧ ϕ ( t1v1,..., trvr)[2r−2] 1+s | {z } ti i=1 i’th term omitted r r viewed as a differential form on D × R>0, where t1, . . . , tr are the coordinates on R>0. r Fix a = (a1, . . . , ar) ∈ (R>0) , consider the path r (4.30) γa : [1, ∞) → (R>0) , γa(t) = (ta1, . . . , tar), and define Z ∗ 0 (4.31) g(v, a; s) := γa νe (v; s)[2r−2]. [1,∞) 48 LUIS E. GARCIA, SIDDARTH SANKARAN

r r Proposition 4.5. Let v = (v1, . . . , vr) ∈ Vσ1 and a = (a1, . . . , ar) ∈ R>0. (i) The integral (4.31) converges to a smooth form on D if Re(s) > 0. (ii) As a current, g(v, a; s) extends meromorphically3 to the right half plane Re(s) > 1 − 2 . (iii) If v is non-degenerate, then the current g(v, a; s) is regular at s = 0 and 1 √ √ g(v, a; s)s=0 = g(v · a 2 ) = g( a1v1,..., arvr). 0 (iv) Let r = dimhv1, . . . , vri. The constant term CT g(v, a; s) satisﬁes the current s=0 equation c r−r0 0 1 dd CT g(v, a; s) + δ ∧ (−Ω) = ϕ (v · a 2 )[2r]. s=0 Dv

Proof.√ Part (i) follows immediately from the expression (2.12), which shows that ν0( tv) and its partial derivatives stay bounded as t → +∞. 0 √ √ To show (ii), consider (say) the first term in (4.29). Let v2 = ( a2v2,..., arvr) 0 and r = dimhv2, . . . , vri; then we may rewrite the corresponding contribution in (4.31) as Z ∞ √ −s 0 √ 0 0 dt a1 ν ( ta1v1)[0] ∧ ϕ ( tv2)[2r−2] s+1 1 t ∞ Z √ √ 0 dt (4.32) = a−s ν0( ta v ) ϕ0( tv0 ) − δ ∧ (−Ω)r−1−r 1 1 1 [0] 2 [2r−2] Dv0 s+1 1 2 t ∞ Z √ dt 0 + a−s ν0( ta v ) ∧ δ ∧ (−Ω)r−1−r 1 1 1 [0] s+1 Dv0 1 t 2 Let h ∈ O(r − 1) (case 1) or h ∈ U(r − 1) (case 2) diagonalizing the moment ma- 0 0 00 00 r0 trix T (v2) and write v2 · h = (v2 , 0r−1−r0 ), where v2 ∈ V is non-degenerate. By Proposition 2.5, we have √ √ √ 0 0 0 0 0 00 r−1−r0 (4.33) ϕ ( tv2)[2r−2] = ϕ ( tv2 · h)[2r−2] = ϕ ( tv2 )[2r0] ∧ (−Ω) √ 0 00 and hence Bismut’s asymptotic estimate [4, (3.9)] applied to ϕ ( tv2 )[2r0] shows that √ 0 1 0 0 r−1−r − 2 (4.34) ϕ ( tv2)[2r−2] − δD 0 ∧ (−Ω) = O(t ) v2 √ 0 −2πta1h(sv ) as currents on D. By (2.45) we have ν ( ta1v1)[0] = e 1 ≤ 1 and we conclude 1 that the first integral in (4.32) converges to a current when Re(s) > − 2 . As to the second integral in (4.32), assume first that v1 ∈/ hv2, . . . , vni. Then this integral equals s r−1−r0 (4.35) x Γ(−s, x) ∧ δD 0 ∧ (−Ω) , v2 +∞ −t s−1 where we have set x := 2πa1h(sv1 ) and Γ(s, u) = ∫u e t dt denotes the incomplete Gamma function. Note that when x > 0 the function xsΓ(−s, x) is entire in s. Thus,

3More precisely, we are asserting that for any compactly supported form η, the expression R g(v, a; s) ∧ η admits a meromorphic extension as a function of s, and is continuous in η in the D sense of distributions. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 49

1 to show meromorphic continuation to Re(s) > − 2 of the current (4.35), it suﬃces s to prove that (2πa1h(sv1 )) Γ(−s, 2πa1h(sv1 )) is locally integrable on D for such s.

First note that, letting z be a local coordinate for the divisor Dv1 in D, we have 2 s 1 h(sv1 ) ∼ |z| and hence h(sv1 ) is indeed locally integrable when Re(s) > − 2 . Next, u −t s−1 let γ(s, u) = ∫0 e t dt, so that Γ(s, u) + γ(s, u) = Γ(s) and we can write

(4.36) xsΓ(−s, x) = xsΓ(−s) − xsγ(−s, x).

The integrability now follows from the fact that the function zsΓ(−s)−1γ(−s, z) is entire in both s and z (see [35, §8.2]). 0 √ Now assume that v1 ∈ hv2, . . . , vni. Then ν ( ta1 v1)[0] ∧δD 0 = δD 0 and the second v2 v2 term becomes

∞ −s Z dt 0 a 0 (4.37) a−s · δ ∧ (−Ω)r−1−r = 1 δ ∧ (−Ω)r−1−r 1 s+1 Dv0 Dv0 1 t 2 s 2

for Re(s) > 0; the right hand expression deﬁnes a meromorphic continuation of this term. The remaining terms in (4.29) can be treated in the same manner, and this proves (ii). When v is non-degenerate, the proof of (ii) shows that both integrals on the RHS of (4.32) converge when s = 0, and this establishes (iii). Pr To prove (iv), write g(v, a; s) = i=1 gi(v, a; s) with   Z ∗ 0 √ 0 √ √ dti (4.38) gi(v, a; s) = γa ν ( tivi)[0] · ϕ ( t1v1,..., trvr)[2r−2] 1+s  . [1,∞) | {z } ti i’th term omitted

0 √ √ Consider, say, the ﬁrst term i = 1, and let v = ( a2v2,..., arvr). As in the proof of (ii), we may write (4.39) Z ∞ √ 0 √ 0 0 r−1−r0 dt CT g1(v, a; s) = ν ( a1tv1)[0] ϕ ( tv )[2r−2] − δDv0 ∧ (−Ω) + µ1, s=0 1 t

where

( r−1−r0 − log a1 · δDv0 ∧ (−Ω) , if v1 ∈ hv2, . . . , vri, (4.40) µ1 = √ r−1−r0 g( a1v1) ∧ δDv0 ∧ (−Ω) , otherwise.

Note that Green’s equation (4.13), together with the fact the Ω is closed, implies that (4.41) ( c 0, if v1 ∈ hv2, . . . , vri, dd µ1 = r−1−r0 0 √ r−1−r0 −δDv ∧ (−Ω) + ϕ ( a1v1)[2] ∧ δDv0 ∧ (−Ω) , otherwise. 50 LUIS E. GARCIA, SIDDARTH SANKARAN

On the other hand, as the integral in (4.39) converges uniformly, we may interchange the integration on t with diﬀerentiation, and apply Proposition 2.6.(b), to obtain " Z ∞ √ # c 0 √ 0 0 r−1−r0 dt dd ν ( a1tv1)[0] ϕ ( tv )[2r−2] − δDv0 ∧ (−Ω) 1 t Z ∞ √ c 0 √ 0 0 r−1−r0 dt = dd ν ( a1tv1)[0] ∧ ϕ ( tv )[2r−2] − δDv0 ∧ (−Ω) 1 t Z ∞ √ √ (4.42) d 0 0 0 r−1−r0 = − ϕ ( a1tv1)[2] ∧ ϕ ( tv )[2r−2] − δDv0 ∧ (−Ω) dt 1 dt Z ∞ √ d 0 √ 0 0 = − ϕ ( a1tv1)[2] ∧ ϕ ( tv )[2r−2]dt 1 dt Z ∞ d 0 √ r−1−r0 + ϕ ( a1tv1)[2] ∧ δDv0 ∧ (−Ω) dt. 1 dt 0 √ Note that if v1 ∈ hv2, . . . , vri, then ϕ ( a1tv1)[2] ∧ δDv0 is independent of t. In any case, (4.43) Z ∞ d 0 √ r−1−r0 ϕ ( a1tv1)[2] ∧ δDv0 ∧ (−Ω) dt 1 dt h 0 √ r−1−r0 0 √ r−1−r0 i = lim ϕ ( a1tv1)[2] ∧ δ 0 ∧ (−Ω) − ϕ ( a1v1)[2] ∧ δ 0 ∧ (−Ω) t→∞ Dv Dv c = − dd µ1, and so Z ∞ √ c d 0 √ 0 0 (4.44) dd CT g1(v, a; s) = − ϕ ( a1tv1)[2] ∧ ϕ ( tv )[2r−2]dt. s=0 1 dt Arguing in a similar manner for i = 2, . . . , r, we ﬁnd that (4.45) Z ∞ c X d 0 √ 0 √ √ dd CT g(v, a; s) = − ϕ ( aitvi)[2] ∧ ϕ ( a1tv1,..., artvr)[2r−2]dt s=0 1 dt | {z } i i’th term omitted Z ∞ d 0 √ √ = − ϕ ( a1tv1,..., artvr)[2r]dt 1 dt √ h 0 1 0 1 i = − lim ϕ ( tv · a 2 )[2r] − ϕ (v · a 2 )[2r] t→∞ 0 1 r−r 0 2 = − δDv ∧ (−Ω) + ϕ (v · a )[2r], establishing (iv).

We can now deﬁne a current g(T, y; ϕf ) for possibly singular T , generalizing the constructions in 4.1 and 4.2. We assume that we are in case 1 (all the results proved until the end of this section generalize to case 2 in an obvious way) and begin with the following invariance property. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 51

r Lemma 4.6. Suppose y ∈ Symr(R)>0, and choose elements a = (a1, . . . , ar) ∈ R>0 t r and k ∈ O(r) such that y = kd(a) k. Then, for any v = (v1, . . . , vr) ∈ Vσ1 , the current g(v · k, a; s) depends only on y, i.e. is independent of the choice of k and a. Proof. Let

(4.46) 0 < λ1 < ··· < λr0

denote the distinct eigenvalues of y, and let mi denote the corresponding multiplicity. Given two decompositions y = k · a · tk = k0 · (a0) · tk0, the entries of a and a0 are simply the eigenvalues of y, and so they differ by conjugation by some permutation r matrix σ ∈ Sr. Note that such a matrix acts on D × R>0 by permuting the latter factors, and it is an immediate consequence of the definitions that ∗ 0 0 (4.47) σ νe (v · σ; s) = νe (v; s). 0 Thus, it suffices to assume that a = a = (λ1, . . . , λk), with each eigenvalue appearing with some multiplicity, and so k0 = k · k00 for some matrix of the form θ 00 1 (4.48) k = ∈ O(m1) × · · · × O(mr0 ). θr0 Write 0 0 0 0 0 0 (4.49) v · k = (v , . . . , v , . . . , v , . . . , v ) = (v ,..., v 0 ), 1 m1 r−mr0 +1 r 1 r | {z } m | {z } 1 mr0 so that 0 0 0 (4.50) v · k = (v1 · θ1,..., vr0 · θr). It follows from the rotational invariance of ν0(·) and ϕ0(·), cf. Proposition 2.6.(f) and Proposition 2.5.(f), that ∗ 0 0 γa0 νe (v · k ; s) r0 √ √ √ X −s 0 0 0 0 0 dt = λ ν (λ tv · θ ) ∧ ϕ (λ tv · θ , . . . , λ 0 tv 0 · θ ) i i i i [2mi−2] 1 1 1 r r r [2r−2mi] ts+1 i=1 | {z } i’th term omitted r0 √ √ √ X −s 0 0 0 0 0 dt (4.51) = λ ν (λ tv ) ∧ ϕ (λ tv , . . . , λ 0 tv 0 ) i i i [2mi−2] 1 1 r r [2r−2mi] ts+1 i=1 | {z } i’th term omitted ∗ 0 = γaνe (v · k; s), which implies the lemma. r K Given T ∈ Symr(F ), a Schwartz function ϕf ∈ S(V(Af ) ) , and a parameter t y ∈ Symr(R)>0, write y = k · d(a) · k and consider the sum X −1 (4.52) gj(T, y, ϕf ; s) = ϕf (hj v) g(σ1(v) · k, a; s); v∈ΩT (V) the preceding lemma ensures the construction is independent of the decomposition of y. 52 LUIS E. GARCIA, SIDDARTH SANKARAN

Proposition 4.7. (i) The sum (4.52) converges for Re(s) > 0 to a Γj-invariant current on D, and has meromorphic continuation to Re(s) > −1/2. In particular, the constant term in the Laurent expansion

gj(T, y, ϕf ) := CT gj(T, y, ϕf ; s) s=0

descends to a current on Xj. (ii) Let g(T, y, ϕf ) be the current on XK whose restriction to Xj is gj(T, y, ϕf ). Then c r−rk(T ) dd g(T, y, ϕf ) + δZ(T,ϕf ) ∧ (−Ω) = ω(T, y, ϕf ).

∗ Proof. We proceed as in the proof of Lemma 3.8. Let η ∈ Ac (D) with compact support U. Write ΩT (V) = S1 t S2 with S1 = {v ∈ ΩT (V)| minz∈U Qz(v, v) < 1}, so that, by Proposition 4.5, the sum Z X −1 (4.53) ϕf (hj v) g(σ1(v) · k, a; s)η v∈S1 D

converges and has meromorphic continuation since S1 is ﬁnite. For the same sum where v now runs over S2 we use estimate (2.9), together with the standard argument for the convergence of theta series. For (ii), it suﬃces to prove the given identity, for each component Xj, at the level of Γj-invariant currents on D. The preceding estimates allow us to write ! c c X −1 dd gj(T, y, ϕf ) = dd CTs=0 ϕ(hj v)g(v · k, a; s) (4.54) v X −1 c = ϕ(hj v) dd CTs=0 g(v · k, a; s) v and hence the result follows from Proposition 4.5.(iv). 0 r 0 r0 0 Example 4.8. Consider a tuple v = (v , 0,..., 0) ∈ Vσ1 with v = (v1, . . . , vr ) ∈ Vσ1 non-degenerate. Then

0 0 0 r−r0 νe (v; s)[2r−2] = νe (v ; s)[2r0−2] ∧ (−Ω) r ! (4.55) √ √ 0 r−r0−1 X dti + ϕ t v ,..., t 0 v 0 ∧ (−Ω) ∧ . 1 1 r r [2r0] ts+1 i=r0+1 i

r Let a = (a1, . . . , ar) ∈ R>0 and set α = (a1, . . . , ar0 ). Then (4.56) r Z ∞ √ 0 r−r0 X −s 0 dt r−r0−1 g(v, a; s) = g(v , α; s) ∧ (−Ω) + a ϕ ( t · w) 0 ∧ (−Ω) , i [2r ] ts+1 i=r0+1 1

0 1 √ √ 0 where w = v · α 2 = ( a1v1,..., ar0 vr0 ). As v is non-degenerate, the ﬁrst term is 0 1 holomorphic at s = 0 by Proposition 4.5.(iii), and its value is the current g(v · α 2 ) ∧ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 53

(−Ω)r−r0 . For the second term, write Z ∞ √ Z ∞ √ 0 dt 0 0 1 dt 1 0 2 0 (4.57) ϕ ( t · w)[2r ] s+1 = ϕ ( t · v · α )[2r ] − δDv s+1 + δDv . 1 t 1 t s The ﬁrst term here converges at s = 0, using the estimate (4.34), and so 0 1 r−r0 CT g(v, a; s) = g(v · α 2 ) ∧ (−Ω) s=0 (4.58) r ! X 0 1 r−r −1 0 2 − log ai · δDv ∧ (−Ω) + γ(v · α ), i=r0+1 where Z ∞ √ 0 1 0 0 0 1 dt 2 2 0 (4.59) γ(v · α ) := (r − r ) ϕ ( t · v · α )[2r ] − δDv . 1 t Now consider a matrix of the form τ (4.60) T = with τ ∈ Symr0 (F ) non-degenerate. 0r−r0 Recall that we assume that V is anisotropic; in particular, any v ∈ Vr with T (v) = T is of the form v = (v0, 0,..., 0) with T (v0) = τ. 0 00 0 r0 00 r−r0 Suppose further that ϕf = ϕf ⊗ ϕf with ϕf ∈ S(V(Af ) ) and ϕf ∈ S(V(Af ) ), and y ∈ Symr(R)>0 is of the form y0 (4.61) y = , where y0 ∈ Sym ( ) and y00 ∈ Sym ( ) . y00 t R >0 r−t R >0 It follows from the above discussion that, after descending to the Shimura variety XK , we have the equation of currents 00 h 0 0 r−r0 00 r−r0−1 g(T, y, ϕ ) = ϕ (0)· g(τ, y , ϕ ) ∧ (−Ω) − log(det y )δ 0 ∧ (−Ω) f f f Z(τ,ϕf ) (4.62) Z ∞ h 0 0 i dt r−r0−1i + ω(τ, ty , ϕ ) − δ 0 ∧ (−Ω) . f Z(τ,ϕf ) 1 t 0 0 Since ω(τ, ty , ϕ ) and δ 0 are cohomologous, the term appearing on the second f Z(τ,ϕf ) line above is exact, so that

00 h 0 0 r−r0 g(T, y, ϕf ) =ϕf (0) g(τ, y , ϕf ) ∧ (−Ω) (4.63) 00 r−r0−1i ∗ − log(det y )δ 0 ∧ (−Ω) ∈ D (X )/im∂ + im∂. Z(τ,ϕf ) K

Finally, we note that for T = 0r, the above computation gives the pleasant expression r−1 (4.64) g(0r, y, ϕf ) = − log(det y)ϕf (0) · (−Ω) . 5. Archimedean heights and derivatives of Siegel Eisenstein series Here we prove Theorem 5.1, which now follows easily from the Siegel-Weil formula. 5.1. Siegel Eisenstein series and the Siegel-Weil formula. 54 LUIS E. GARCIA, SIDDARTH SANKARAN

5.1.1. Recall that we have fixed a totally real number field F of degree d and a × × CM extension E; we denote by η : F \AF → {±1} the corresponding quadratic character. We set F, case 1, (5.1) k = E, case 2 and denote by σ1, . . . , σd the archimedean places of k. We fix an m-dimensional k hermitian -vector space (V,Q) such that Vσi is positive definite when i > 1 and (p, 2), case 1, (5.2) sig = Vσ1 (p, q), case 2. with pq 6= 0. From now on we assume that V is anisotropic. We also fix a unitary × × × m character χ : E \ → such that χ| × = η . AE C AF We let Mp2r(AF ) be the metaplectic double cover of Sp2r(AF ) and for a positive integer r we set 0 Mp2r(AF ), case 1, (5.3) Gr(A) = U(r, r)(AF ), case 2. 0 We denote by Pr(A) the standard Siegel parabolic of Gr(A); we have Pr(A) = Mr(A)n Nr(A), with

Mr(A) = {(m(a), )|a ∈ GLr(AF ), = ±1}, (5.4) Nr(A) = {(n(b), 1)|b ∈ Symr(AF )} in case 1 and

(5.5) Mr(A) = {m(a)|a ∈ GLr(AE)},Nr(A) = {n(b)|b ∈ Herr(AE)}

in case 2. Deﬁne a character χV of Mr(A) by (5.6) ( · γ (det a, ψ)−1, if m is odd, χ ((m(a), )) = det a, (−1)m(m−1)/2 det V · A A 1, if m is even,

(here γA denotes the Weil index, see [21]) in case 1 and by χ(m(a)) = χ(det a) in case 2. We extend it to a character of Pr(A) whose restriction to Nr(A) is trivial and 0 set I( , s) = IndGr(A)(χ| · |s) (smooth induction), where the induction is normalized V Pr(A) so that s = 0 belongs to the unitary axis. We say that a section Φ(s) ∈ I(V, s) is standard if its restriction to the standard maximal compact K0 of G0 ( ) is K0 - r,A r A r,A ﬁnite and independent of s.

0 0 5.1.2. Given a standard section Φ(s) ∈ I(V, s) and g ∈ Gr(A), the sum X (5.7) E(g0, Φ, s) = Φ(γg0, s) 0 γ∈Pr(F )\Gr(F ) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 55 converges for Re(s) 0 and admits meromorphic continuation to s ∈ C; the function E(g0, Φ, s) is the Siegel Eisenstein series. It admits a Fourier expansion

0 X 0 (5.8) E(g , Φ, s) = ET (g , Φ, s)

T ∈Xr(F )

with Xr(F ) as in (4.24), and Z 0 0 (5.9) ET (g , Φ, s) = E(n(b)g , Φ, s)ψ(−tr(T b))dn(b), Nr(Q)\Nr(A)

with dn(b) the Haar measure on Nr(AF ) that is self-dual with respect to the pairing (b, b0) 7→ ψ(tr(bb0)). Q 0 Q 0 If det T 6= 0 and Φ = v Φv, then ET (g , Φ, s) = v WT,v(gv, Φv, s), where Z 0 −1 0 (5.10) WT,v(gv, Φv, s) = Φv(wr n(b)g , s)ψ(−tr(T n(b)))dn(b), Re(s) 0. Xr(Fv)

0 For nonsingular T , each WT,v(gv, Φv, s) admits analytic continuation to s ∈ C, and we can write

d 0 X 0 (5.11) ET (g , Φ, s) = ET (g , Φ, s0)v, ds s=s0 v where the sum runs over places of F and we set

0 0 Y 0 d 0 (5.12) E (g , Φ, s0)v = WT,w(g , Φw, s0) · WT,v(g , Φv, s0) T w ds v w6=v s=s0

0 0 Let Kr be the standard maximal compact subgroup of Gr(Fv)(v archimedean) 0 l l described in 2.5.1 and l be the weight of Kr in (3.47). If Φ∞ = Φ ⊗ · · · ⊗ Φ ∈ I(V(R), s) is the normalized vector of parallel weight l, we set

c(s) −ιm/4 0 l (5.13) E(τ, Φf , s) := (det y1 ··· yd) E(gτ , Φf ⊗ Φ∞, s) c(s0) in case 1 and

c(s) Y −1 −ιm/4 0 l (5.14) E(τ, Φf , s) := χσi (yi) (det y1 ··· yd) E(gτ , Φf ⊗ Φ∞, s) c(s0) 1≤i≤d in case 2, where

−ιrs/2 (5.15) c(s) = π Γr(ι(s − s0 + m)/2). Note that the normalization factor c(s) is natural from the point of view of the functional equation of Eisenstein series (cf. [27, Lemma 4.6]). We deﬁne similarly 0 ET (τ, Φf , s) and ET (τ, Φf , s)v. 56 LUIS E. GARCIA, SIDDARTH SANKARAN

0 r 5.1.3. Let ω = ωψ,χ be the Weil representation of Gr(AF ) × U(V(A)) on S(V(AF ) ). r 0 0 For φ ∈ S(V(AF ) ), g ∈ Gr(AF ) and h ∈ U(V(A)), deﬁne the theta series X (5.16) Θ(g0, h; φ) = ω(g0, h)φ(v). v∈V(F )r r 0 0 For any φ ∈ S(V(AF ) ), the function Φ(g ) = ω(g )φ(0) belongs to I(V, s0), where 0 s0 is given by (3.19) and (3.37); these maps deﬁne a Gr(A)-intertwining map λ : r 0 S(V(AF ) ) → I(V, s0). Since V is anisotropic, E(g , λ(φ), s) is regular at s = s0 and Z 0 0 (5.17) E(g , λ(φ), s0) = κ0 Θ(g , h; φ)dh, U(V(F ))\U(V(AF )) where 1, if s0 > 0, (5.18) κ0 = 2, if s0 = 0,

and the Haar measure dh is normalized so that Vol(U(V(F ))\U(V(AF )) = 1; this is the Siegel-Weil formula (see [15, 17, 27, 40]). Comparing the constant terms on both sides shows that 0 0 (5.19) E0(g , λ(φ), s0) = κ0 · ω(g )φ(0).

5.2. Archimedean height pairings. In this section we assume that rk(E) = 1. For a real number M ≥ 1 and a non-degenerate vector v ∈ Vr (r ≤ p + 1), let Z M 0 1/2 dt (5.20) g(σ1(v))≤M = ν (t σ1(v))[2r−2] 1 t

and g(T, y, ϕf )≤M be the diﬀerential form on XK deﬁned in the same way as g(T, y, ϕf ), but replacing g(σ1(v)) with g(σ1(v))≤M . By Lemma 3.8.(b), we have

M→+∞ (5.21) g(T, y, ϕf )≤M −−−−−→ g(T, y, ϕf ) as currents on XK . Let Z p (5.22) Vol(XK , Ω) = Ω . [XK ]

Note that, for any open compact subgroup K ⊂ H(Af ), the line bundle E on D descends to a positive line bundle L on XK with ﬁrst Chern form Ω, and so Vol(XK , Ω) = degL(XK ) is a positive rational number (a positive integer if K is neat). Theorem 5.1. Assume that r ≤ p + 1 and rk(E) = 1 and that det T 6= 0. If T is not totally positive deﬁnite, then r Z (−1) ικ0 p+1−r T 0 g(T, y1, ϕf ) ∧ Ω q = ET (τ, λ(ϕf ), s0)σ1 . 2Vol(XK , Ω) [XK ] GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 57

If T is totally positive definite, then (−1)rικ Z 0 g(T, y , ϕ ) ∧ Ωp+1−rqT 2Vol(X , Ω) 1 f K [XK ] 1 ι = E 0 (τ, λ(ϕ ), s ) − E (τ, λ(ϕ ), s ) log det σ (T ) T f 0 σ1 T f 0 2 1 Proof. By (5.21), in both cases the left hand side equals the limit of ∫ g(T, y , ϕ ) ∧ [XK ] 1 f ≤M Ωp+1−r as M → ∞. For t ≥ 1, define the Schwartz function (5.23) φ = ω(m(t1/2 · 1 ))˜ν ⊗ ϕ ⊗ · · · ⊗ ϕ ∈ S( r ⊗ ) = ⊗ S( r ), t r + + V F R i≤i≤d Vσi where ϕ is the Gaussian for r if i > 1 andν ˜ ∈ S( r ) is given by (3.90). Consider + Vσi Vσ1 0 0 the theta series Θ(gτ , ·; ϕf ⊗φt) in (5.16) and its T -th Fourier coefficient ΘT (gτ , ·; ϕf ⊗ + φt). Note that, for z = hz0 ∈ D (h ∈ U(Vσ1 )), we have p+1−r T g(T, y1, ϕf )≤M (z) ∧ Ω (z)q (5.24) Z M dt 2πtr((t−1)σ1(y1T )) 0 p = e ΘT (gτ , h; ϕf ⊗ φt)α(ty1, . . . , yd) ∧ Ω (z), 1 t where we write −ιm/4 1, case 1, (5.25) α(ty1, . . . , yd) = (det ty1 ··· yd) · Q −1 1≤i≤d χσi (yi) , case 2. Applying the Siegel-Weil formula (5.17) shows that Z 0 p dt 0 (5.26) ΘT (gτ , h; ϕf ⊗ φt)Ω = cET (gτ , λ(ϕf ⊗ φt), s0) [XK ] t for a constant c. To determine this constant we consider the term corresponding to T = 0: we have Z Z 0 p 0 p (5.27) Θ0(g , h; ϕf ⊗ φt)Ω = ω(g )(ϕf ⊗ φt)(0) Ω ; [XK ] [XK ] 0 0 by (5.19), this equals cE0(g , λ(ϕf ⊗ φt), s0) = cκ0ω(g )(ϕf ⊗ φt)(0), and we conclude −1 1/2 that c = κ0 Vol(XK , Ω). Using (3.43) and writing ϕf ⊗φt = ω(m(t ·1r))˜ν ⊗v6=σ1 φv, we find that Z M dt 2πtr((t−1)σ1(y1T )) 0 e ET (g , λ(ϕf ⊗ φt), s)α(ty1, . . . , yd) 1 t Y 0 (5.28) = α(y1, . . . , yd) WT,v(gv, λ(φv), s) v6=σ1 Z M dt 2πtr((t−1)σ1(y1T )) −ιrm/4 · e WT,σ1 (gτ1 , λ(φt), s)t . 1 t l r−1 1/2 For i ≥ 2 we have λ(φσi ) = λ(ϕ+) = Φ (s0). By (3.93) we have λ(φt) = (−1) r(m(t · ˜ 1r))Φ(s0). The limit of the integral as M → ∞ can be computed using Lemma 3.6 and (3.89). 58 LUIS E. GARCIA, SIDDARTH SANKARAN

Note that the sum in Theorem 4.4 runs over non-singular matrices T by the hypothesis on ϕf,1 ⊗ ϕf,2. In particular, if r1 + r2 = p + 1, then all such T are indeﬁnite. Thus Theorem 5.1 has the following corollary generalizing the main result of [23] to arbitrary compact orthogonal Shimura varieties (for U(p, 1) this corollary is due to Liu [33]).

Corollary 5.2. Assume that rk(E) = 1 and r1 + r2 = p + 1. Under the hypotheses of Theorem 4.4, we have

p+1 (−1) ι T1 T2 hg(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2), 1iq1 q2 Vol(XK , Ω)

X 0 τ1 = E (( τ ) , λ(ϕ ⊗ ϕ ), 0) . T 2 f,1 f,2 σ1 T ∈Xr(F ) T = T1 ∗ ∗ T2

References [1] Nicole Berline, Ezra Getzler, and Michèle Vergne. Heat kernels and Dirac operators, volume 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992. URL: http://dx.doi.org/10.1007/ 978-3-642-58088-8, doi:10.1007/978-3-642-58088-8. [2] J.-M. Bismut, H. Gillet, and C. Soulé.Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion. Comm. Math. Phys., 115(1):49–78, 1988. URL: http: //projecteuclid.org/euclid.cmp/1104160849. [3] J.-M. Bismut, H. Gillet, and C. Soulé.Bott-Chern currents and complex immersions. Duke Math. J., 60(1):255–284, 1990. URL: https://doi.org/10.1215/S0012-7094-90-06009-0. [4] Jean-Michel Bismut. Superconnection currents and complex immersions. Invent. Math., 99(1):59–113, 1990. URL: http://dx.doi.org/10.1007/BF01234412, doi: 10.1007/BF01234412. [5] Jean-Michel Bismut, Henri Gillet, and Christophe Soulé.Complex immersions and Arakelov geometry. In The Grothendieck Festschrift, Vol. I, volume 86 of Progr. Math., pages 249–331. BirkhäuserBoston, Boston, MA, 1990. [6] J.-B. Bost, H. Gillet, and C. Soulé.Heights of projective varieties and positive Green forms. J. Amer. Math. Soc., 7(4):903–1027, 1994. URL: https://doi.org/10.2307/2152736. [7] J. Bruinier, B. Howard, S. S. Kudla, M. Rapoport, and T. Yang. Modularity of generating series of divisors on unitary Shimura varieties. URL: https://arxiv.org/abs/1702.07812. [8] J. Bruinier, B. Howard, S. S. Kudla, M. Rapoport, and T. Yang. Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications. URL: https://arxiv.org/ abs/1710.00628. [9] J. I. Burgos Gil, J. Kramer, and U. Kühn.Cohomological arithmetic Chow rings. J. Inst. Math. Jussieu, 6(1):1–172, 2007. URL: https://doi.org/10.1017/S1474748007000011. [10] Stephan Ehlen and Siddarth Sankaran. On two arithmetic theta lifts. URL: https://arxiv. org/abs/1607.06545. [11] Luis E. Garcia. Superconnections, theta series, and period domains. To appear in Advances in Mathematics. URL: https://arxiv.org/abs/1604.03897. [12] Henri Gillet and Christophe Soulé.Arithmetic intersection theory. Inst. Hautes Etudes´ Sci. Publ. Math., (72):93–174 (1991), 1990. URL: http://www.numdam.org/item?id=PMIHES_1990__72_ _93_0. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 59

[13] Michael Harris and Stephen S. Kudla. Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2). Duke Math. J., 66(1):59–121, 1992. URL: http://dx.doi.org/10. 1215/S0012-7094-92-06603-8, doi:10.1215/S0012-7094-92-06603-8. [14] Michael Harris, Stephen S. Kudla, and William J. Sweet. Theta dichotomy for unitary groups. J. Amer. Math. Soc., 9(4):941–1004, 1996. URL: http://dx.doi.org/10.1090/ S0894-0347-96-00198-1, doi:10.1090/S0894-0347-96-00198-1. [15] Atsushi Ichino. A regularized Siegel-Weil formula for unitary groups. Math. Z., 247(2):241–277, 2004. URL: https://doi.org/10.1007/s00209-003-0580-5. [16] Atsushi Ichino. Pullbacks of Saito-Kurokawa lifts. Invent. Math., 162(3):551–647, 2005. URL: http://dx.doi.org/10.1007/s00222-005-0454-z, doi:10.1007/s00222-005-0454-z. [17] Atsushi Ichino. On the Siegel-Weil formula for unitary groups. Math. Z., 255(4):721– 729, 2007. URL: http://dx.doi.org/10.1007/s00209-006-0045-8, doi:10.1007/ s00209-006-0045-8. [18] M. Kashiwara and M. Vergne. On the Segal-Shale-Weil representations and harmonic polynomials. Invent. Math., 44(1):1–47, 1978. URL: http://dx.doi.org/10.1007/BF01389900, doi:10.1007/BF01389900. [19] Stephen Kudla and Michael Rapoport. Special cycles on unitary Shimura varieties I. Un- ramiﬁed local theory. Invent. Math., 184(3):629–682, 2011. URL: https://doi.org/10.1007/ s00222-010-0298-z. [20] Stephen Kudla and Michael Rapoport. Special cycles on unitary Shimura varieties II: Global theory. J. Reine Angew. Math., 697:91–157, 2014. URL: https://doi.org/10.1515/ crelle-2012-0121. [21] Stephen S. Kudla. Splitting metaplectic covers of dual reductive pairs. Israel J. Math., 87(1-3):361–401, 1994. URL: http://dx.doi.org/10.1007/BF02773003, doi:10.1007/ BF02773003. [22] Stephen S. Kudla. Algebraic cycles on Shimura varieties of orthogonal type. Duke Math. J., 86(1):39–78, 1997. URL: http://dx.doi.org.ezproxy.cul.columbia.edu/10.1215/ S0012-7094-97-08602-6, doi:10.1215/S0012-7094-97-08602-6. [23] Stephen S. Kudla. Central derivatives of Eisenstein series and height pairings. Ann. of Math. (2), 146(3):545–646, 1997. URL: http://dx.doi.org/10.2307/2952456, doi:10.2307/2952456. [24] Stephen S. Kudla and John J. Millson. The theta correspondence and harmonic forms. I. Math. Ann., 274(3):353–378, 1986. URL: http://dx.doi.org/10.1007/BF01457221, doi:10.1007/ BF01457221. [25] Stephen S. Kudla and John J. Millson. The theta correspondence and harmonic forms. II. Math. Ann., 277(2):267–314, 1987. URL: http://dx.doi.org/10.1007/BF01457364, doi:10.1007/ BF01457364. [26] Stephen S. Kudla and John J. Millson. Intersection numbers of cycles on locally symmetric spaces and Fourier coeﬃcients of holomorphic modular forms in several complex variables. Inst. Hautes Etudes´ Sci. Publ. Math., (71):121–172, 1990. URL: http://www.numdam.org/item?id= PMIHES_1990__71__121_0. [27] Stephen S. Kudla and Stephen Rallis. On the Weil-Siegel formula. J. Reine Angew. Math., 387:1–68, 1988. URL: http://dx.doi.org/10.1515/crll.1988.391.65, doi:10.1515/crll. 1988.391.65. [28] Stephen S. Kudla and Stephen Rallis. Degenerate principal series and invariant distributions. Israel J. Math., 69(1):25–45, 1990. URL: http://dx.doi.org/10.1007/BF02764727, doi:10. 1007/BF02764727. [29] Stephen S. Kudla, Michael Rapoport, and Tonghai Yang. Derivatives of Eisenstein series and Faltings heights. Compos. Math., 140(4):887–951, 2004. URL: http://dx.doi.org/10.1112/ S0010437X03000459, doi:10.1112/S0010437X03000459. 60 LUIS E. GARCIA, SIDDARTH SANKARAN

[30] Stephen S. Kudla, Michael Rapoport, and Tonghai Yang. Modular forms and special cycles on Shimura curves, volume 161 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2006. [31] Soo Teck Lee. On some degenerate principal series representations of U(n, n). J. Funct. Anal., 126(2):305–366, 1994. URL: http://dx.doi.org/10.1006/jfan.1994.1150, doi:10.1006/ jfan.1994.1150. [32] Soo Teck Lee and Chen-Bo Zhu. Degenerate principal series and local theta correspondence. Trans. Amer. Math. Soc., 350(12):5017–5046, 1998. URL: http://dx.doi.org/10. 1090/S0002-9947-98-02036-4, doi:10.1090/S0002-9947-98-02036-4. [33] Yifeng Liu. Arithmetic theta lifting and L-derivatives for unitary groups, I. Algebra Number Theory, 5(7):849–921, 2011. URL: https://doi.org/10.2140/ant.2011.5.849. [34] Varghese Mathai and Daniel Quillen. Superconnections, Thom classes, and equivariant differential forms. Topology, 25(1):85–110, 1986. URL: http://dx.doi.org/10.1016/0040-9383(86) 90007-8, doi:10.1016/0040-9383(86)90007-8. [35] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark, editors. NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). [36] Daniel Quillen. Superconnections and the Chern character. Topology, 24(1):89–95, 1985. URL: http://dx.doi.org/10.1016/0040-9383(85)90047-3, doi:10.1016/0040-9383(85) 90047-3. [37] S. Rallis. On the Howe duality conjecture. Compositio Math., 51(3):333–399, 1984. URL: http: //www.numdam.org/item?id=CM_1984__51_3_333_0. [38] Goro Shimura. Confluent hypergeometric functions on tube domains. Math. Ann., 260(3):269– 302, 1982. URL: http://dx.doi.org/10.1007/BF01461465, doi:10.1007/BF01461465. [39] C. Soulé. Lectures on Arakelov geometry, volume 33 of Cambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 1992. With the collabora- tion of D. Abramovich, J.-F. Burnol and J. Kramer. URL: http://dx.doi.org/10.1017/ CBO9780511623950, doi:10.1017/CBO9780511623950. [40] William Jay Sweet, Jr. The metaplectic case of the Weil-Siegel formula. ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Ph.D.)–University of Maryland, College Park. URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info: ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9110358. [41] Nolan R. Wallach. Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals. In Representations of Lie groups, Kyoto, Hiroshima, 1986, volume 14 of Adv. Stud. Pure Math., pages 123–151. Academic Press, Boston, MA, 1988. [42] Raymond O. Wells, Jr. Differential analysis on complex manifolds, volume 65 of Graduate Texts in Mathematics. Springer, New York, third edition, 2008. With a new appendix by Oscar Garcia-Prada. URL: https://doi.org/10.1007/978-0-387-73892-5.