Green Forms and the Arithmetic Siegel-Weil Formula 2

Home , Induced representation, Tautological bundle

arXiv:1808.09313v1 [math.NT] 28 Aug 2018 RE OM N H RTMTCSEE-ELFORMULA SIEGEL-WEIL ARITHMETIC THE AND FORMS GREEN .Lclacieenhihsaddrvtvso iglEise Siegel of derivatives and heights archimedean Local 5. varieties Shimura on cycles special for forms Green 4. functi Whittaker of derivatives and heights Archimedean 3. domains symmetric hermitian on forms Green 2. .Introduction 1. ..BhvorudrteWi representation Weil the under forms Behaviour the of properties 2.5. Basic O( unitary for and formulas 2.4. orthogonal Explicit of c domains Koszul symmetric 2.3. of Hermitian forms characteristic and 2.2. Superconnections 2.1. Acknowledgments conventions and 1.3. Notation results 1.2. Main 1.1. Sh compact for and, codimensions, all in varieties, Shimura Abstract. ..Fuircecet fsaa egtEsnti series Eisenstein weight scalar formula of Siegel-Weil coeﬃcients the Fourier and series Eisenstein 5.2. Siegel 5.1. on products Star varieties Shimura 4.5. cycles Unitary special for forms 4.4. Green cycles 4.3. Special varieties Shimura 4.2. Orthogonal 4.1. of integral The functionals 3.4. U( Whittaker of series principal 3.3. Mp Degenerate of series principal 3.2. Degenerate 3.1. products Star forms 2.7. Green 2.6. h ato i ojcueivliglclacieenheig archimedean local involving conjecture his s of of part intersections A the arithmetic to series. derivatives Eisentein these Siegel relates of derivatives of values special n U( and p, ) eso httersliglclacieenhih pair height archimedean local resulting the that show we 1), ecntutntrlGenfrsfrseilcce northo in cycles special for forms Green natural construct We USE ACAADSDAT SANKARAN SIDDARTH AND GARCIA E. LUIS g o X K p, Contents )adU( and 2) ϕ 1 and ,r r, 2 r ( ) p, R ν ) 1) eilcce,adorrslssettle results our and cycles, pecial ojcueptfradb Kudla by forward put conjecture hts. mr aite ftp O( type of varieties imura senseries nstein onals nsaerltdto related are ings omplexes oa n unitary and gonal groups p, 2) 59 56 55 54 51 47 47 45 45 40 33 30 28 27 25 21 18 15 14 12 8 7 7 6 2 2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 2

5.3. Archimedean height pairings 75 5.4. Classes in arithmetic Chow groups 80 5.5. Kudla’s arithmetic height conjecture 81 References 83

1. Introduction The Arakelov theory of Shimura varieties has been intensively studied since, about twenty years ago, Kudla [23] launched a program relating families of special cycles in their arithmetic Chow groups with derivatives of Eisenstein series and Rankin-Selberg L- functions. This paper is a contribution to the archimedean aspects of this theory. Building upon previous work of the first author [11], we use Quillen’s formalism of superconnections [39] as developed by Bismut-Gillet-Soulé[2, 3] to construct natural Green forms for special cycles, in all codimensions, on orthogonal and unitary Shimura varieties. We show that these forms have good functorial properties and are compatible with star products. Furthermore, for compact Shimura varieties of type GSpin(p, 2) or U(p, 1), we compute the corresponding local archimedean heights of special cyles explicitly. These results provide compelling evidence for Kudla’s conjectural identity, termed the arithmetic Siegel-Weil formula, between special derivatives of Eisenstein series and generating series of arithmetic heights of special cycles. More precisely, we show that the non-holomorphic terms on both sides are equal for these Shimura varieties. Our methods combine Quillen’s extension of Chern-Weil theory with the theory of the theta correspondence. This allows us to use representation theoretic arguments and the Siegel-Weil formula when computing archimedean local heights. In this way, we avoid the highly involved computations that feature in prior work, and give a conceptual explanation for the equality of non-holomorphic terms in Kudla’s conjectural identities. 1.1. Main results. The remainder of the introduction outlines our results in more detail. Throughout the paper, we treat GSpin(p, 2) and U(p,q) Shimura varieties in parallel; these are referred to as the orthogonal and unitary cases, respectively. Let F denote a totally real field of degree [F : Q] = d and let E be a CM extension of F equipped with a fixed CM type. Suppose that V is a quadratic space over F in the orthogonal case (resp. a Hermitian space over E in the unitary case). We assume that there is one archimedean place σ1 such that Vσ1 satisfies the signature condition (p, 2) with p> 0 orthogonal case, signature(Vσ1 )= (1.1.1) ((p,q) with p,q > 0 unitary case, and V is positive definite at all other archimedean places. Let Res GSpin(V), orthogonal case, H = F/Q (1.1.2) (ResF/Q U(V), unitary case, GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 3

and D be the hermitian symmetric domain attached to H(R); concretely, D parametrizes oriented negative-deﬁnite real planes (resp. negative deﬁnite complex q-dimensional sub-

spaces) in Vσ1 in the orthogonal (resp. unitary) case. For a fixed compact open subgroup K H(A ), let X be the corresponding Shimura variety, which has a canonical model ⊂ f V,K over σ1(F ) (resp. σ1(E)). Then XV,K is a finite disjoint union X = Γ D (1.1.3) V,K \ of quotients of D by certain arithmetic subgroupsa Γ H(Q). Let denote the variety ⊂ XK obtained by viewing the canonical model of XV,K as a variety over Spec(Q). The complex points (C)= X (1.1.4) XK V[k],K are a finite disjoint union of Shimura varietiesa attached to V[1] := V and its nearby spaces V[k] (see Section 4). The variety is equipped with a family of rational special cycles XK Z(T, ϕ ) , (1.1.5) { f } as defined by Kudla [22], that are parametrized by pairs (T, ϕf ) consisting of a matrix r K T Symr(F ) (resp. T Herr(E)) and a K-invariant Schwartz function ϕf (V(Af ) ) . These∈ cycles generalize∈ the construction of Heegner points on modular curves∈ and S Hirzebruch- Zagier cycles on Hilbert modular surfaces. The irreducible components of the cycle Z(T, ϕf ) on XV,K (say) admit a complex uni- formization by certain complex submanifolds Dv of D defined as follows: for a collection of vectors v = (v , . . . , v ) Vr satisfying 1 r ∈ v 1 T ( ) := 2 Q(vi, vj) i,j=1,...,r = T, (1.1.6) let Dv := z D z v for all i = 1, . . . , r , (1.1.7) { ∈ | ⊥ i } so that Dv, if non-empty, is a hermitian symmetric subdomain of D of codimensionr ˜ := rk(T ) in the orthogonal case (resp.r ˜ := q rk(T ) in the unitary case). · As a first step towards defining a Green current for Z(T, ϕf ), we construct a current go(v) on D satisfying the equation

c o r−rk(T ) o dd g (v) + δ v Ω ∨ = [ϕ (v)]. (1.1.8) D ∧ E KM Here δDv is the current deﬁned by integration along Dv and ΩE∨ is the top Chern form o of the dual of the tautological bundle , see Section 2.2 below; the form ϕKM(v) := 2πtr(T (v)) E 1 r,˜ r˜ e ϕKM(v) is, up to a normalizing factor, the Schwartz form ϕKM(v) A (D) introduced by Kudla and Millson [27]. ∈ In recent work [11], the ﬁrst author introduced a superconnection v on D and showed ∇ o that the component of degree (˜r, r˜) of the corresponding Chern form agrees with ϕKM(v).

1 This means that ϕKM(v) and all its derivatives are of exponential decay in v. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 4

This allows us to apply the general results of [2] to obtain an explicit natural form νo(v) satisfying the transgression formula o d o ddc ν (√t v) = t ϕ (√t v), t R . (1.1.9) − dt KM ∈ >0 o o Moreover, the forms ν (tv) and ϕ (tv) are of exponential decay in t on D Dv. To define go(v), assume first that T is a non-degenerate matrix (so that\r = rk(T )) and consider the integral ∞ o o dt g (v) := ν (√tv) , (1.1.10) 1 t Z o initially defined on D Dv. The estimates in [3] show that g (v) is smooth on D Dv, locally integrable on D, and\ satisfies (1.1.8), which in this case reduces to Green’s equation;\ in o other words, g (v) is a Green form for Dv, in the terminology of [6, 1.1]. When T is degenerate, the expression (1.1.10) is not locally integrable§ on D in general. We will circumvent this problem by regularizing the integral and obtain a current go(v) satisfying (1.1.8) for every v, see Section 2.6.1. Returning to the special cycles Z(T, ϕf ), we define currents g(T, y, ϕf ) on K(C) as weighted sums of go(v) over vectors v satisfying T (v)= T ; these currents also dependX on a parameter y Symr(F Q R)≫0 or y Herr(E Q R)≫0 in the orthogonal or unitary cases, respectively.∈ ⊗ ∈ ⊗

Theorem 1.1.1. The current g(T, y, ϕf ) satisﬁes c r−rk(T ) dd g(T, y, ϕ ) + δ Ω ∨ = ω(T, y, ϕ ), (1.1.11) f Z(T,ϕf )(C) ∧ E f where ω(T, y, ϕf ) is the T ’th coeﬃcient in the q-expansion of the theta function attached to ϕ ϕ . KM ⊗ f Moreover, if T1 and T2 are non-degenerate and Z(T1, ϕ1) and Z(T2, ϕ2) intersect properly, then

y1 c g(T , y , ϕ ) g(T , y , ϕ ) g (T, ( y ) , ϕ ϕ ) (modim d + im d ). 1 1 1 ∗ 2 2 2 ≡ 2 1 ⊗ 2 T =( T1 ∗ ) X∗ T2 Note that our construction is valid for all T ; for example, when V is anisotropic we obtain ∨ ∗ r−1 g(0, y, ϕf ) X (C) = ϕf (0) log(det σk(y)) crk(E)−1( , ) Ω ∨ (1.1.12) | K,σk − · E ∇ ∧ E for each real embedding σk of F . When T is non-degenerate, (1.1.11) is Green’s equation for the cycle Z(T, ϕf ) and hence g(T, y, ϕf ) is a Green current (in fact, a Green form) for Z(T, ϕf ). When T is degenerate, the cycle Z(T, ϕf ) appears in the “wrong” codimension; following [22], this deficiency can be rectified by intersecting with a power of the tautological bundle and, as we discuss in Section 5.4, solutions to (1.1.11) correspond naturally to Green currents for this modified cycle. Our main result computes local archimedean heights of special cycles in terms of Siegel Eisenstein series. We restrict our attention to the case that V is anisotropic, so that the GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 5

corresponding Shimura variety is compact; our assumption on the signature of V ensures that this is the case whenever F = Q. We further assume that q = 1 in the unitary case. Fix an integer r p + 1 and let 6 ≤ p + 1 r s := − , (1.1.13) 0 2 r K and, for a Schwartz function ϕf (V(Af ) ) , consider the corresponding genus r Siegel Eisenstein series ∈ S E(ττ, Φf ,s) = ET (ττ, Φf ,s) (1.1.14) XT of parallel scalar weight l = dimF (V)/2 (resp. l = dimE(V)), see Section 5.2; here τ = x + iy Hd, where H is the Siegel (resp. Hermitian) upper half-space of genus r. Let ∈ r r ′ d ET (ττ, Φf ,s0) := ET (ττ, Φf ,s) (1.1.15) ds s=s0 denote the derivative of its Fourier coefficient E (ττ, Φ ,s) at s = s . T f 0 Theorem 1.1.2. Suppose that V is anisotropic and, in the unitary case, that q = 1. Then for any T , there is an explicit constant κ(T, Φf ), given by Definition 5.2.11, such that ( 1)rκ − 0 g(T, y, ϕ ) Ωp+1−r qT = E′ (ττ, Φ ,s ) κ(T, Φ ) qT . (1.1.16) 2Vol(X , Ω ) f ∧ E T f 0 − f V,K E Z[XK (C)] T 2πitr(Tττ ) Here q = e , and κ0 = 1 if s0 > 0 and κ0 = 2 if s0 = 0. As a special case, suppose that T is non-degenerate, so that there is a factorization E (ττ, Φ ,s) = W (ττ, Φl ,s) W (e, Φ ,s), (1.1.17) T f T,∞ ∞ · T,f f where the factors on the right are the products of the archimedean and non-archimedean local Whittaker functionals, respectively. Let E′ (ττ, Φ ,s ) = W ′ (ττ, Φl ,s ) W (e, Φ ,s ) (1.1.18) T f 0 ∞ T,∞ ∞ 0 · T,f f 0 denote the archimedean contribution to the special derivative. Then Theorem 1.1.2 specializes to the identity ( 1)rκ − 0 g(T, y, ϕ ) Ωp+1−rqT = E′ (ττ, Φ ,s ) (1.1.19) 2Vol(X , Ω ) f ∧ E T f 0 ∞ V,K E Z[XK (C)] if T is not totally positive definite, and to ( 1)rκ − 0 g(T, y, ϕ ) Ωp+1−rqT 2Vol(X , Ω ) f ∧ E V,K E Z[XK (C)] ιd Γ′ (ιm/2) ι = E′ (ττ, Φ ,s ) E (ττ, Φ ,s ) r log π r + log N det T T f 0 ∞ − T f 0 2 − Γ (ιm/2) 2 F/Q r (1.1.20) if T is totally positive definite; here ι = 1 (resp. ι = 2) in the orthogonal (resp. unitary) case. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 6

When T is non-degenerate, the proof of the theorem can be summarized as follows: the current g(T, y, ϕf ) is given by a sum of integrals of the form (1.1.10), for vectors v with T (v)= T . Interchanging the order of integration, the Siegel-Weil formula relates the left hand side of (1.1.16) to the Fourier coeﬃcient ET (ττ, Φ(ν),s0) of an Eisenstein series attached to the Schwartz form o ν(v)= e−2πtr(T (v))ν (v). (1.1.21)

We then analyze the behaviour of ν(v) under the action of the metaplectic group Mp2r(R) (resp. the unitary group U(r, r)) via the Weil representation. A multiplicity one argument ′ allows us to identify Φ(ν) explicitly, and in turn relate ET (ττ, Φ(ν),s0) to ET (ττ, Φf ,s0) via a lowering operator. To conclude the proof, we apply work of Shimura [43] to derive asymptotic estimates for the Fourier coefficients ET (ττ, Φf ,s) as y . When T is degenerate, the idea is roughly the same, though addition→∞al care is required in handling the regularization, as well as establishing the required asymptotics of ET (ττ, Φf ,s). Prior results of this form have appeared in only a few special cases in the literature. For divisors, the Green function we define specializes to the one defined by Kudla [23], and Theorem 1.1.2 was proved in [31] for Shimura curves; a related result for U(p, 1) Shimura varieties over imaginary quadratic fields was proved by Ehlen and the second author [9]. In higher codimension much less was known. For (arithmetic) codimension two cycles on Shimura curves, Kudla [23] defined Green currents using star products; this construction does not coincide with ours, but does agree modulo exact currents by Theorem 1.1.1, and Theorem 1.1.2 reduces to results proved by elaborate explicit computations in [23] and [32]. Similar methods were used by Liu [35] for arithmetic codimension p + 1 cycles on U(p, 1) to prove a star product version of the particular case of the theorem given by (1.1.19). Again in the non-degenerate case, a recent preprint of Bruinier and Yang [14] proves a star product version of (1.1.19) for arithmetic codimension p+1 cycles on O(p, 2) by a different argument involving induction on p. Finally, we place our results in the context of Kudla’s conjectures on special cycles in arithmetic Chow groups. Putting aside the difficult issues involved in constructing integral models, the integral appearing in Theorem 1.1.2 is the archimedean contribution to the height of an arithmetic cycle lifting Z(T, ϕf ); according to Kudla’s conjectural arithmetic Siegel-Weil formula, this height should equal the Fourier coefficient of an appropriately normalized version of the Eisenstein series appearing above. The remaining contribution to the arithmetic height is purely algebro-geometric in nature, and in particular should be independent of y; thus Theorem 1.1.2 asserts that the non-holomorphic terms in Kudla’s conjectural identity coincide. Put another way, our theorem reduces Kudla’s conjecture to a relatively explicit conjectural formula for the analogue of the Faltings height (as in [6]) of a special cycle Z(T, ϕf ) in terms of T and ϕf ; we discuss this point in more detail in Section 5.5.

1.2. Notation and conventions. Let K be a ﬁeld endowed with a (possibly trivial) involution a a. We write Symr(K) (resp. Herr(K)) for the group of symmetric (resp. hermitian) r-by-7→ r matrices with coeﬃcients in K under matrix addition. For a GL (K) ∈ r GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 7

and b Her (K), let ∈ r a 0 1 b 1 m(a)= , n(b)= r , w = r . (1.2.1) 0 ta¯−1 0 1 r 1 r − r For x = (x ,...,x ) Kr, we write 1 r ∈ x1 . d(x) = diag(x1,...,xr)=  ..  . (1.2.2) x  r   × We fix the following standard choice of additive character ψ = ψF : F C when F is a 2πix → local field of characteristic zero. If F = R we set ψ(x)= e ; if F = Qp we choose ψ = ψQp −1 −2πi/p 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 A× for the ideles of F and set ψ : F × A× C×, where F F \ F → 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) on a smooth manifold X. Given α A∗(X) = Ak(X), we write α for its component of ∈ ⊕k≥0 [k] 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∂∂. − − We denote by ∗ the operator on Ak,k(X) acting by multiplication by ( 2πi)−k on ⊕k≥0 − 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, which we will denote by [ω] or simply 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.

1.3. Acknowledgments. This work was done while L. G. was at the University of Toronto and IHES and S.S. was at the University of Manitoba; the authors thank these institutions for providing excellent working conditions. An early draft, with a full proof of the main identity for non-degenerate Fourier coeﬃcients, was circulated and posted online in January 2018; we are grateful to Daniel Disegni, Stephen Kudla and Shouwu Zhang for comments on it, and other helpful conversations. S.S. acknowledges ﬁnancial support from NSERC.

2. Green forms on hermitian symmetric domains Let X be a complex manifold and Z X be a closed irreducible analytic subset of codimension c. A Green current for Z is a⊂ current g Dc−1,c−1(X) such that Z ∈ c dd gZ + δZ = [ωZ], (2.0.1) 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 (see [6, 1.1]). − § GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 8

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 (or as the orthogonal case) and to the case involving U(p,q) as case 2 (or as the unitary case). Our methods apply uniformly in both cases. There is a natural hermitian holomorphic vector bundle over D, and the submanifolds of D that we consider are the zero loci Z(s) of certain naturalE holomorphic sections s H0(( ∨)r) (r 1). In this setting, the results in [4, 2, 3] (reviewed in Section 2.1) can∈ be appliedE to≥ construct some natural differential forms on D, 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, 26]), and then (Sections 2.4 and 2.5) we establish the main properties of these forms. Using these results, in Section 2.6 we define some currents related to Z(s), including a Green form for Z(s). The final Section 2.7 considers star products and will be used in the proof of Theorem 4.5.1.

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 ( , u), where is a holomorphic hermitian vector bundle and u is a holomorphic section of itsE dual. TheE results in this section are due to Quillen [39], Bismut [4] and Bismut-Gillet- Soulé[2, 3]. We will use Quillen’s formalism of superconnections and related notions of superalgebra. For more details, the reader is referred to [39, 1]. We briefly recall that a super vector space V is just a complex Z/2Z-graded vector space; we write V = V V and refer to V and 0 ⊕ 1 0 V1 as the even and odd part of V respectively. We write τ for the endomorphism of V deg(v) determined by τ(v) = ( 1) v. The supertrace trs : End(V ) C is the linear form defined by − →

trs(u) = tr(τu), (2.1.1) a b where tr denotes the usual trace. Thus if u = c d with a End(V0), d End(V1), b Hom(V ,V ) and c Hom(V ,V ), then tr (u) = tr(a) tr(d∈). ∈ ∈ 1 0 ∈ 0 1 s − 2.1.1. Let be a holomorphic vector bundle on a complex manifold X and u H0( ∨) be a holomorphicE section of its dual ∨. Let K(u) be the Koszul complex of u: its∈ underlyingE vector bundle is the exterior algebraE and its diﬀerential u : k k−1 is deﬁned by ∧E ∧ E→∧ E u(e e )= ( 1)i+1u(e )e eˆ e . (2.1.2) 1 ∧···∧ k − i 1 ∧···∧ i ∧···∧ k 1≤Xi≤k The grading on K(u) is given by K(u)−k = k , so that K(u) is supported in non-positive ∧ E degrees. We identify K(u) with the corresponding complex of sheaves of sections of k , and say that u is a regular section if the cohomology of K(u) vanishes in negative degrees.∧ E If u is regular with zero locus Z(u), then K(u) is quasi-isomorphic to (regarded as OZ(u) a complex supported in degree zero). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 9

2.1.2. Assume now that is endowed with a hermitian metric E . This induces a E k k · k j hermitian metric on : for any x X, the subspace x is orthogonal to x if j = k, and an orthonormal∧E basis of ∈k is given by all∧ elementsE e e ,∧ whereE 6 ∧ Ex i1 ∧···∧ ik 1 i1 < < ik rk and e1,...,erkE is an orthonormal basis of x. Let be the corresponding≤ · · · Chern≤ connectionE { on . We} regard as a super vectorE bundle, with∇ even part even and odd part odd , and∧E u as an odd∧E endomorphism of . Let u∗ be the adjoint∧ of Eu, deﬁne the superconnection∧ E ∧E = + i√2π(u + u∗) (2.1.3) ∇u ∇ on , and consider Quillen’s Chern form ∧E o o 2 ϕ (u)= ϕ ( , , u)=tr (e∇u ) Ak,k(X). (2.1.4) E k · kE s ∈ Mk≥0 We recall some properties of ϕo(u) established (in greater generality) by Quillen [39]. The form ϕo(u) is closed and functorial: given a holomorphic map of complex manifolds f : X′ X, consider the pullback bundle (f ∗ ,f ∗ ) and the pullback section f ∗u H0(f ∗ →∨). Then E k · k ∈ E o o ϕ (f ∗u)= f ∗ϕ (u). (2.1.5) ∗ k,k −k k,k Let be the operator on k≥0A (X) acting by multiplication by ( 2πi) on A (X). Writing [ϕo(u)∗] for the cohomology⊕ class of ϕo(u)∗ and ch( ) for the− Chern character, we have · o [ϕ (u)∗] = ch( ) = ch( even ) ch( odd ). (2.1.6) ∧E ∧ E − ∧ E 2.1.3. In particular, [ϕo(u)∗] depends on , but not on u. Thus the forms ϕo(tu) for E t R>0 all belong to the same cohomology class, but as t + they concentrate on the zero∈ locus of u. More precisely, recall that 2 is an even element→ ∞ of the (super)algebra ∇u ∗ A(X, End( )) := A (X) ˆ ∞ Γ(End( )). (2.1.7) ∧E ⊗C (X) ∧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 on A(X, End( )) k · kU,E,k ∧E measuring uniform convergence on U of partial derivatives of order at most k. We will 2 ∇tu 2 2 need an estimate of e U,E,k for large t. To obtain it, write tu = ( tu)[0] + Rtu, where 2 k k ∇ ∇ ( tu)[0] has form-degree zero and Rtu has form-degree 1. Note that Rtu is nilpotent and that∇ ≥ 2 ∗ 2 2 2 0 ( ) = 2π(tu + tu ) = 2πt u ∨ id A (X) End( ) (2.1.8) ∇tu [0] − − k kE ⊗ ∈ ⊗ ∧E ∨ ∨ (here E∨ denotes the unique metric on such that the isomorphism induced by k · kis an isometry). In particular, ( 2E) and R commute. HenceE we≃ haveE k · kE ∇tu [0] tu 2 2 e∇tu = e(∇tu)[0] eRtu N (2.1.9) −2πt2kuk2 1 k = e k! Rtu Xk=0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 10

2 with N dim X. Let a be any positive real number strictly less than min u(x) ∨ . ≤ R x∈U {k kE } Since Rtu is polynomial in t, it follows from (2.1.9) that

2 2 e∇tu Ce−2πat , t R (2.1.10) k kU,E,k ≤ ∈ >0 for some positive real number C. Thus a similar bound holds for ϕo(tu).

d o 1/2 2.1.4. It follows from (2.1.6) 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 Soulé [2] construct such a transgression, and the resulting form is key to our results. To define it, let N End( ) be the number operator acting on k by multiplication by k and set ∈ ∧E ∧ E − o 2 ν (u)=tr (Ne∇u ) Ak,k(X). (2.1.11) s ∈ Mk≥0 Then νo(u) is functorial with respect to holomorphic maps f : X′ X and satisfies ([2, Thm. 1.15]) →

1 o d o ∂∂ν (t1/2u)= ϕ (t1/2u), t> 0. (2.1.12) − t dt

2.1.5. Assume that the section u has no zeroes on X. Then the Koszul complex K(u) is acyclic and (2.1.6) shows that ϕo(u) is exact. In this case one can deﬁne a characteristic form ξo(u) giving a ∂∂-transgression of ϕo(u) by setting

+∞ o o dt ξ (u)= ν (t1/2u) . (2.1.13) t Z1 The bound (2.1.10) implies that all partial derivatives of νo(t1/2u) decrease rapidly as t k,k → + . In particular, the integral converges and deﬁnes a form in k≥0A (X); moreover, one∞ can diﬀerentiate under the integral sign, so that by (2.1.12) we⊕ have

o o ∂∂ξ (u)= ϕ (u). (2.1.14)

The form ξo is functorial with respect to holomorphic maps X′ X. In addition, ξo(tu) (for t R ) is rapidly decreasing as t . More precisely, given→ a relatively compact ∈ >0 → ∞ open subset U of X with closure U and a positive integer k, we denote by any k · kU,k seminorm on A∗(X) measuring uniform convergence on U of partial derivatives of order at most k. The bound (2.1.10) shows that

o 2 ξ (tu) Ce−2πat for all t 1, (2.1.15) k kU,k ≤ ≥ for some positive constants a and C. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 11

2.1.6. We now go back to the general case where is a hermitian holomorphic vector bundle on X and u H0( ∨), and now make the assumptionE that u is regular (see Sec- tion 2.1.1) and the zero∈ locusE Z(u) is smooth; thus K(u) is a resolution of and the OZ(u) codimension of Z(u) (if non-empty) in X equals rk( ). Define o o ∗ E g (u)= ξ (u)[2rk(E)−2] ∞ rk(E)−1 o dt (2.1.16) = i ν (t1/2u) . 2π [2rk(E)−2] t Z1 The following proposition is contained in [3]. It shows that go(u) is a Green form for Z(u). Proposition 2.1.7. (1) The integral (2.1.16) converges to a smooth differential form go(u) Ark(E)−1,rk(E)−1(X Z(u)). (2) The form∈ go(u) is locally integrable− on X. (3) As currents on X we have c o o ∗ dd g (u)+ δZ(u) = ϕ (u)[2rk(E)], c i where dd = 2π ∂∂. 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]. In that paper, one considers an immersion i: M ′ M of complex manifolds, a vector bundle η on M ′ and a complex (ξ, v) of holomorphic→ hermitian vector bundles on M that gives a resolution of i ′ (η). Assume that Z(u) is non-empty. We set M = X, M ′ = Z(u), η = , and ∗OM OZ(u) for the complex (ξ, v) we take the Koszul complex ( , u); with these choices, the form o 1/2 ∧E ν (t u) agrees with the form αt defined in [3, (3.12)]. Let z Z(u), choose local coordinates z ,...,z (N = dim X) around z such that 0 ∈ 1 N 0 Z(u) is defined by the equations z = = z = 0 and let y = ( z 2 + + z 2)1/2. 1 · · · rk(E) | | | 1| · · · | rk(E)| Since the complex normal bundle to Z(u) in X is canonically identified with , the equations [3, (3.24), (3.26)] show that E +∞ o du y 2rk(E)−1 ν (t1/2u) (2.1.17) | | u Z1 is bounded in a neighborhood of z ; part (2) follows since y −(2rk(E)−1) is locally integrable 0 | | around z0. To prove (3), let β be a compactly supported form on X. Then a o c o 1/2 ∗ c dt g (u) dd β = lim ν (t u)[2rk(E)−2] dd β X ∧ a→+∞ 1 X ∧ t Z Z a Z c o 1/2 ∗ dt = lim dd ν (t u)[2rk(E)−2] β a→+∞ 1 X ∧ t Z a Z (2.1.18) d o dt = lim ( t ϕ (t1/2u)∗ ) β a→+∞ − dt [2rk(E)] ∧ t Z1 ZX o o = ϕ (u)∗ β lim ϕ (a1/2u)∗ β. [2rk(E)] ∧ − a→+∞ [2rk(E)] ∧ ZX ZX GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 12

Here the first equality follows from dominated convergence since the integrals a o du y 2rk(E)−1 ν (t1/2u)p (2.1.19) | | ∫1 u for a 1 are uniformly bounded in a neighborhood of z0, as shown in the proof of [3, Thm. ≥ o 1/2 ∗ 3.3]. The third equality follows from (2.1.12). This establishes (3) since ϕ (a u)[2rk(E)] approaches δ as a + , as shown by [3, (1.14), (1.19)]. Z(u) → ∞ 2.2. Hermitian symmetric domains of orthogonal and unitary groups. 2.2.1. Let R case 1 K = (2.2.1) C case 2 and id case 1 σ = (2.2.2) 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 = 0 and in case 1 we further assume that q = 2. Let U(V ) := Aut(V,Q) denote its isometry6 group, and set SO(p, 2)0 case 1 G = U(V )0 = (2.2.3) ∼ U(p,q) case 2. We fix an orthogonal decomposition V = V + V − 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 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, K = × (2.2.4) U(V +) U(V −), case 2. ( × Let Gr(q, V ) be the Grassmannian of oriented two-dimensional subspaces of V in case 1, which consists of two copies of the usual Grassmannian. In case 2, we take Gr(q, V ) to be the space of q-dimensional (complex) subspaces of V . Let D = z Gr(q, V ) Q < 0 (2.2.5) { ∈ | |z } 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 D [v] P(V (C)) Q(v, v) = 0, Q(v, v) < 0 . (2.2.6) ≃ { ∈ | } ′ The correspondence between both models sends z D to the line [ez + iez] P(V (C)), ′ ∈ ∈ ′ ′ where ez and ez form an oriented orthogonal basis of z satisfying Q(ez, ez)= Q(ez, ez). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 13

2.2.2. Let be the tautological bundle on D, whose fiber over z D is the subspace z V . ThusE is a holomorphic line bundle in case 1 (for it corresponds∈ to the pullback of⊂ ( 1)E under the isomorphism (2.2.6)), and a holomorphic vector bundle of rank OP(V (C)) − q in case 2. It carries a natural hermitian metric hE defined by Q(v , v ) case 1 h (v )= z z (2.2.7) E z −Q(v , v ) case 2 − z z for vz z = z. This metric is equivariant for the natural U(V )-equivariant structure on . We∈ denote E by the corresponding Chern connection on and by E ∇E E rkE Ω=Ω := ctop( , )∗ = i det( 2 ) ArkE,rkE (D) (2.2.8) E E ∇E 2π ∇E ∈ its Chern-Weil form of top degree. We denote by ΩD the Kähler form

ΩD = ∂∂ log kD(z, z), (2.2.9) where kD is the Bergmann kernel function of D. As shown in [48, p. 219], the invariant form i Ω agrees with the first Chern form c (Ωtop) of the canonical bundle Ωtop on any − 2π D 1 X X quotient X = Γ D+ by a discrete torsion free subgroup Γ G. When has rank one, the canonical bundle\ on D is naturally isomorphic to ⊗p in⊂ case 1 (as anE application of E the adjunction formula shows) and to ⊗(p+1) in case 2, and so in both cases i Ω is a E − 2π D positive integral multiple of ΩE . An element v V defines a global holomorphic section s of ∨: for v′ , we define ∈ v E z ∈Ez ′ ′ sv(vz)= Q(vz, v). (2.2.10) Let D be the zero locus of s on D and set D+ = D D+. We have v v v v ∩ D = z D v z (2.2.11) v { ∈ | ⊥ } 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 for the stabilizer of v 0 ⊥ 0 − + 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) in case 2. Thus Dv is the symmetric − × 0 − × ∨ domain attached to Gv. We conclude that codimDDv = rk and hence that sv is a regular section of ∨. In fact, this shows that s is regular wheneverE v = 0, since in the remaining E v 6 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 is a ∨ r ∈ holomorphic section sv = (s ,...,s ) of ( ) , with zero locus v1 vr E Dv := Z(sv)= D . (2.2.12) ∩1≤i≤r vi Note that Dv depends only on the span v , . . . , v . It is non-empty if and only if h 1 ri v1, . . . , vr is a a positive definite subpace of (V,Q) of positive dimension; in that case, its h i + + (complex) codimension in D is rk( ) dim v , . . . , v . We set D = Dv D . E · Kh 1 ri v ∩ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 14

2.2.3. We will now specialize the constructions in Section 2.1 to the setting of hermitian symmetric domains. 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 = (sv1 ,...,svr ) of ( r)∨. On its underlying vector bundle ( r), we consider the superconnection E ∧ E ∗ v := v = + i√2π(sv + s ), (2.2.13) ∇ ∇s ∇ v and we deﬁne forms

2 o o ∗ i k ∇v ϕ (v) := ϕ (sv) = 2π trs(e )[2k], Xk≥0 2 (2.2.14) o o ∗ i k ∇v ν (v) := ν (sv) = 2π trs(Ne )[2k], k≥0 X where N is the number operator on ( r) acting on k( r) by multiplication by k. ∧ E ∧ E − r Deﬁnition 2.2.4. For v = (v1, . . . , vr) V , write Q(v, v) = Q(v1, v1)+ + Q(vr, vr) and deﬁne ∈ · · · o ϕ(v)= e−πQ(v,v) ϕ (v), o ν(v)= e−πQ(v,v) ν (v).

Thus ϕ(v) and ν(v) belong to Ak,k(D). ⊕k≥0 The forms ϕ(v) and ν(v) were already deﬁned and studied in the setting of general period domains in [11].

2.3. Explicit formulas for O(p, 2) and U(p, 1). Let us give some explicit formulas when the tautological bundle is a line bundle. Thus V is a either a real vector space of signature (p, 2) (case 1) orE a complex vector space of signature (p, 1) (case 2). There is a unique hermitian metric on ∨ making the isomorphism = ∨ induced by h an isometry; E E ∼ E E we denote this metric by h and its Chern connection by ∨ . For v V we have ∇E ∈

−π(Q(v,v)+2h(sv )) ∂h(sv) ∂h(sv) ϕ(v)[2] = e i ∧ ΩE , h(sv) − (2.3.1) ϕ(v)= ϕ(v) Td−1( ∨, )∗, [2] ∧ E ∇ where 2 −∇E∨ −1 ∨ 1 e Td ( , ) = det − 2 (2.3.2) E ∇ ∨ ∇E ! ∨ denotes the inverse Todd form of ( , E∨ ). This is a special case of the Mathai-Quillen formula [36, Thm. 8.5]; see also [11E, 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 [26]. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 15

Let us now consider the form ν(v) for v V . Here the Koszul complex K(v) has just sv ∈ two terms: K(v) = ( D), and the operator N acts by zero on D and by 1 on . For the component ofE degree−→O zero of νo(v) we obtain O − E o 2 ∇v ν (v)[0] = trs(Ne )[0] 2 (∇ )[0] = trs(Ne v ) (2.3.3) −2πh(sv ) = trs(Ne ) = e−2πh(sv ). 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

Q (v, v)= Q(v ⊥ , v ⊥ ) Q(v , v ). (2.3.4) z z z − z z 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) ν(v)[0] = ϕ (v) := e . (2.3.5) o If v = 0, then using (2.3.3) we also obtain an explicit formula for the Green function g (sv) defined6 in (2.1.16) (see Example 2.6.3 below). 2.4. Basic properties of the forms ϕ and ν. 2.4.1. We 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 may identify Dw 0 ′ ′ with the hermitian symmetric domain attached to Gw. We write ϕDw (v ) and νDw (v ) for the forms on Dw given in Definition 2.2.4. Lemma 2.4.2. Let v, w V with Q(w, w) > 0 and write v = v′ + v′′ with v′ w⊥ and v′′ w . Then ∈ ∈ ∈ h i o o ′′ ′′ ν (v) = ν (v′), ν(v) = e−πQ(v ,v )ν (v′), |Dw Dw |Dw Dw o o ′′ ′′ ϕ (v) = ϕ (v′), ϕ(v) = e−πQ(v ,v )ϕ (v′). |Dw Dw |Dw Dw ⊥ Proof. Let w be the tautological bundle on Dw whose fiber over z Dw is z w . The restrictionE of to D is isometric to . For any v V , this isometry∈ induces⊂ an E w Ew ∈ isomorphism K(v) = K(v′), where K(v′) denotes the Koszul complex with underlying |Dw ∼ vector bundle and differential s ′ . Thus = ′ and the lemma follows. ∧Ew v ∇v|Dw ∇v 2.4.3. The proof of the next proposition is a straightforward consequence of general properties of Koszul complexes and Chern forms. Proposition 2.4.4. Let r 1 and v = (v , . . . , v ) V r. Then: ≥ 1 r ∈ (a) ϕ(v)= ϕ(v1) . . . ϕ(vr). (b) ϕ(v) is closed.∧ ∧ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 16

(c) ϕ(v)[k] = 0 if k< 2r rk( ). · E ∗ (d) For every g Aut(V,Q), we have g ϕ(gv1, . . . , gvr)= ϕ(v1, . . . , vr). (e) ϕ(0) = c ∈( ∨, )∗ Td−1( ∨, )∗ (here we assume r = 1). rk(E) E ∇ ∧ E ∇ (f) Let O(r), case 1 h ∈ U(r), case 2. Then ϕ((v , . . . , v ) h)= ϕ(v , . . . , v ). 1 r · 1 r Proof. Except for (c), which follows from (a) and the Mathai-Quillen formula [36, 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 c(F, ) = det(t 2 + 1 )=1+ c (F, )t + . . . + c (F, )trk(F ) (2.4.1) ∇F ∇F rk(F ) 1 ∇F rk(F ) ∇F for the total Chern-Weil form of a vector bundle F with connection . ∇F Proposition 2.4.5. Let r 1 and v = (v , . . . , v ) V r. ≥ 1 r ∈ (a) We have ν(v)= 1≤i≤r νi(v), where ν (v) := ν(v ) ϕ(v ,..., vˆ , . . . , v ). P i i ∧ 1 i r (b) For t> 0: o d o ddcν (t1/2v)= t ϕ (t1/2v). − dt (c) ν(v)[k] = 0 if k< 2r rk( ) 2. · E − ∗ (d) For any g Aut(V,Q), we have g ν(gv1, . . . , gvr)= ν(v1, . . . , vr). (e) For the zero∈ vector 0 V r, we have ∈ ν(0) = r c ( ∨, )∗ c ( ∨, )∗ r−1 . [2r·rk(E)−2] · rk(E)−1 E ∇ ∧ rk(E) E ∇ In particular, ν(0)[0] = 1 when r = rk( ) = 1. E (f) Let O(r), case 1 h ∈ U(r), case 2. Then ν((v , . . . , v ) h)= ν(v , . . . , v ). 1 r · 1 r 2 −πQ(v,v) ∇v Proof. Recall that ν(v) = e trs(Ne ), where N is the number operator on the Koszul complex K(v) 1≤i≤rK(vi). Letting Ni be the number operator on K(vi), we can write N = N + ≃+ ⊗N and 2 = 2 + + 2 , where [N , 2 ] = [ 2 , 2 ] = 0 1 · · · r ∇v ∇v1 · · · ∇vr i ∇vj ∇vi ∇vj for i = j; this proves (a). Part6 (b) follows from (2.1.12). By part (a) and Proposition 2.4.4.(c), it suﬃces to prove (c) when r = 1. Then (c) is vacuously true if rk( ) = 1, and in general it follows from [5, (3.72),(3.35), Thm. 3.10]. E For any v and any g U(V ), the U(V )-equivariant structure on induces an isomorphism g∗K(gv) K(v)∈ preserving the metric; this proves (d). E ≃ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 17

For part (e), first consider the case r = 1. When also has rank one, the desired relation follows immediately from (2.3.5). For general , byE taking u = 0 in [5, (3.35)] we find that E ∇2 ∗ d i 2 tr (Ne ) = det ∨ b1 s [2rk(E)−2] −db 2π ∇E − rk(E) b=0 d = ( 1)r−1 det i 2 + b1 − db 2π ∇E rk(E) b=0 (2.4.2) = ( 1)r−1c ( , )∗ − rk(E)−1 E ∇ = c ( ∨, )∗. rk(E)−1 E ∇ Note that the number operator N in op. cit. has sign opposite from ours. The formula for general r follows from part (a) above together with Proposition 2.4.4(a),(e). To prove (f), note that h induces an isometry i(h) K(v , . . . , v ) K((v , . . . , v ) h) (2.4.3) 1 r −−→ 1 r · that commutes with N and such that = i(h)−1 i(h). Thus (f) follows ∇(v1,...,vr)·h ∇(v1,...,vr) from the conjugation invariance of trs. The properties of ϕ(v) and ν(v) established in Propositions 2.4.4 and 2.4.5 also hold for ϕo(v) and νo(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 hold (note o for example that the restriction of ϕ (tv) to Dv is independent of t R). Let (V r) be the Schwartz space of complex-valued smooth functions∈ on V r all whose derivativesS are rapidly decreasing. Lemma 2.4.6. For fixed z D and r 1, we have ∈ ≥ ϕ( , z),ν( , z) (V r) T ∗D. · · ∈ S ⊗∧ z Proof. By Proposition 2.4.4.(a) and Proposition 2.4.5.(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. Write 2(z) = ( 2) (z)+ S(v, z). By Duhamel’s formula ([44, p. 144]) we have ∇v ∇v [0] 2 e−πQ(v,v)e∇v(z) = e−2πQz(v)

k −2π(1−tk )Qz(v) −2π(tk −tk−1)Qz(v) −2πt1 Qz(v) + ( 1) e S(v, z)e S(v, z)e dt1 dtk. − ∆k · · · · · · Xk≥1 Z (2.4.4) 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 ≤ ≤ be≤ an} algebra seminorm as in Section 2.1.2 k · kU,k and let Q a positive deﬁnite quadratic form on V such that Q

(with diﬀerent C) and hence that the same bound holds for φ(v) where φ(v) is any k kU,k 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 identity r r-by-r matrices respectively. Consider the vector space Wr := K endowed with the σ- skew-Hermitian form determined by 0 1 r . (2.5.1) 1 0 − r Its isometry group is the symplectic group Sp (R)= g GL (R) g 0 1r tg = 0 1r (2.5.2) 2r ∈ 2r −1r 0 −1r 0 in case 1 and the quasi-split unitary group

U(r, r)= g GL (C) g 0 1r tg = 0 1r (2.5.3) ∈ 2r −1r 0 −1r 0 in case 2. Denote by Mp (R) the metaplectic double cover of Sp (R ), and identify it (as 2r 2r a set) with Sp (R) 1 as in [40]. Deﬁne 2r × {± }

′ Mp2r(R), orthogonal case Gr = (2.5.4) (U(r, r), unitary case. ′ Let Nr and Mr be the subgroups of Gr given by N = (n(b), 1) b Sym (R) (2.5.5) r { | ∈ r } M = (m(a),ǫ) a GL (R),ǫ = 1 (2.5.6) r { | ∈ r ± } in case 1 and by

Nr = n(b) b Herr , { | ∈ } (2.5.7) M = m(a) a GL (C) r { | ∈ r } in case 2. ′ ′ We also ﬁx a maximal compact subgroup Kr of Gr as follows. In the orthogonal case, ′ let Kr be the inverse image under the metaplectic cover of the standard maximal compact subgroup a b − a + ib U(r) = U(r) (2.5.8) b a ∈ ∼ ′ ′ ′ of Sp2r(R). In the unitary case, we deﬁne Kr = Gr U(2r). Thus in this case Kr ≃ ∩ ≃ U(r) U(r); an explicit isomorphism U(r) U(r) K′ is given by × × −→ r 1 k1 + k2 ik1 + ik2 (k1, k2) [k1, k2] := − . (2.5.9) 7→ 2 ik1 ik2 k1 + k2 − GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 19

2.5.2. Recall that we have ﬁxed the additive character ψ(x)= e2πix; in the unitary case, × m ′ we also ﬁx a character χ = χV of C such that χ R× = sgn( ) . Then Gr U(V ) acts on (V r) via the Weil representation | · × S ω , orthogonal case, ω = ψ (2.5.10) (ωψ,χ, unitary case. (see [13, 1]). Here the action of U(V ) is in both cases given by § ω(g)φ(v)= φ(g−1v), g U(V ), φ (V r). (2.5.11) ∈ ∈ S ′ To describe the action of Gr, we write (m(a), 1), for a GL (R) in case 1, m(a)= ∈ r m(a), for a GL (C) in case 2, ( ∈ r (n(b), 1), for b Sym (R) in case 1, n(b)= ∈ r (2.5.12) n(b), for b Her in case 2, ( ∈ r (wr, 1), case 1, wr = (wr, case 2. Let denote the normalized absolute value on K; thus z = z and z = zz. Then | · |K | |R | | | |C m/2 χψ(det a) case 1 ω(m(a))φ(v) = det a K φ(v a) | | · · (χ(det a) case 2, (2.5.13) ω(n(b))φ(v) = ψ (tr(bT (v))) φ(v), ˆ ω(wr)φ(v) = γV r φ(v);

see [21] and also [17, 4.2] for explicit formulas for χψ and γV r . Here for v = (v1, . . . , vr) 1 § ˆ we deﬁne T (v)= 2 (Q(vi, vj)) and φ(v) denotes the Fourier transform ˆ 1 φ(v) = φ(w) ψ 2 trC/Rtr(Q(v, w)) dw, (2.5.14) r ZV where dw is the self-dual Haar measure onV r with respect toψ.

′ 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 [26], where it is shown that ϕ spans a ′ one-dimensional representation of Kr. We will show that ν also generates an irreducible ′ representation of Kr. To describe it we next recall the parametrization of irreducible ′ representations of Kr by highest weights; we denote by πλ the (unique up to isomorphism) representation with highest weight λ. ′ Consider ﬁrst the orthogonal case. Then Kr is a double cover of U(r) and its irreducible representations are πλ with λ = (l ,...,l ) Zr ( 1 + Z)r, l l . (2.5.15) 1 r ∈ ∪ 2 1 ≥···≥ r GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 20

k/2 ′ 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×. Kudla and Millson [26, Thm. 3.1.(ii)] show that → ω(k′)ϕ = det(k′)m/2ϕ, k′ K′ , (2.5.16) ∈ r ′ and so ϕ aﬀords the one-dimensional representation πl of Kr with highest weight m l := 2 (1,..., 1). (2.5.17) Now consider the unitary case. Using the isomorphism (2.5.9), any irreducible represen- ′ ⊠ tation of Kr is isomorphic to 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 | | ω([k , k ])ϕ = (det k )(m+k(χ))/2(det k )(−m+k(χ))/2ϕ, k , k U(r), (2.5.18) 1 2 1 2 1 2 ∈ ′ and so ϕ generates the one-dimensional representation πl of Kr with highest weight m+k(χ) −m+k(χ) l := ( 2 (1,..., 1), 2 (1,..., 1)). (2.5.19) ′ We will now determine the Kr representation generated by the Schwartz form ν(v)[2r−2]. Recall that by Proposition 2.4.5 we can write

ν(v)= νi(v), (2.5.20) 1≤Xi≤r where −πQ(v ,v ) ∇2 ν (v)= e i i tr (N e vi ) ϕ(v ,..., vˆ , . . . , v ). (2.5.21) i s i ∧ 1 i r Let ǫ = 1 and, for 1 i < r, set r r ≤ 1i−1 0 1 ǫi =   SO(r). (2.5.22) 1r−i−1 ∈  1 0  −  Then   νi(v)= ω(m(ǫi))νr(v), 1 i r; (2.5.23) ′ ≤ ≤ thus ν(v)[2r−2] belongs to the Kr-representation generated by νr(v)[2r−2]. ′ Deﬁne a weight λ0 of Kr by setting λ := ( m ,..., m , m 2) (2.5.24a) 0 2 2 2 − in the orthogonal case, and λ := (( m+k(χ) ,..., m+k(χ) , m+k(χ) 1), ( −m+k(χ) ,..., −m+k(χ) , −m+k(χ) + 1)) (2.5.24b) 0 2 2 2 − 2 2 2 in the unitary case. Lemma 2.5.4. Let r 1 and assume that rk( ) = 1. Under the action of K′ via the Weil ≥ E r representation, the form ν(v)[2r−2] generates an irreducible representation πλ0 with highest weight λ0. The form νr(v)[2r−2] is a highest weight vector in πλ0 . GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 21

Proof. Assume ﬁrst that r = 1. By (2.3.5), ν(z)[0] is the Siegel gaussian, which has weight λ = p−q = m 2 (2.5.25) 0 2 2 − in case 1 and λ = ( p−q+k(χ) , q−p+k(χ) ) = ( m+k(χ) 1, −m+k(χ) + 1) (2.5.26) 0 2 2 2 − 2 in case 2. Now assume that r> 1. By Proposition 2.4.4.(c), we have ν (v) = ν(v ) ϕ(v , . . . , v ) (2.5.27) r [2r−2] r [0] · 1 r−1 [2r−2] and hence by (2.5.16), (2.5.18), (2.5.25) and (2.5.26), 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 ∆+ (see (3.1.4) and (3.2.6)). This can be done by a ∈ c direct computation using (2.5.13) and the explicit formulas (2.3.5) and (2.3.1) for ν[0] and ϕ, or alternatively as follows. Evaluating at z0, (2.3.5) and (2.3.1) show that ν (v, z ) (S(V r) p∗)K; (2.5.28) r 0 ∈ ⊗∧ r r −πQ (v,v) here S(V ) (V ) is the subspace spanned by functions of the form e z0 p(v), ⊂ S 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 [19, III.6] (see also [12, − ′ § Prop. 4.2.1] in case 1) that the only Kr-representation containing the weight λ0 realized in S(V r) in Howe degree 2r 2 is π , and the statement follows. − λ0 2.6. Green forms.

2.6.1. In this section we will construct certain currents on D depending on a parameter r v = (v1, . . . , vr) V and having singularities at Dv. We begin with the following special case. ∈ Deﬁnition 2.6.2. A tuple v = (v , . . . , v ) V r is non-degenerate if v , . . . , v is linearly 1 r ∈ { 1 r} independent. We say that v is regular if Dv is either empty or of codimension r in D. Note that the latter occurs if and only if v is non-degenerate and v1, . . . , vr span a positive deﬁnite subspace of V .

r ∨ If v is regular, then sv is a regular section of ( ) in the sense of Section 2.1.1, and Dv is smooth. Therefore, setting E ∞ o o dt r·rk(E)−1,r·rk(E)−1 g (v) := ν (√tv) A (D Dv), (2.6.1) [2r·rk(E)−2] t ∈ − Z1 o o Proposition 2.1.7 shows that g (v)= g (sv) is smooth on D Dv, locally integrable on D and, as a current, satisﬁes − c o o dd g (v)+ δDv = ϕ (v)[2r rk(E)], (2.6.2) o so that g (v) is a Green form for Dv. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 22

Example 2.6.3. Let v = 0 V and assume that rk( ) = 1. Then, using (2.3.3), we compute 6 ∈ E +∞ +∞ o o dt dt g (v)= ν (t1/2v) = e−2πth(sv ) = Ei( 2πh(s )), (2.6.3) [0] t t − − v Z1 Z1 ∞ −zt dt o where Ei( z) = 1 e t denotes the exponential integral. Thus g (v) coincides with the Green− form defined−∫ in [23, (11.24)]. ⋄ 2.6.4. If v is no longer assumed regular, then the integral in (2.6.1) is often no longer convergent. We shall overcome this deficiency by regularization: for any v V r and ρ C with Re(ρ) > 0, define ∈ ∈ +∞ o o dt g (v; ρ)= ν (√tv) . (2.6.4) [2r·rk(E)−2] tρ+1 Z1 As νo(tv) is bounded as t , locally uniformly on D, this integral defines a smooth form on D. We will show that→ ∞ it admits a meromorphic continuation (as a current) to a neighbourhood of ρ = 0, beginning first with the case r = 1. Lemma 2.6.5. For v V with v = 0 and a complex parameter ρ, consider the integral on ∈ 6 D Dv: ∞ − o o dt g (v; ρ) := ν (√tv) . [2 rk(E)−2] t1+ρ Z1 For Re(ρ) > 1/2, this integral converges to a locally integrable form on D, and the convergence is locally− uniform on D and in ρ. o ∇2 Proof. Recalling that ν (v)=trs(Ne v ), it follows by taking s = 1 in (2.1.9) that we may write rk(E)−1 o k −2πth(sv) ν (√tv)[2rk(E)−2] = t e ηk(v) (2.6.5) Xk=0 for some differential forms ηk(v) that are smooth on D. For convenience, set x = 2πh(sv); then rk(E)−1 ∞ rk(E)−1 ∞ o k−ρ −tx dt ρ−k k−ρ −t dt g (v; ρ)= t e ηk(v)= x t e ηk(v). (2.6.6) 1 t · x t Xk=0 Z Xk=0 Z The forms ηk(v) are locally bounded since they are smooth, and a straightforward computation in local coordinates, as in the proof of Proposition 2.1.7, implies that xρ−k is locally integrable for Re(ρ) > 1/2 and k rk( ) 1. As for the integrals, write − ≤ E − ∞ dt 1 dt 1 dt ∞ dt tk−ρe−t = tk−ρ tk−ρ 1 e−t + tk−ρe−t t t − − t t Zx Zx Zx Z1 (2.6.7) 1 xk−ρ 1 dt ∞ dt = − tk−ρ 1 e−t + tk−ρe−t . k ρ − − t t − Zx Z1 1 1 The latter two integrals are absolutely bounded uniformly in x and in ρ for 2 < Re(ρ) < 2 , say, while the first term is evidently holomorphic on this region. In particular,− multiplying GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 23

ρ−k 1 1 each of these terms by x yields locally integrable functions for 2 < Re(ρ) < 2 ; since the more direct estimate covers the case Re(ρ) > 0, this proves the− lemma. Proposition 2.6.6. Let v = (v , . . . , v ) V r and let r′ = dim v , . . . , v . 1 r ∈ h 1 ri (i) The integral (2.6.4) converges to a smooth form on D if Re(ρ) > 0. (ii) Let k O(r) (case 1) or k U(r) (case 2). Then ∈ ∈ o o g (v k; ρ)= g (v; ρ). · o 2 1 (iii) As a current, g (v; ρ) extends meromorphically to the right half plane Re(ρ) > 2 . (iv) If v is regular, then the current go(v; ρ) is regular at ρ = 0 and − o o g (v;0) = g (v). o +∞ o 1/2 dt Similarly, for any v, the identity g (v;0) = 1 ν (t (v)) t holds on D Dv. (v) The constant term of go(v; ρ)at ρ = 0 is given∫ by − ∞ o o 1/2 ′ ∨ ∗ r−r′−1 dt CTρ=0 g (v; ρ)= ν (t v)[2r·rk(E)−2] (r r )δDv crk(E)−1( , ) ΩE∨ 1 − − ∧ E ∇ ∧ t Z (vi) The constant term CT go(v; ρ) satisﬁes the equation ρ=0

c o r−r′ o dd CT g (v; ρ)+ δDv Ω ∨ = ϕ (v)[2r·rk(E)] ρ=0 ∧ E of currents on D, where Ω ∨ = ctop( ∨, )∗. E E ∇ Proof. Part (i) follows immediately from the expression (2.1.9), which shows that (locally on D) νo(√tv) and its partial derivatives stay bounded as t + . Part (ii) follows from Proposition 2.4.5.(f). → ∞ ′ ′ To show (iii), let k be as in (ii) such that v k = (0r−r′ , v ) with v non-degenerate. For convenience, set q′ = rk( ). By Propositions 2.4.4· .(c) and 2.4.5.(c),(f) we can write o o E ν (v) ′ = ν (v k) ′ [2rq −2] · [2rq −2] o ′ o o o ′ = ν (v ) ′ ′ ϕ (0 ′ ) ′ ′ + ν (0 ′ ) ′ ′ ϕ (v ) ′ ′ . [2r q −2] ∧ r−r [2(r−r )q ] r−r [2(r−r )q −2] ∧ [2r q ] (2.6.8) The same propositions also show that o r−r′ ϕ (0r−r′ )[2(r−r′)q′] =Ω ∨ E (2.6.9) o ′ ∨ ∗ r−r′−1 ν (0 ′ ) ′ ′ = (r r )c ( , ) Ω ∨ r−r [2(r−r )q −2] − rk(E)−1 E ∇ ∧ E and hence o o ′ r−r′ ν (v)[2rq′−2] = ν (v )[2r′q′−2] Ω ∨ ∧ E (2.6.10) o ′ ′ ∨ ∗ r−r′−1 + ϕ (v ) ′ ′ (r r )c ( , ) Ω ∨ . [2r q ] ∧ − rk(E)−1 E ∇ ∧ E 2 o v More precisely, we are asserting that for any compactly supported form η, the expression RD g ( ; ρ) ∧ η admits a meromorphic extension as a function of ρ, and is continuous in η in the sense of distributions. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 24

o ′ Consider the contribution of each term in the last expression to g (v; ρ). Writing vi = ′ ′ ′ (v1,..., vi, . . . , vr′ ), the first term contributes +∞ o ′ o ′ dt r−r′ √ ′ √ v ′ ′ ∨ b ν ( tvi)[2q −2] ϕ ( t i)[2(r −1)q ] ρ+1 ΩE . (2.6.11) ′ 1 ∧ t ∧ 1≤Xi≤r Z o ′ Since ϕ (√tvi) stays bounded as t + , Lemma 2.6.5 shows that (2.6.11) converges to a locally integrable form on D for Re(→ρ)∞> 1/2. The contribution of the second term is − +∞ o ′ dt ′ ∨ ∗ r−r′−1 ϕ (√tv ) ′ ′ (r r )c ( , ) Ω ∨ , (2.6.12) [2r q ] tρ+1 ∧ − rk(E)−1 E ∇ ∧ E Z1 hence it suffices to prove meromorphic continuation of the integral in this expression. We rewrite this integral as +∞ +∞ o ′ dt o ′ dt 1 ϕ (√tv ) ′ ′ = (ϕ (√tv ) ′ ′ δ v ) + δ v ; (2.6.13) [2r q ] tρ+1 [2r q ] − D tρ+1 ρ D Z1 Z1 then Bismut’s asymptotic estimate [4, (3.9)] implies that o ′ − 1 ϕ (√tv ) ′ ′ δ v = O(t 2 ) (2.6.14) [2r q ] − D as currents on D, and hence the right hand side gives the desired meromorphic continuation to Re(ρ) > 1/2, proving (iii). − When v is regular, or for general v upon restriction to D Dv, the proof of (iii) shows that the integral defining go(v; ρ) converges when Re(ρ) > − 1/2; thus we can set ρ = 0 and obtain (iv). − o To prove (v), we proceed as in (iii) and analyze the contribution to CTρ=0 g (v; ρ) of each summand in the right hand side of (2.6.10). Observe that the constant term at ρ = 0 of (2.6.11) is simply +∞ o ′ o ′ dt r−r′ √ ′ √ ′ ′ ν ( tvi)[2q −2] ϕ ( tvi)[2(r −1)q ] ΩE∨ , (2.6.15) ′ 1 ∧ t ∧ 1≤Xi≤r Z whereas the constant term of (2.6.13) is +∞ o ′ dt (ϕ (√tv ) ′ ′ δ v ) . (2.6.16) [2r q ] − D t Z1 Substituting in (2.6.12) and adding these two contributions gives (v). Finally, note that (by (ii) and Proposition 2.4.4.(f)) all terms in (vi) are invariant under replacing v with v k for any matrix k O(r) (case 1) or k U(r) (case 2). Thus we can · ′ ′ ∈ r′ ∈ o assume that v = (0r−r′ , v ), where v V is non-degenerate. Then ν (v)[2rq′−2] is given o ∈ by (2.6.10); since ϕ (v′) is closed, we conclude that c o c o ′ r−r′ √ ′ √ ′ ′ dd ν ( tv)[2rq −2] = dd ν ( tv )[2r q −2] ΩE∨ ∧ (2.6.17) d o ′ r−r′ = t ϕ (√tv ) ′ ′ Ω ∨ . − dt [2r q ] ∧ E GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 25

Using (v) gives ∞ c o c o 1/2 dd CTρ=0 g (v; ρ)= dd ν (t v)[2rq′−2] 1 Z ′ ∨ ∗ r−r′−1 dt (r r )δ v c ( , ) Ω ∨ − − D ∧ rk(E)−1 E ∇ ∧ E t ∞ c o 1/2 dt = dd ν (t v)[2rq′−2] 1 t (2.6.18) Z ∞ d o 1/2 ′ dt r−r′ = t ϕ (t v ) ′ ′ Ω ∨ − dt [2r q ] t ∧ E Z1 o o 1/2 ′ r−r′ = ϕ (v)[2rq′] lim ϕ (t v )[2r′q′] Ω ∨ − t→∞ ∧ E o r−r′ = ϕ (v) ′ δ v Ω ∨ , [2rq ] − D ∧ E where the last equality follows from (2.6.14), proving (vi). Definition 2.6.7. Let v = (v , . . . , v ) V r. Define 1 r ∈ o o g (v)=CT g (v; ρ) D∗(D). ρ=0 ∈ Example 2.6.8. Suppose that v V r is degenerate and choose k in O(r) or U(r) such that ′ ′ r′∈ v k = (0r−r′ , v ), where v V is non-degenerate. Then the proof of Proposition 2.6.6.(iii) shows· that ∈ o o ′ r−r′ ′ g (v)= g (v ) Ω ∨ + µ(v ), (2.6.19) ∧ E where +∞ ′ ′ o ′ dt ∨ ∗ r−r′−1 √ ′ ′ µ(v ) = (r r ) ϕ ( tv )[2r q ] δDv crk(E)−1( , ) ΩE∨ . (2.6.20) − 1 − t · E ∇ ∧ Z 2.7. Star products. Let k, l be positive integers with (k + l)rk( ) dim(D) + 1. Fix regular tuples v′ = (v′ , . . . , v′ ) and v′′ = (v′′, . . . , v′′) V l such thatE the≤ tuple 1 k 1 l ∈ v = (v , . . . , v ) := (v′ , . . . , v′ , v′′, . . . , v′′) V k+l (2.7.1) 1 k+l 1 k 1 l ∈ is also regular. This implies that Dv = Dv′ Dv′′ , if non-empty, is a proper intersection, see Definition 2.6.2. Define the star product∩ of the Green forms go(v′) and go(v′′), which for the regular case are given by (2.6.1), by

o ′ o ′′ o ′ o ′ o ′′ 2(k+l)rk(E)−2 g (v ) g (v )= g (v ) δ ′′ + ϕ (v ) g (v ) D (D). (2.7.2) ∗ ∧ Dv [2rk(E)k] ∧ ∈ Our next goal is to compare the currents go(v) and go(v′) go(v′′). For t ,t R , deﬁne ∗ 1 2 ∈ >0 i o o dt dt α(t ,t , v′, v′′) = ν (t1/2v′) ∂(ν (t1/2v′′) ) 1 2 , 1 2 2π 1 [2k·rk(E)−2] ∧ 2 [2l·rk(E)−2] ∧ t t 1 2 (2.7.3) ′ ′′ i o 1/2 ′ o 1/2 ′′ dt1 dt2 β(t1,t2, v , v ) = ∂(ν (t1 v )[2k·rk(E)−2]) ν (t2 v )[2l·rk(E)−2] . 2π ∧ ∧ t1 t2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 26

We set ′ ′′ ′ ′′ α(v , v )= α(t1,t2, v , v ), 1≤t ≤Zt ≤+∞ 1 2 (2.7.4) ′ ′′ ′ ′′ β(v , v )= β(t1,t2, v , v ).

1≤t1≤Zt2≤+∞

The estimate (2.1.10) shows that the integrals converge to smooth forms on D (Dv′ − ∪ Dv′′ ). These forms are locally integrable on D by a straightforward adaptation of the proof of Proposition 2.1.7 and so they define currents [α(v′, v′′)] and [β(v′, v′′)] on D. ′ ′′ ′ ′′ ′ ′′ ′ ′′ Theorem 2.7.1. The currents [α(v , v )] and [β(v , v )] satisfy g∗α[(v , v )] = [α(gv , gv )] ′ ′′ ′ ′′ and g∗[β(v , v )] = [β(gv , gv )] for g G and o o o∈ g (v′) g (v′′) g (v)= ∂[α(v′, v′′)] + ∂[β(v′, v′′)]. ∗ − ′ 2 Proof. Set q = rk( ). Denote by t1,t2 the coordinates on R>0 and consider the form o E ν (v) A∗(D R2 ) defined by ∈ × >0 o o ′ o ′′ dt1 ν (v) = ν (√t1v )[2kq′−2] ϕ (√t2v )[2lq′] e ∧ t1 o ′ o ′′ dt2 e + ϕ (√t1v )[2kq′] ν (√t2v )[2lq′−2] . (2.7.5) ∧ t2 2 ∗ 2 For a piecewise smooth path γ : I R>0 (I R a closed interval) and α A (D R>0), let → ⊂ ∈ × α A∗−1(D) (2.7.6) ∈ Zγ 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), γM = (1,t), and γM = (t, M) (2.7.7) with t [1, M]. By Proposition 2.4.5.(a), we have ∈ M o o 1/2 dt o lim ν (v) = lim ν (t v)[2(k+l)q′−2] = g (v). (2.7.8) M→∞ M→∞ t Zγd,M Z1 Next, note that e ′ ∗ o o 1/2 ′′ o ′ dt (id γM ) ν (v)= ν (t v )[2lq′−2] ϕ (v )[2kq′] , × ∧ ∧ t (2.7.9) ′′ ∗ o o 1/2 ′ o 1/2 ′′ dt (id γ ) ν (v)= ν (t v ) ′ ϕ (M v ) ′ . × M e [2kq −2] ∧ [2lq ] ∧ t o 1/2 ′′ By [4, Thm. 3.2], as M we have ϕ (M v ) ′ δ ′′ as currents, and hence e→∞ [2lq ] → Dv o o o lim ν (v)= g (v′) g (v′′). (2.7.10) M→∞ ∗ ′ Z ′′ γM +γM e GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 27

Let ∆ = (t ,t ) 1 t t M R2 , (2.7.11) M { 1 2 | ≤ 1 ≤ 2 ≤ } ⊂ >0 ′ ′′ 2 oriented so that ∂∆M = γd,M γM γM . Let d = d1 + d2 be the differential on D R>0, − − 2 × where d1 = ∂ + ∂ is the differential on D and d2 is the differential on R>0. Then we have

o o o o ν (v) ν (v) = ν (v) = d2 ν (v). (2.7.12) γ − γ′ +γ′′ ∂∆ ∆ Z d,M Z M M Z M Z M Applying Propositione 2.4.5.(b)e we obtain e e

o d o 1/2 ′ o 1/2 ′′ d2 ν (v)= t1 ϕ (t1 v )[2kq′] ν (t2 v )[2lq′−2] dt1 ∧ o 1/2 d o 1/2 dt1dt2 ′ ′ ′′ ′ e ν (t1 v )[2kq −2] t2 ϕ (t2 v )[2lq ] − ∧ dt2 ∧ t1t2 −1 o 1/2 ′ o 1/2 ′′ (2.7.13) = ( 2πi) (∂∂ν (t v )) ′ ν (t v ) ′ − − 1 [2kq ] ∧ 2 [2lq −2] o 1/2 o 1/2 dt1dt2 ′ ′ ′′ ′ +ν (t1 v )[2kq −2] (∂∂ν (t2 v ))[2lq ] ∧ ∧ t1t2 ′ ′′ ′ ′′ = ∂α(t1,t2, v , v )+ ∂β(t1,t2, v , v ). The statement follows by taking the limit as M + in (2.7.12), and the equivariance property under g G follows from Proposition 2.4.5→.(d).∞ ∈ As a corollary, we obtain the following invariance property of star products. Corollary 2.7.2. Let k O(k + l) (case 1) or k U(k + l) (case 2) and suppose that v = (v′, v′′) V k+l is non-degenerate.∈ Let v′ , v′′ be∈ deﬁned by v k = (v′ , v′′) and set ∈ k k · k k [α(k; v′, v′′)] = [α(v′, v′′)] [α(v′ , v′′)], [β(k; v′, v′′)] = [β(v′, v′′)] [β(v′ , v′′)], − k k − k k with α and β as in (2.7.4). Then o o o o g (v′) g (v′′) g (v′ ) g (v′′)= ∂[α(k; v′, v′′)] + ∂[β(k; v′, v′′)] D2(k+l)rk(E)−2(D). ∗ − k ∗ k ∈ Proof. This is a consequence of the theorem and the invariance property go(v k)= go(v), which follows from Proposition 2.4.5.(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 [35] 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 5.3.3), generalizes these results to arbitrary hermitian symmetric spaces of orthogonal or unitary type and gives a conceptual proof.

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 v ,≤ . . . , v are linearly independent)∈ and denote by Stab v , . . . , v the pointwise 1 r Gh 1 ri GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 28

o stabilizer of v1, . . . , vr in G. Let g (v) be the form given in Section 2.6. Assuming that rk( ) = 1, weh will computei the integral E o g (v) Ωp−r+1, (3.0.1) + ∧ ZΓv\D where Γv is a discrete subgroup of ﬁnite covolume in StabG v1, . . . , vr , under some additional conditions ensuring that the integral converges. The resulth is statedi in Theorem 3.4.9 and relates this integral to the derivative of a Whittaker functional deﬁned on a degen- ′ erate principal series representation of Gr. Despite the length of this section, its proof is conceptually simple and follows easily from Lemma 2.5.4, which determines the weight of ν(v)[2r−2], together with results in [29, 33] 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 [43] 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 ′ ′ ′ ≥ G = Mp2r(R). We abbreviate N = Nr, M = Mr and K = Kr (see 2.5.1). 3.1.1. We denote by g′ the complexiﬁed Lie algebra of G′. We have the Harish-Chandra decomposition g′ = p p k′, (3.1.1) + ⊕ − ⊕ with

′ X1 X2 t t k = X1 = X1, X2 = X2 , X2 X1 − − (3.1.2) 1 X iX p = p (X)= Xt = X , p = p . + + 2 iX X − + − ′ ′ ′ ′ Note that k = Lie(K )C, where K is the maximal compact subgroup of G in Section 2.5.1. For x = (x ,...,x ) Cr, let d(x) denote the diagonal matrix diag(x ,...,x ), write 1 r ∈ 1 r i d(x) h(x)= (3.1.3) i d(x) − · · ′ r ′ 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 ∆+ =∆{ + |∆+∈ given} by c ⊔ nc + ∆c = ei ej 1 i < j r , { − | ≤ ≤ } (3.1.4) ∆+ = e + e 1 i j r , nc { i j| ≤ ≤ ≤ } + + 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 G′, 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, χ(m(a),ǫ)= ǫ (3.1.5) · (1, if det a> 0. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 29

For α Z/4Z and s C, consider the character ∈ ∈ χα s : M R, (m(a),ǫ) χ(m(a),ǫ)α det a s (3.1.6) | · | → 7→ | | and the smooth induced representation ′ Iα(s) = IndG χα s (3.1.7) P | · | with its ∞ topology, where the induction is normalized so that Iα(s) is unitary when Re(s) = 0.C In concrete terms, Iα(s) consists of smooth functions Φ: G′ C satisfying → Φ (m(a),ǫ) n(b) g′ = χ(m(a),ǫ)α det a s+ρr Φ(g′), ρ := r+1 , (3.1.8) | | r 2 with the action of G′ deﬁned byr(g′)Φ(x)=Φ(xg′). Note that, by the Cartan decomposition G′ = P K′, any such function is determined by its restriction to K′; in particular, given α α Φ(s0) I (s0), there is a unique family (Φ(s) I (s))s∈C such that Φ(s) K′ = Φ(s0) K′ for all ∈s. Such a family is called a standard section∈ of Iα(s). | |

3.1.3. We denote by χα the character of K′ whose diﬀerential restricted to h′ has weight α ′ α 2 (1,..., 1). The K -types appearing in I (s) were determined by Kudla and Rallis [29] to ′ be precisely those 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 l + 2Z, 1 i r. (3.1.9) i ∈ 2 ≤ ≤ Moreover, these K′-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) = 0 (op. cit.,· Prop.∈ 1.1), hence from now on we normalize all such highest weight vectors6 so that Φλ(e, s) = 1. Note that the restriction of Φλ to K′ is independent of s. For scalar weights λ = l(1,..., 1) with l α + 2Z, we have ∈ 2 Φl(k′,s) :=Φl(1,...,1)(k′,s) = (det k′)l, k′ K′. (3.1.10) ∈ 3.1.4. Suppose that X g′ is a highest weight vector for K′ of weight λ. Then XΦl(s) is a highest weight vector of∈ weight λ + l and hence multiplicity one of K′-types implies that XΦl(s)= c(X,l,s)Φλ+l(s) (3.1.11) for some constant c(X,l,s) C. We will need to determine this constant explicitly for certain choices of X; let ∈

ei = (0,..., 0, 1 , 0,..., 0) (3.1.12) i−th − − ′ and deﬁne Xi = p−(d(ei)). When i = r, the vector Xr is a highest weight vector for K of weight 2e . Let ι : Mp (R) G′ be the embedding deﬁned by − r r 2 → a b 1r−1 0r−1 , ǫ a b , ǫ . (3.1.13) c d 7→ 0r−1 1r−1 c d ! ! GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 30

α − Then, for any Φ I (s), the value of Xr Φ(e, s) only depends on the pullback function ∗ ∈ ′ ιrΦ(s) : Mp2(R) C. Note that every element g of Mp2(R) can be written uniquely in the form → ′ 1/2 g = n(x)m(y ), 1 k˜θ, (3.1.14) where x R, y R , θ R/4πZ and we deﬁne ∈ ∈ >0 ∈

(kθ, 1), if π<θ π, cos θ sin θ k˜θ = − ≤ where kθ = . (3.1.15) (k , 1), if π<θ 3π, sin θ cos θ ( θ − ≤ −

We think of x, y and θ as coordinates on Mp2(R). In terms of these coordinates, we have d i d X−Φ = 2iy + ι∗Φ, (3.1.16) r − dτ 2 dθ r for any Φ Iα(s), where d = 1 d + i d . Taking Φ = Φl, the following lemma is a ∈ dτ 2 dx dy straightforward computation using the explicit expression 1 r+1 ∗ l 2 s+ 2 iθl ιrΦ (x,y,θ,s)= y e . (3.1.17) Lemma 3.1.5. Let l α + 2Z. Then ∈ 2 X−Φl(1,...,1)(s)= 1 s + r+1 l Φl(1,...,1)−2er (s). r 2 2 − In other words, this shows that c(X−,l,s)= 1 s + r+1 l . r 2 2 − 3.1.6. We now return to the quadratic space V over R of signature (p, 2). Define p r + 1 I (V,s)= I dim V (s) and s = − ; (3.1.18) r 0 2 we assume that r p + 1, so that ≤ s 0. (3.1.19) 0 ≥ ′ r Consider the Weil representation ω = ωψ of G O(V ) on (V ), as in Section 2.5.2, × +S − and let Rr(V ) be its maximal quotient on which (so(V ), O(V ) O(V )) acts trivially. The map × λ: (V r) I (V,s ), λ(ϕ)(g′) := ω(g′)ϕ (0) (3.1.20) S → r 0 ′ is G -intertwining and factors through Rr(V ). By [29], it defines an embedding (R (V ) ֒ I (V,s ). (3.1.21 r → r 0 3.2. Degenerate principal series of U(r, r). In this section we fix r 1 and let G′ = ′ ′ ≥ U(r, r). We abbreviate N = Nr, M = Mr and K = Kr (see Section 2.5.1). Our setup follows [18]. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 31

′ ′ ′ ′ 3.2.1. We denote by g the complexiﬁed Lie algebra of G and let gss = X g trX = 0 . Let { ∈ | } 1 1 1 u = r r U(2r), (3.2.1) √ i1r i1r ∈ 2 − −1 ′ 1r 0 and note that u G u is the isometry group of the Hermitian form determined by 0 −1r . In addition, we have ′ k1 0 −1 Kr = [k1, k2]= u u k1, k2 U(r) , (3.2.2) 0 k2 | ∈ see (2.5.9), and the Harish-Chandra decomposition g′ = p p k′, (3.2.3) + ⊕ − ⊕ ′ with g = Mat2r(C) and

′ ′ X1 0 −1 k = Lie(Kr)C = u u X1, X2 Matr(C) , 0 X2 ∈ 0 X p = p (X) := u u−1 X Mat (C) , (3.2.4) + + 0 0 ∈ r 0 0 p = p (X) := u u−1 X Mat (C) . − − X 0 ∈ r ′ ′ ′ Let kss = k gss and ∩ h = u d(x) u−1 x = (x ,...,x ) C2r, x + . . . + x = 0 . (3.2.5) · · | 1 2r ∈ 1 2r Then h is a Cartan subalgebra of k′ . For 1 i 2r, the assignment u d(x) u−1 x ss ≤ ≤ · · 7→ i deﬁnes a functional ei : h C. We write ∆ for the set of roots of (gss, h) and ﬁx the set of positive roots ∆+ =∆+ →∆+ given by c ⊔ nc + ∆c = ei ej 1 i < j r ei + ej r

to K′; in particular, given Φ(s ) I(χ,s ), there is a unique family (Φ(s) I(χ,s)) 0 ∈ 0 ∈ s∈C such that Φ(s) ′ = Φ(s ) ′ for all s. Such a family is called a standard section of I(χ,s). |K 0 |K 3.2.3. Some useful facts on K′-types of I(χ,s) were proved by Lee [33], namely that I(χ,s) is multiplicity free as a representation of K′, andifΦ(λ1,λ2)( ,s) I(χ,s) is a highest weight · ∈ vector of weight (λ , λ ), then Φ(λ1,λ2)(e, s) = 0. Hence from now on we normalize all such 1 2 6 highest weight vectors so that Φ(λ1,λ2)(e, s) = 1. Note that the restriction of Φ(λ1,λ2) to K′ is independent of s, and for scalar weights l = (l (1,..., 1), l (1,..., 1)), l , l Z, (3.2.10) 1 2 1 2 ∈ we have

(l1,l2) (l1(1,...,1),l2(1,...,1)) l1 l2 Φ ([k1, k2],s) :=Φ ([k1, k2],s) = (det k1) (det k2) , k1, k2 U(r). ∈(3.2.11)

− − 3.2.4. Let Xr p− be as in Section 3.1.4; that is, Xr = p−(d(er)) with er = (0,..., 0, 1) Cr. Then X− is∈ a highest weight vector for K′ of weight ( e , e ) and so ∈ r − r r − l − l+(−er,er) Xr Φ (s) = c(Xr ,l,s)Φ (s) (3.2.12) − for some constant c(Xr ,l,s) C. To compute this constant, note that U(r, r) GL2r(R)= − sp ∈ ∩ Sp2r(R) and Xr 2r,C. Moreover, if Φ I(χ,s), then the restriction Φ Sp2r(R) belongs α ′ ∈′ ∈ | to I (s ), where s = 2s + (r 1)/2 and α =0 if χ × is trivial and α = 2 otherwise (this − |R follows directly from a comparison of (3.1.8) and (3.2.9)). If l = (l1(1,..., 1), l2(1,..., 1)) is a scalar weight, then Φl (s)=Φ(l1−l2)·(1,...,1)(s′) Iα(s′) (3.2.13) |Sp2r(R) ∈ and so Lemma 3.1.5 and the above remarks show that the constant in (3.2.12) is given by l l c(X−,l,s)= s + ρ 1 − 2 . (3.2.14) r r − 2 3.2.5. We now return to the m-dimensional hermitian space V over C of signature (p,q) × m with pq = 0. From now on we ﬁx a character χ = χV of C such that χ R× = sgn( ) and deﬁne 6 | · Ir(V,s)= I(χ,s), m r (3.2.15) s = − . 0 2 We assume that r p + 1, so that ≤ q 1 s − 0, (3.2.16) 0 ≥ 2 ≥ with equality only when q = 1 and r = p + 1. Consider the Weil representation ω = ω of G′ U(V ) on (V r), as in Section 2.5.2, ψ,χ × S and let Rr(V ) be its maximal quotient on which (u(V ), K) acts trivially. The map λ: (V r) I (V,s ), λ(ϕ)(g′) := ω(g′)ϕ (0) (3.2.17) S → r 0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 33

′ is G -intertwining and factors through Rr(V ); by [34, Thm. 4.1], it deﬁnes an embedding (R (V ) ֒ I (V,s ). (3.2.18 r → r 0 3.3. Whittaker functionals. 3.3.1. Let Sym (R), case 1 T r (3.3.1) ∈ (Herr, case 2 be a non-singular matrix and let ψ : N C× denote the character deﬁned by T r → 2πi tr(T b) ψT (n(b)) = ψ(tr(T b)) = e . (3.3.2)

Let Ir(V,s) and s0 be as in (3.1.18) (resp. (3.2.15)) in case 1 (resp. in case 2). A continuous functional l : I (V,s ) C is a called a Whittaker functional if it satisﬁes r 0 → l(r(n)Φ) = ψ (n) l(Φ) for all n N , Φ I (V,s ). (3.3.3) T · ∈ r ∈ r 0 Such a functional can be constructed as follows. Let dn be the Haar measure on Nr that is self-dual with respect to the pairing (n(b),n(b′)) ψ(tr(bb′)). Embed Φ I (V,s ) in a 7→ ∈ r 0 (unique) standard section (Φ(s))s∈C and deﬁne

W (Φ,s)= Φ(w−1n,s) ψ ( n) dn (3.3.4) T r T − ZNr The integral converges for Re(s) 0 and admits holomorphic continuation to all s [47]; ≫ its value at s0 deﬁnes a Whittaker functional WT (s0). It is shown in op. cit. that WT (s0) spans the space of Whittaker functionals. For g′ G′ and Φ I (V,s ), deﬁne ∈ r ∈ r 0 W (g′, Φ,s)= Φ(w−1ng′,s) ψ ( n) dn. (3.3.5) T r T − ZNr and d W ′ (g′, Φ,s )= W (g′, Φ,s) . (3.3.6) T 0 ds T s=s0

3.3.2. In this section, we apply results of [43] to extract some necessary asymptotic estimates for Whittaker functionals. Assume throughout this section that T is non-degenerate. As in Section 2.2.1, let R, orthogonal case K = and ι := [K : R]. (3.3.7) (C, unitary case,

It will be useful for us to work in symmetric space coordinates, as follows. Let Hr denote the Siegel (resp. Hermitian) upper half space of genus r, so that in the orthogonal case, H = τ = x + iy Sym (C) y > 0 (3.3.8) r { ∈ r | } and in the unitary case 1 H = τ Mat (C) (τ tτ) > 0 ; (3.3.9) r ∈ r | 2i − GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 34

in the latter case, write τ = x + iy with y = 1 (τ tτ) Her (C) and x = τ iy. 2i − ∈ r >0 − For a point τ = x + iy Hr, fix a matrix α GLr(K) with det α R>0 and such that y = α tα and let ∈ ∈ ∈ · g′ := n(x)m(α) G′ . (3.3.10) τ ∈ r l Let Φ (s) be the normalized highest weight vector of Ir(V,s) of scalar weight m , orthogonal case 2 l :=  m + k(χ) m + k(χ) (3.3.11)  , − , unitary case  2 2 (see (3.1.10) and (3.2.11)) and define (y,s) := (det y)−ιm/4 W (g′ , Φl,s) e2πtr(Ty); (3.3.12) WT T iy note that this is independent of the choice of α in (3.3.10). Proposition 3.3.3. Suppose T is positive definite of rank r. (i) For any integer k 0 and any fixed s C, ≥ ∈ ∂k lim T (λy, s) < . λ→∞ ∂sk W ∞ (ii) There exists a constant C > 0, depending on s,y,T and k, such that ∂ ∂k (λy, s) = O(λ−1−C ) ∂λ ∂sk WT as λ . →∞ Let κ =1+ ι (r 1) and 2 − m−(r+1) 2 , orthogonal case s0(r)= m−r ( 2 , unitary case.

Then we have the following more precise results for the value and derivative at s0(r): (iii) ιrm/2 ( 2πi) ιs0(r) T (y,s0(r)) = r(κ−−1)/2 (det T ) W 2 Γr(ιm/2) (iv) There is an asymptotic formula

ι ( 2πi)ιrm/2 (det T )ιs0(r) Γ′ (ιm/2) ′ (λy, s (r)) = log det πT r T 0 − r(κ−1)/2 W 2 · 2 Γr(ιm/2) − Γr(ιm/2) + O(λ−1) as λ , where the implied constant depends on y and T . →∞ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 35

1 t 1 1 Proof. Write T = (T 2 ) (T 2 ), for some matrix T 2 GL (K). If we define · ∈ r t 1 1 g := 4π (T 2 ) y (T 2 ), (3.3.13) · · then applying [43, (1.29), (3.3), (3.6)] gives ( 2πi)ιrm/2 πrβ (y,s) = (det T )β+ιm/2−κ ω(g; β + ιm/2, β) (3.3.14) T r(κ−−1)/2 W 2 Γr(β + ιm/2) where β = ι (s s (r)), • 2 − 0 Γ (β)= πιr(r−1)/4 r−1 Γ(β ιk/2), and • r k=0 − ω(g; α, β) is an entire function in (α, β) C2, initially defined by the formula • Q ∈ −1 β −tr(gx) α−κ β−κ ω(g; α, β)=Γr(β) det(g) e det(x + 1) det(x) dx (3.3.15) NZ+ on the region Re(β) > κ 1, where the integral is absolutely convergent; here we abuse notation and write− Sym (R) , orthogonal case N + = r >0 (3.3.16) (Herr(C)>0, unitary case. and take the measure dx to be the standard Euclidean measure, following [43, Section 1]. The analytic continuation of ω(g; α, β) to (α, β) C2 is proven in [43, Theorem 3.1]. ∈ Replacing y by λy corresponds to replacing g by λg; thus, in light of (3.3.14), in order to prove parts (i) and (ii) of the proposition, it suffices to prove the corresponding estimates for ω(g; α, β). More precisely, we shall show that for fixed g, integers k, k′ 0, and (α, β) C2, ≥ ∈ ′ ∂k+k lim k k′ ω(λg; α, β) < (3.3.17) λ→∞ "∂α ∂β # ∞ and k+k′ ∂ ∂ −2 k k′ ω(λg; α, β) = O(λ ); (3.3.18) ∂λ "∂α ∂β # the implied constants may depend on k, k′, α, β and g. To prove these estimates, we would like to use (3.3.15), but our choice of parameters (α, β) may place us outside the range of absolute convergence of the integral; to circumvent this, consider the differential operator

1 ∂ det (1 + δij) , orthogonal case ∆= 2 ∂gij (3.3.19) det( ∂ ), unitary case  ∂gij  GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 36

+ as in [43, (3.10.I-II)], where gij are the coordinates for the entries of g N ; it satisﬁes the identity ∈ ∆ e−tr(g) = ( 1)r e−tr(g). (3.3.20) − Then (3.12) and (3.7) of [43] together imply that for N Z , ∈ >0 ω(g; α, β) = ( 1)rN etr(g) det(g)β ∆N e−tr(g) det(g)−βω(z; α N, β) − · − =( 1)rN etr(g) det(g)β ∆N e−tr(g) det(g)−βω (g; κ β, κ α + N) . (3.3.21) − · − − h i Fixing N > Re(α) 1, the ﬁnal incarnation of ω(...) in (3.3.21) is now in the range of convergence of (3.3.15−); for convenience, rewrite (3.3.15) for these parameters as

ω(g; α, β) := ω (g; κ β, κ α + N) . (3.3.22) − − By passing partial derivativese in the coordinates of g under the integral (3.3.15) deﬁning ω(g; α, β), it follows that ω(g; α, β) can be written as a ﬁnite sum of terms of the form

e f(α, β) (det g)κ−α+N−m F (g) e−tr(gx) F (x) det(x + 1)−β det(x)−α+N dx (3.3.23) · 1 · 2 NZ+ where f(α, β) is a holomorphic function independent of g, m 0 is an integer, and F (g) ≥ 1 and F2(x) are homogeneous polynomials with deg F1 < mr; here F1 and F2 arise as products of iterated partial derivatives, of det(g) and e−tr(gx) respectively, with respect to the entries of g. For a parameter λ> 0 and ﬁxed g, replacing g by λg and applying a change of variables in the previous display implies that ω(λg; α, β) can be written as a sum of terms of the form

λ−M f(α, β) (det g)κ−α+N−m F (g) e−tr(gx) F (x) det(λ−1x + 1)−β det(x)−α+N dx · 1 · 2 NZ+ (3.3.24) ′ ∂k+k where M = mr deg F +deg F 0. It follows (again, for ﬁxed g, etc.) that ′ ω(λg; α, β) − 1 2 ≥ ∂αk∂βk can be written as a ﬁnite sum of terms of the form

−M A κ−α+N−m λ f(α, β) (log det g) (det g) F1(g)

e−tr(gx) F (x) log(det(λ−1x + 1))B det(λ−1x + 1)−β log(det x)C det(x)−α+N dx × 2 NZ+ (3.3.25) where f(α, β) is independent of λ, and A,B,C are integers. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 37

For λ> 1, we have the (rather crude) estimates

log(det(λ−1x + 1))B det(λ−1x + 1)−β det(λ−1x + 1)−β+B | | ≤ | | det(x + 1)−Re(β)+B , if B Re(β) ≥− ≤ (1, otherwise. (3.3.26)

Hence, by dominated convergence, we may pass the limit λ inside the integral in (3.3.25), and, by comparison with (3.3.15), conclude that the→ limit ∞ of this integral exists ′ ∂k+k as λ . Since ′ ω(λg; α, β) is a sum of terms as in (3.3.25), this shows the desired →∞ ∂αk∂βk existence of the limit (3.3.17). To prove the second estimate (3.3.18), diﬀerentiate (3.3.25) with respect to λ; dominated convergence again allows us to pass the derivative under the integral sign, and the estimate in the same way. These two estimates in turn imply statements (i) and (ii) of the proposition. We now turn to the more precise versions at s = s0(r). To prove (iii), recall that β = 0 when s = s0(r); since ω(g; ιm/2, 0) 1 by [43, (3.15)], evaluating (3.3.14) at s = s0(r) yields the desired formula. ≡ Finally to prove (iv), take the logarithmic derivative with respect to s in (3.3.14) to obtain r(κ−1)/2 2 Γr(ιm/2) ′ T (λy, s0(r)) (3.3.27) ( 2πi)ιrm/2(det T )ιs0(r) ·W − ′ (λy, s (r)) ι Γ′ (ιm/2) ∂ = WT 0 = log det πT r + ω(λ g; β + ιm/2, β) . (λy, s (r)) 2 − Γ (ιm/2) ∂β · T 0 r β=0! W

It will therefore suﬃce to show that ∂ ω(λ g; β + ιm/2, β) = O(λ−1). ∂β · |β=0 To this end, for suﬃciently large integer N, we have

ω(g; β+ιm/2, β) = ( 1)rN etr(g) det(g)β ∆N e−tr(g) det(g)−βω (g; β + ιm/2, β) (3.3.28) − · h i with e ω(g; β + ιm/2, β) := ω (g; κ β, κ β ιm/2+ N) . (3.3.29) − − − ∂ Consider expanding ∂β ω(λg, β +ιm/2, β) as a sum of terms as in (3.3.25), and note that e −1 any terms with either F1 or F2 non-constant are O(λ ). As these terms arise from the partial derivatives of det g or ω(g, β) respectively, it follows that ∂ ∂ ω(λg; β + ιm/e 2, β)= ω(λg; β + ιm/2, β)+ O(λ−1). (3.3.30) ∂β ∂β

We are reduced to proving that ∂ ω(λg; β +eιm/2, β) is itself O(λ−1); taking N large ∂β |β=0 enough to ensure the convergence of (3.3.15), and applying a change of variables in the e GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 38

integral, gives (det g)κ−β−ιm/2+N ω(λ g; β+ιm/2, β)= e−tr(gx) det(λ−1x+1)−β det(x)−β−ιm/2+N dx. Γr(κ β ιm/2+ N) − − NZ+ e (3.3.31) Now substitute the Taylor expansion det(λ−1x + 1)−β = 1 log(det(λ−1x + 1)) β + . . . (3.3.32) − at β = 0 into the previous expression, and note that the integral in (3.3.31) is uniformly convergent for β in a neighbourhood of β = 0. Moreover, there is a useful integral representation [43, (1.16)]:

e−tr(gx) det(x)−β−ιm/2+N dx = Γ (κ β ιm/2+ N) det(g)−κ+β+ιm/2−N (3.3.33) r − − NZ+ Combining these observations, the Taylor expansion of ω(λg; β + ιm/2, β) around β =0 is of the form e (det g)κ−ιm/2+N ω(λg; β + ιm/2, β) =1 e−tr(gx) log(det(λ−1x + 1)) det(x)−ιm/2+N dx β − Γr(κ ιm/2+ N)  − NZ+ e  + higher order terms in β. (3.3.34)  For fixed g, the coefficient of β is O(λ−1) as λ , as can be easily deduced from the estimate → ∞ log(det(λ−1x + 1)) < λ−1tr(x) for x N +. (3.3.35) ∈ This proves ∂ ω(λg; β + ιm/2, β) = O(λ−1) as required. ∂β |β=0 Proposition 3.3.4. Suppose T is non-degenerate of signature (p,q) with q > 0. e (i) Let s0(r) be as in Proposition 3.3.3. Then T (y,s0(r)) = 0 for all y. W ′ (ii) For any fixed s0 C, k N and y > 0, there are positive constants C and C such that ∈ ∈ ∂k (λy, s) = O(e−Cλ) ∂sk WT s=s0 and k ∂ ∂ ′ (λy, s) = O(e−C λ); ∂λ ∂sk WT s=s0 ! ′ here C,C and the implied constants also depend on T,s0, k and y.

1 Proof. Following the notation of [43], choose a symmetric positive matrix y 2 such that 1 2 y = (y 2 ) , consider the collection

(µ1,...µr) (3.3.36) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 39

1 1 of eigenvalues (repeated with multiplicity) of the matrix y 2 Ty 2 , and deﬁne

δ (y, T ) := µ , and δ (y, T ) := µ . (3.3.37) + i − | i| µ >0 µ <0 Yi Yi Then, by [43, (4.34K)] there is a function ω(y, T ; α, β) that is holomorphic in (α, β) C2 such that ∈ det(y)−β−ιm/2+κ (y,s) = C (β) δ (y, T )−κ+β+ιm/2+ιq/4δ (y, T )−κ+β−ιp/4 WT p,q Γ (β + ιm/2) Γ (β) + − p · q ! ω(2πy,T ; β + ιm/2, β) e2π tr(yT ) (3.3.38) × · where Cp,q(β) is an entire, nowhere vanishing, function depending only on p and q; here ι −1 β = 2 (s s0(r)). Note that Γq(β) vanishes at β = 0, while the remaining terms are holomorphic;− this proves (i). Moreover, for ﬁxed y and λ R , we may write ∈ >0 −ιmq/2 λ 2πλ tr(yT ) T (λy, s)= f(y, β) ω(2πλy, T ; β + ιm/2, β) e (3.3.39) W Γp(β + ιm/2)Γq(β) for some function f(y, β) that is entire and nowhere-vanishing in β. By [43, Theorem 4.2], for any compact subset U of C, there are positive constants A and B (depending only on T and U) such that the estimate ω(2πy,T ; β + ιm/2, β) e2π tr(yT ) < Ae−(τ−(y,T )) 1+ µ(y, T )−B (3.3.40) | | holds for all y and β U; here τ−(y, T ) = µi and µ(y, T ) = min( µi ). Since T ∈ ∈ µi<0 | | | | is not positive deﬁnite, at least one eigenvalue µ is negative, so τ (y, T ) > 0. Replacing y Pi − by λy and noting that

τ−(λy, T ) = λτ−(y, T ) and µ(λy, T ) = λµ(y, T ) (3.3.41) it follows easily that for fixed y, there is a constant C such that (λy, s) = O(e−Cλ) (3.3.42) WT uniformly for s in some neighbourhood of s = s0, say. This proves (ii) for the case k = 0. The estimates for k 1 follow immediately from Cauchy’s integral formula ≥ k ∂ k! T (λy, s) k T (λy, s) = W k+1 ds. (3.3.43) ∂s W 2πi (s s0) s=s0 I − Finally, we turn to the derivative with respect to λ. The results of [43, Section 5], see especially Lemma 5.7, imply that for fixed y, the function F (λ, s) := (λy, s) (3.3.44) WT is entire in s and extends to a complex function in λ that is holomorphic on Re(λ) > 0 and k ′ satisfies, in the same manner as before, the asymptotic ∂ F (λ, s) = O(e−C Re(λ)) for ∂sk |s=s0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 40

some constant C′. The proposition follows from another application of Cauchy’s integral formula: for a point λ0 > 1, k k ∂ ∂ 1 1 ∂ ′ (λy, s) = F (λ, s) dλ = O(e−C λ0 ), ∂λ ∂sk WT 2πi (λ λ )2 ∂sk |s=s0 s=s0 ! 0 λ=λ0 I − (3.3.45) where the integral is taken around a circle of radius one (say) around λ0. 3.4. The integral of go. For the rest of Section 3 we assume that rk( ) = 1. Recall that we had constructed a Schwartz form E r ν(v) = νi(v) (3.4.1) Xi=1 for v V r, as in Proposition 2.4.5; note that for the permutation matrices ǫ (1 i r) ∈ i ≤ ≤ as in (2.5.22), we have νi(v)= ω(m(ǫi))νr(v).

3.4.1. Let z0 D be the base point ﬁxed in Section 2.2.1, and consider the Schwartz functionsν ˜ , ν˜ ∈ (V r) deﬁned by i ∈ S ν (v, z ) Ωp−r+1(z ) =ν ˜ (v)Ωp(z ), 1 i r (3.4.2) i 0 ∧ 0 i 0 ≤ ≤ and ν˜ :=ν ˜1 + ..., +˜νr; (3.4.3) i here Ω = ΩE = 2π c1( , ). E ∇ r The next lemma computes the images ofν ˜1,..., ν˜r under the map λ: (V ) Ir(V,s0) described in Sections 3.1.6 and 3.2.5. S →

′ λ0 Lemma 3.4.2. Let λ0 be the weight of K given by (2.5.24) and Φ (s) be the unique vector λ0 λ0 in Ir(V,s) of weight λ0 with Φ (e, s) = 1. For 1 i r, let Φ˜ i(s) = r(m(ǫi))Φ (s). Then ≤ ≤ λ (˜ν ) (g′) = ω(g′)(˜ν )(0) = ( 1)r−1Φ˜ (g′,s ), g′ G′ . i i − i 0 ∈ r r ′ Proof. Since the map λ: (V ) Ir(V,s0) is Gr-intertwining, it suﬃces to consider the case i = r. By PropositionS 2.4.5 →we have ν (0) = ( Ω)r−1, and hence r [2r−2] − ω(g′)˜ν (0) = ( 1)r−1Φλ0 (g′,s ), g′ G′ , (3.4.4) r − 0 ∈ r since both sides deﬁne highest weight vectors of weight λ0 in Ir(V,s0) (see Lemma 2.5.4) that moreover agree for g′ = 1, and the K′-types in this representation appear with multiplicity one. Thus, setting Φ(˜ s)= r(m(ǫ ))Φλ0 (s) I (V,s), (3.4.5) i ∈ r 1X≤i≤r we have ω(g′)˜ν(0) = ( 1)r−1Φ(˜ g′,s ). (3.4.6) − 0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 41

λ0 The following lemma relating the Whittaker functional WT ( ,s0) evaluated at Φ with ′ l · the derivative WT (Φ ,s0) is the main ingredient in the proof of Theorem 3.4.9. It will be convenient to work in classical coordinates: for any Φ Ir(V,s) and y Symr(R)>0 (resp. y Her (C) ) in the orthogonal (resp. unitary) case,∈ let ∈ ∈ r >0 (y, Φ,s) := (det y)−ιm/4 W (g′ , Φ˜,s) e2πtr(Ty) (3.4.7) WT T iy where g′ = m(α) for any matrix α GL (K) with det α R and y = α tα. iy ∈ r ∈ >0 · Lemma 3.4.3. Let (y,s) = (y, Φl,s) be given by (3.3.12) and write ′ (y,s) = WT WT WT d (y,s). For any t R we have ds WT ∈ >0 2 d (ty, Φ˜,s )= t ′ (ty,s ). WT 0 ι · dtWT 0 Proof. We begin with the orthogonal case. If k O(r), a change of variables in (3.3.5) shows that ∈

W (m(k)g, Φ,s)= Wt (g, Φ,s), Φ I (V,s). (3.4.8) T kTk ∈ r Hence it suﬃces to consider the case where y is diagonal, i.e. we may assume

y = d(y1,...,yr). (3.4.9) The statement is now a direct computation using (3.1.16), as follows. For 1 j r, let λ0 ≤ ≤ Φ˜ j(s)= r(m(ǫj))Φ (s). By Lemma 3.1.5, we can write Φ˜ (s)=2(s s )−1 Ad(m(ǫ ))X−Φl(s). (3.4.10) j − 0 j r l ˜ 1/2 1/2 1/2 Let F (g)= WT (g, Φ ,s) and Fj(g) = WT (g, Φj ,s) and set y := d(y1 ,...,yr ). Using coordinates xj, yj and θj for the embedding ι := m(ǫ )ι m(ǫ )−1 : Mp (R) G′ = Mp (R) (3.4.11) j j r j 2 → r 2r as in (3.1.13)-(3.1.16), we compute

2−1(s s )F (m(y1/2)) = Ad(m(ǫ ))X−F (m(y1/2)) − 0 j j r d d i d (3.4.12) = iy + y + F (m(y1/2)). − j dx j dy 2 dθ j j j Note that for n(x) N and k′ K′ we have ∈ r ∈ r 1 ′ l 2πi tr(Tx) ′ l 1 l WT n(x)m(y 2 )k , Φ ,s = e (det k ) WT m(y 2 ), Φ ,s (3.4.13) and hence, in coordinates (xj,yj,θj) as above, we ﬁnd d d i d d m iy + y + F (m(y1/2)) = 2πy T + y F (m(y1/2)). − j dx j dy 2 dθ j jj j dy − 4 j j j j (3.4.14) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 42

Substituting this expression in (3.4.12) we conclude that r −1 ˜ −m/4 d m 1/2 2πtr(Ty)) 2 (s s0) T (y, Φj,s)= yi (2πyjTjj + yj )F (m(y )) e − W · dyj − 4 · Yi=1 d = yj T (y,s). (3.4.15) dyj W Adding these equations for j = 1, . . . , r, the lemma follows in the orthogonal case by comparing the Taylor expansions around s = s0. The unitary case follows from analogous considerations, using (3.2.14) instead of Lemma 3.1.5.

3.4.4. Supppose det T = 0 and let 6 Ω (V )= (v , . . . , v ) V r (Q(v , v )) = 2T . (3.4.16) T { 1 r ∈ | i j i,j } Thus Ω (V ) = if and only if (V,Q) represents T , and in this case U(V ) = Aut(V,Q) acts T 6 ∅ transitively on ΩT (V ); assuming this, let dµ(v) be an U(V )-invariant measure on ΩT (V ) and consider the functional (V r) C defined by S → φ φ(v) dµ(v). (3.4.17) 7→ ZΩT (V ) This functional is obviously U(V )-invariant and non-zero, and hence defines a Whittaker SW functional on Rr(V ). We denote by dµ(v) the unique U(V )-invariant measure on ΩT (V ) such that, for any φ (V r), ∈ S SW −1 φ(v) dµ(v) = γ r W (e, Φ,s ), where Φ(g) := ω(g)φ(0). (3.4.18) V · T 0 ZΩT (V ) We denote by dg(2) the invariant measure on U(V ) defined as dg(2) = dp dk, where dk 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 dp is× the left Haar measure on×P induced by the invariant p + volume form Ω on D ∼= G/K.

Lemma 3.4.5. Let λ0 be given by (2.5.24) and let v = (v1, . . . , vr) ΩT (V ), where ∈ 0 det T = 0. Let Gv U(V ) be the pointwise stabilizer of v1, . . . , vr and Γv Gv be a torsion6 free subgroup⊂ of ﬁnite covolume. Then, for any g′ h G′ , we havei ⊂ ∈ r ′ p−r+1 ι ′ ˜ ω(g )ν(v) Ω = CT,Γv WT (g , Φ,s0), + ∧ −2 ZΓv\D ˜ where Φ is as in (3.4.5) and CT,Γv is the non-zero constant given by (2) 2 r dg /dgv −1 C v = ( 1) Vol(Γv Gv, dgv) γ r . T,Γ ι − \ dµ(v)SW V (2) Here dgv is an arbitrary Haar measure on Gv and dg /dgv denotes the quotient measure (2) on ΩT (V ) ∼= U(V )/Gv induced by dg and dgv. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 43

0 Proof. The hypothesis that det T = 0 implies that Gv is reductive, hence the invariant (2) 6 measure dg /dgv exists. By (3.4.2) and Proposition 2.4.5.(d), we haveν ˜(gv) =ν ˜(v) for any g U(V ) stabilizing V −. We compute ∈ ω(g′)ν(v) Ωp−r+1 = ω(g′)˜ν(g−1v) dg(2) + ∧ ZΓv\D ZΓv\U(V ) (2) ′ −1 dg = Vol(Γv Gv, dgv) ω(g )˜ν(g v) \ dgv Gv\ZU(V ) (2) dg /dgv ′ SW = Vol(Γv Gv, dgv) ω(g )˜ν(v) dµ(v) \ dµ(v)SW ZΩT (V ) (2) r−1 dg /dgv −1 ′ = ( 1) Vol(Γv Gv, dgv) γ r W (g , Φ˜,s ), − \ dµ(v)SW V T 0 (3.4.19) where the last equality follows from (3.4.18) and (3.4.6). 3.4.6. We will now apply the results obtained in this section to compute the integral (3.0.1) under our standing assumption that rk( ) = 1. Rather than aiming for the most general result, we restrict to cocompact arithmeticE subgroups Γ and integral vectors v (see below); this suffices for the application to compact Shimura varieties in Section 5. Fix a lattice L V and a torsion free cocompact arithmetic subgroup Γ G stabilizing ⊂ + ⊂ L, and let Γv = Stab v , . . . , v and X =Γ D . Γh 1 ri Γ \ r Lemma 3.4.7. Assume that v L is non-degenerate and let Z(v)Γ be the image of the + ∈ natural map Γv D X . \ v → Γ (a) The sum o o −1 g (v)Γ = g (γ v) γ∈XΓv\Γ converges absolutely to a smooth form on X Z(v) that is locally integrable on X . Γ − Γ Γ (b) As currents on XΓ we have M o 1/2 −1 dt M→+∞ o ν (t γ v)[2r−2] g (v)Γ. 1 t −−−−−→ γ∈XΓv\Γ Z Proof. Let us first show that go(v) converges as claimed. Let z D+ and pick a relatively Γ ∈ compact neighborhood U of z. Recall that there is a positive definite form Qz defined by (2.3.4) that varies continuously with z. Write Γv = S S (disjoint union), with 1 ⊔ 2 ′ S1 = v Γv min hz(sv′ ) < 1 ; (3.4.20) { ∈ | z∈U } r o then S1 is finite since Γv L . We can split the sum defining g (v)Γ accordingly; the ⊂ + + sum over S converges to a locally integrable form on D that is smooth outside v′ D ′ 1 ∪ ∈S1 v by Proposition 2.1.7. The sum over S2 converges to a smooth form on U by the estimate GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 44

(2.1.10) and the standard argument of convergence of theta series. By Proposition 2.4.5.(d), o the current deﬁned by g (v)Γ is invariant under Γ, and this shows (a). For part (b), note that for each v′ S we have M νo(t1/2v′) dt go(v′) as M + (as currents on ∈ 1 ∫1 [2r−2] t → → ∞ D+) by dominated convergence, as remarked in the proof of Proposition 2.1.7. For the sum over S2 we apply the bound (2.4.6), and (b) follows. Lemma 3.4.8. Assume that v Lr is non-degenerate. Then the integral (3.0.1) converges and ∈ M o o dt g (v) Ωp−r+1 = lim ν (t1/2v) Ωp−r+1 . + ∧ M→+∞ + ∧ t ZΓv\D Z1 ZΓv\D o p−r+1 Proof. By Lemma 3.4.7.(a), the integral (3.0.1) equals + g (v) Ω and so it ∫Γ\D Γ ∧ converges since XΓ is compact. Applying Lemma 3.4.7.(b) and unfolding the sum and the integral proves the claim.

r Theorem 3.4.9. Suppose det T = 0 and v ΩT (V ) L . Let CT,Γv be as in Lemma 3.4.5 and l be the weight in (3.3.11). Recall6 that∈ we assume∩ that rk( ) = 1. E (a) If T is not positive deﬁnite (so that Dv = ), then ∅ −2πtr(T ) o p−r+1 ′ l e g (v) Ω = CT,Γv WT (e, Φ ,s0). + ∧ ZΓv\D (b) If T is positive deﬁnite (so that Dv = ), then 6 ∅ −2πtr(T ) o p−r+1 ′ l e g (v) Ω = CT,Γv WT (e, Φ ,s0) + ∧ ZΓv\D ′ l ι Γr(ιm/2) C v W (e, Φ ,s ) log det(πT ) . − T,Γ T 0 2 − Γ (ιm/2) r Proof. The integral converges by Lemma 3.4.8. We compute

∞ o o dt g (v) Ωp−r+1 = ν (t1/2v) Ωp−r+1 + ∧ + ∧ t ZΓv\D Z1 ZΓv\D ! ∞ dt = e2πtr(tT ) ν(t1/2v) Ωp−r+1 + ∧ t Z1 ZΓv\D ! ∞ ιrm 2πtr(tT ) 1/2 p−r+1 − dt = e ω(m(t 1r))ν(v) Ω t 4 + · ∧ t Z1 ZΓv\D ! ∞ ιrm ι 2πtr(tT ) 1/2 ˜ − dt = CT,Γv e WT (m(t 1r), Φ,s0) t 4 , −2 1 · t Z (3.4.21) where we have used Lemma 3.4.8 and Lemma 3.4.5 for the ﬁrst and last equality respectively. The last integrand equals (t 1 , Φ˜,s ), and so applying Lemma 3.4.3 we conclude WT · r 0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 45

that o p−r+1 ′ ′ g (v) Ω = CT,Γv ( T (e, s0) lim T (t 1r,s0)). (3.4.22) + ∧ W − t→∞ W · ZΓv\D If T is not positive deﬁnite, then the limit in the above expression vanishes by part (ii) of Proposition 3.3.4; this proves (a). If T is positive deﬁnite, then (b) follows from Proposi- tion 3.3.3.(iii)-(iv).

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, and apply the results of the previous sections to construct\ Green forms for the special cycles Z(T, ϕf ) on orthogonal and unitary Shimura varieties introduced by Kudla in [22].

4.1. Orthogonal Shimura varieties. Let us brieﬂy recall the deﬁnition and basic properties of orthogonal Shimura varieties attached to quadratic spaces over F ; see [22] for more detail.

4.1.1. Let F be a totally real field with real embeddings σ1,...,σd. 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 signature(Vσ ) = (4.1.1) i (p + 2, 0) if i = 1, ( 6 where we abbreviate V := V R for the real quadratic spaces at each place. σi ⊗F,σi Attached to Vσ1 is the symmetric space D(V)= D(Vσ1 ) defined in (2.2.6); recall that it is defined to be D(V)= D(V ) = [v] P(V (C)) Q(v, v) = 0, Q(v, v) < 0 . (4.1.2) σ1 { ∈ σ1 | i } Here Q( , ) is the C bilinear extension of the bilinear form on V . Let · · − σ1 HV := ResF/Q GSpin(V). (4.1.3) Then H (R) GSpin(V ) GSpin(V ) (4.1.4) V ≃ σ1 ×···× σd acts transitively on D(V) via the first factor. Definition 4.1.2. For a compact open subgroup K H(A ), consider the Shimura variety ⊂ f X := H (Q) D(V) H (A )/K V,K V \ × V f If K is neat, then XV,K is a complex quasi-projective algebraic variety. If V is moreover 3 anisotropic, then XV,K is projective.

3Our assumptions on the signature of V imply that this is always the case when F =6 Q. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 46

The space XV,K may be written in a perhaps more familiar fashion: fix a connected + + + component D D(V) and let HV(R) denote its stabilizer in HV(R). Setting HV(Q) = H (R)+ H (Q⊂), there exist finitely many elements h ,...h H (A ) such that V ∩ V 1 t ∈ V f + HV(Af )= HV(Q) hjK; (4.1.5) aj then we may write X Γ D+ =: X (4.1.6) V,K ≃ j V,j j j a a as a disjoint union of quotients of D+ by discrete subgroups Γ = H (Q)+ h Kh−1 . (4.1.7) j V ∩ j j + In general, we regard the quotients XV,K and Γj D as orbifolds. In particular, for a + \ Γj-invariant differential form η of top degree on D , we define

′ −1 η = [Γj : Γ ] η, (4.1.8)

+ ′ + [Γj \ZD ] Γ \ZD ′ where Γ Γj is any neat subgroup of finite index, and set = + . ⊂ ∫[XK ] j [Γj \D ] 4.1.3. The theory of canonical models of Shimura varieties (see [42])P impliesR the existence of a quasi-projective model K over Spec(F ), which is projective when V is anisotropic, such that X C X . XK ⊗F,σ1 ≃ V,K From the point of view of arithmetic intersection theory, it will be important to work with all the complex fibres of K simultaneously; the remaining fibres have the following concrete description. X For each k = 2,...,d, let V[k] denote a quadratic space over F such that (i) V[k] V , i.e. the signature of V[k] is (p, 2); σk ≃ σ1 σk (ii) V[k]σ1 Vσk ; (iii) and (V≃[k]) V at all other places. w ≃ w The space V[k] is unique up to isometry, and we have V[1] V. Fix, once and for all, identifications ≃ V[k] A V A (4.1.9) ⊗F f ≃ ⊗F f inducing identifications H (A ) H (A ) V[k] f ≃ V f for all k, and so in particular we may view K H (A ). Then, setting D(V[k]) = ⊂ V[k] f D(V[k]σk ) D(V), the theory of conjugation of Shimura varieties (see [37, 38], as well as [8, Section≃ 7] for our particular situation) gives identifications C X = H (Q) D(V[k] ) H (A ) K. (4.1.10) XK ×F,σk ≃ V[k],K V[k] σk × V[k] f / . GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 47

In particular, viewing as a scheme over Q via the map Spec(F ) Spec(Q), we have XK → d (C) = X . (4.1.11) XK V[k],K ka=1 4.2. Special cycles. Recall that in Section 2.2.2 we have deﬁned the tautological bundle r ∨ r over D(V[k]) and a global section sv of ( ) for any v (V[k]σk ) , whose zero locus E E ∈ r Z(sv) we denote by Dv. Given a rational vector v = (v , . . . , v ) V[k] , we set 1 r ∈

Dv = Z(sσk(v)) (4.2.1) + + and D = Dv D(V[k]) . v ∩ Let Hv(Q) be the pointwise stabilizer of spanF v1, . . . , vr in HV[k](Q). Given a com- + { } ponent Xj =Γj D XV[k],K associated to hj as in (4.1.6), let Γj(v)=Γj Hv(Q); then the natural map\ ⊂ ∩ Γ (v) D+ Γ D+ = X (4.2.2) j \ v → j\ j defines a (complex algebraic) cycle on Xj that we denote by c(v, Xj ). In addition, recall that for a matrix T Sym (F ), we had defined ∈ r Ω (V[k]) = v V[k]r T (v)= T ; (4.2.3) T { ∈ | } where T (v) = ( 1 Q(v , v )) Sym (F ) is the moment matrix of v. 2 i j i,j ∈ r r K Definition 4.2.1 ([22]). Let ϕf (V(Af ) ) be a K-invariant Schwartz function which, ∈ S r for each k = 1,...,d, may be viewed as a Schwartz function on V[k](Af ) via (4.1.9). For T Sym (F ), define the weighted special cycle Z(T, ϕ , K) on (C)= X by ∈ r f XK V[k],K d ` −1 Z(T, ϕf , K) = ϕf (hj v) c(v, Xj ). k=1 Xj ⊂XV[k],K v∈ΩT (V[k]) X X modX Γj This is a complex algebraic cycle on (C) that is in fact defined over F . XK It follows from the discussion after (2.2.12) that if Z(T, ϕf , K) is non-empty, then T is totally positive semi-definite and the codimension of Z(T, ϕf , K) is equal to the rank of T . Note that this definition is independent of all choices. Moreover, if K′ K is an ⊂ open subgroup of finite index and π : K′ K is the natural covering map, then ∗ ′ X → X π Z(T, ϕf , K) = Z(T, ϕf , K ). See [22, 5] for a proof of this and further properties of these cycles. §

To lighten notation, we will once and for all ﬁx a compact open subgroup K HV(Af ), and write, for example, Z(T, ϕ )= Z(T, ϕ , K), X = X and = ,⊂ etc. f f V[k] V[k],K X XK 4.3. Green forms for special cycles. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 48

4.3.1. For the moment, fix a real embedding σ : F R and a component k → X =Γ D+ X (C) j j\ ⊂ V[k] ≃Xσk + + with D = D(V[k]σk ) ; here Γj is attached to hj HV(Af ) as in (4.1.7). ∈ r Let T Symr(F ) with det T = 0. Any collection of vectors v = (v1, . . . , vr) V[k] with T (v)=∈ T is necessarily linearly6 independent. For such v, we defined in Section∈ 2.6.1 a form satisfying the equation c o o dd g (σk(v)) + δDv = ϕ (σk(v))[2r] (4.3.1) of currents on D+. We introduce Green forms that depend on an auxiliary parameter y Sym (F R) ∈ r ⊗Q ≫0 as follows: fix some element α GLr(FR) with totally positive determinant such that t ∈ r K y = α α and, for a Schwartz function ϕf S(V(Af ) ) , define a form gj(T, y, ϕf )σk on D+ by· setting ∈ o g (T, y, ϕ ) := ϕ (h−1v) g (σ (v) σ (α)); (4.3.2) j f σk f j k · k v∈ΩXT (V[k]) here we view σk(α) GLr(R) via the R-linear map σk : F Q R R induced by σk. The convergence of this sum∈ to a locally integrable form on D+⊗follows→ from Lemma 3.4.7.(a) and the fact that the number of orbits of Γj on Supp(ϕf ) ΩT (V[k]) is finite. Note that the definition is independent of the choice of α,∩ by Proposition 2.4.5.(f). More- over, the form gj(T, y, ϕf )σk is invariant under the action of Γj by Proposition 2.4.5.(d), and so it descends to a form on the connected component Xj that, abusing notation, we also denote by gj(T,y,ϕf )σk . Finally, let

g(T, y, ϕf )σk (4.3.3) denote the form on X (C) whose restriction to X is g (T, y, ϕ ) . Essentially by V[k] ≃Xσk j j f σk construction, it is a Green form for the cycle Z(T, ϕf )σk ; more precisely, let ω(T, y, ϕf )σk be the diﬀerential form on XV[k] whose restriction to the component Xj is

−1 o ω(T, y, ϕf )σ = ϕf (h v) ϕ (σk(v) σk(α))[2r], (4.3.4) k Xj j · v∈Ω (V[k]) XT

Then the form ω(T, y, ϕf )σk is the T ’th coeﬃcient of the theta function T ΘKM(τ ; ϕf )σk := ω(T, y, ϕf )σk q , (4.3.5) T ∈SymXr(F ) d T d 2πitr(τiσi(T )) where τ (Hr) and q := i=1 e . This theta function was considered (in much greater∈ generality) by Kudla and Millson [27]. Applying the identity (4.3.1Q) of currents on D, summing over x with T (x) = T and descending to the Shimura variety XV[k] yields the equation ddcg(T, y, ϕ ) + δ = ω(T, y, ϕ ) (4.3.6) f σk Z(T,ϕf )σk f σk GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 49

of currents on X (C). The collection V[k] ≃Xσk g(T, y, ϕ ) k = 1,...,d (4.3.7) { f σk | } deﬁnes a Green form g(T, y, ϕ ) for the cycle Z(T, ϕ ) on , in the sense of [44, Chap. II]. f f X 4.3.2. We will next construct a current g(T, y; ϕ ) = g(T, y, ϕ ) for an arbitrary f { f σk } matrix T Symr(F ) and y Symr(FR)≫0. For the moment, choose an embedding σk and ∈ ∈ r a component Xj = Γj D σk (C). Recall that for any v = (v1, . . . , vr) V[k] , we had deﬁned the current \ ⊂ X ∈ ∞ o o dt g (σ (v); ρ) := ν (√tv) (4.3.8) k [2r·rk(E)−2] tρ+1 Z1 on D, see (2.6.4). here ρ is a complex parameter. t Let y Symr(FR)≫0 and let α GLr(FR)≫0 such that y = α α. Given a Schwartz function ∈ϕ S(V(A )r)K, consider∈ the sum · f ∈ f o o g (T, y, ϕ ; ρ) := ϕ (h−1v) g (σ (v) σ (α); ρ) , (4.3.9) j f σk f j k · k ΩTX(V[k]) + + viewed as a current on D = D(V[k]σk ) . Note that the right hand side is independent of the choice of α by Proposition 2.6.6.(ii).

Proposition 4.3.3. The sum (4.3.9) converges for Re(ρ) 0 to a Γj-invariant current on D+ that has a meromorphic continuation to Re(ρ) > 1≫/2. In particular, the constant term in the Laurent expansion − o o gj (T, y, ϕf )σk := CTρ=0 gj (T, y, ϕf ; ρ)σk descends to a current on X =Γ D+. j j\ Proof. For convenience, we take σk = σ1 and suppress this index from the notation, writing y = y1 etc. The proof for the other embeddings σk is identical. ∗ + + Let η Ac(D ) with compact support. Choose an open set U D with compact closure such∈ that supp(η) U. Let ⊂ ⊂ r −1 S := v V T (v)= T, ϕ (h v) = 0, and Dv U = , (4.3.10) 1 { ∈ | f j 6 ∩ 6 ∅} and r −1 S := v V T (v)= T, ϕ (h v) = 0, and Dv U = , (4.3.11) 2 { ∈ | f j 6 ∩ ∅} so that S S indexes the non-zero terms appearing on the right hand side of (4.3.9). 1 ⊔ 2 Note also that S1 is ﬁnite, while there exists a bound C > 0 such that r min hz(σ(vi),σ(vi)) >C (4.3.12) z∈U Xi=1 for all v = (v , . . . , v ) S . 1 r ∈ 2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 50

By Proposition 2.6.6, the ﬁnite sum

o ϕ (h−1v) g (σ(v) σ(α); ρ) η (4.3.13) f j · ∧ x D X∈S1 Z converges for ρ large, and has meromorphic continuation to Re(ρ) > 1/2. For the same − sum where v now runs over S2, the exponential decay estimate (2.1.10) and standard arguments for the convergence of theta series imply that the sum

−1 o ϕf (hj v) g (σ(v) σ(α); ρ) η D+ · ∧ v∈S2 Z X (4.3.14) ∞ dt −1v o √ v = ϕf (hj ) ν tσ( ) σ(α) η ρ+1 + · ∧ t v 1 D X∈S2 Z Z converges absolutely and locally uniformly in ρ, and hence deﬁnes an entire function of ρ.

o We next show that by patching together the gj (T, y, ϕf )σk on XV[k], we obtain a current satisfying an analogue of Green’s equation (4.3.6). o Proposition 4.3.4. Let g (T, y, ϕf ) denote the current on (C) = σ (C) whose o X k X k restriction to a connected component Xj σk (C) is gj (T, y, ϕf )σk . Then there is an identity of currents ⊂ X ` c o r−rk(T ) dd g (T, y, ϕ )+ δ Ω ∨ = ω(T, y, ϕ ) (4.3.15) f Z(T,ϕf ) ∧ E f ∨ ∗ where ΩE∨ = c1( , ) and ω(T, y, ϕf ) is the diﬀerential form on (C) whose restriction to (C) is ω(T,E y,∇ ϕ ) . X Xσk f σk Proof. It suﬃces to prove the given identity, for each component Xj, at the level of Γj- invariant currents on D. The estimates in the proof of Proposition 4.3.3 allow us to write

c o c −1 o dd gj (T, y, ϕf )σk = dd CTρ=0 ϕf (hj v) g (σk(v) σk(α); ρ) X j " v · # X −1 c o = ϕf (hj v) dd CTρ=0 g (σk(v) σk(α); ρ) . (4.3.16) v · X The proposition follows immediately from Proposition 2.6.6.(vi).

Finally, in order to obtain agreement with the derivatives of Eisenstein series in our main o ′ theorem, we introduce a modiﬁed version of g (T, y, ϕf ). We write det A for the product of non-zero eigenvalues of a square matrix A, with the convention det′(0) = 1.

Definition 4.3.5. Let y = (yv)v|∞ Symr(FR)≫0 and T Symr(F ), and define a current ∗ ∈ ∈ g(T, y, ϕf ) D ( (C)) as follows: if T is not totally positive semidefinite, set ∈ X o g(T, y, ϕf ) := g (T, y, ϕf ) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 51

and if T is totally positive semideﬁnite, set o g(T, y, ϕf ) := g (T, y, ϕf ) ′ det σv(T ) det yv r−rk(T )−1 ∨ log ′ · δZ(T,ϕf )σv ΩE − det (σv(T )yv) ∧ Xv|∞

∨ ∨ ∗ i ∨ where ΩE = c1( , ) = 2π c1( , ). Note that the additional term is closed, and vanishes if T is non-degenerate.E ∇ E ∇ Note also that when det T = 0, the current deﬁned by the Green form (4.3.7) agrees with the one in Deﬁnition 4.3.56 by Proposition 2.6.6.(iv).

4.4. 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 with [F : Q]= d, and that V is a Hermitian space over E, with Hermitian form Q( , ). · · Fix a CM type Φ = σ1,...,σd Hom(E, C). For each i, the space Vσi = V σi,E C is a complex Hermitian{ space; we assume} ⊂ ⊗

(p,q), if i = 1 signature Vσi = (4.4.1) ((p + q, 0), if i = 2, . . . d. for some integers p,q > 0. Let D(V ) := z V negative-deﬁnite subspace, dim z = q , σ1 { ⊂ σ1 C } (see (2.2.5)) be the symmetric space attached to the real points of the unitary group

HV := ResF/Q U(V). (4.4.2)

Just as in Section 4.1, a ﬁxed compact open subgroup K HV(Af ) determines a complex Shimura variety ⊂ X = X := H (Q) D(V ) H (A )/K (4.4.3) V V,K V \ σ1 × V f which is quasi-projective, and projective when V is anisotropic; choosing representatives h ,...,h for the double coset space H (Q) H (A )/K gives a decomposition 1 t V \ V f X Γ D =: X , where Γ := H (Q) h Kh−1 . (4.4.4) V ≃ j\ j j V ∩ j j aj aj

Let denote the canonical model over E, so that σ1 (C) XV. For the other complex embeddings,X the story is similar to Section 4.1. ForX each k =≃ 1,...,d, let V[k] denote the (unique up to isometry) E-Hermitian space such that V[k] V ; • σk ≃ σ1 V[k]σj is positive deﬁnite, for j = k; and • V[k] V at all ﬁnite places v.6 • v ≃ v GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 52

Identifying V[k] A V A , and in particular, viewing K as a subgroup of H (A ), ⊗Q f ≃ ⊗Q f V[k] f we may define the complex Shimura variety XV[k] = XV[k],K in the same way as XV,K. Now suppose ρ Hom(E, C) and let σk be the element of the CM type such that ρ = σ . Then there∈ is an identification |F k|F (C) X ; (4.4.5) Xρ ≃ V[k] this follows from the general considerations of [38] and [37, Section II.4], or [35, Section 3A] for the case at hand. The special cycles are defined just as in Section 4.2: recall that a tuple v = (v1, . . . , vr) (V[k])r determines a section of ( r)∨, where is the rank q tautological bundle on D(V[k]).∈ E E Its vanishing locus Dv D(V[k]) determines a cycle c(v, Xj ) on each component Xj = ⊂ ′ ′ Γj D(V[k]), which is either empty or of codimension r q, where r = dimspan v1, . . . , vr . \ r K { } Given a K-invariant Schwartz function ϕf (V(Af ) ) and a matrix T Herr(E), define the (complex) special cycle ∈ S ∈

−1 Z(T, ϕf ) = ϕf (hj v) c(v, Xj ) (4.4.6) Xj ⊂XV[k]⊂X (C) ΩT (V[k]) X modX Γj exactly as in Definition 4.2.1, with the sum taken over all connected components of (C); as before, these are the complex points of a rational cycle. X Given y Her (E R) , use the CM type Φ to identify ∈ r ⊗Q ≫0 y = (y ,...,y ) (Her (C) )d Her (E R) . (4.4.7) 1 d ∈ r >0 ≃ r ⊗Q ≫0 For ρ: E ֒ C, let g(T, y, ϕf )ρ denote the current on ρ(C) XV[k] whose restriction to a component→X X is given by X ≃ j ⊂ V[k] o −1 o g (T, y, ϕf )ρ = CTs=0 ϕf (h v) g (ρ(v) σk(α); s) . (4.4.8) Xj j · v∈Ω (V[k]) XT t Here α GLr(ER) is any matrix with totally positive determinant such that y = α α. The independence∈ of the choice of α follows again from Proposition 2.6.6.(ii). · The analogue of Proposition 4.3.4, which can be proved with straightforward modifica- tions to the arguments in the previous section, is: Proposition 4.4.1. The following equation of currents on (C) holds: Xρ c o r−rk T dd g (T, y, ϕ ) + δ Ω ∨ = ω(T, y, ϕ ) . f ρ Z(T,ϕf )ρ ∧ E f ρ Here Ω ∨ = ctop( ∨, )∗ is the top Chern-Weil form of the Hermitian bundle ∨. E E ∇ E o As in the orthogonal case, we introduce a modified version of g (T, y, ϕf ) by adding a closed current. Definition 4.4.2. Let y = (y ) Her (E ) and T Her (E). We set v v|∞ ∈ r R ≫0 ∈ r o g(T, y, ϕf ) := g (T, y, ϕf ) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 53

if T is not totally positive semidefinite and o g(T, y, ϕf ) := g (T, y, ϕf ) ′ det σv(T ) det yv ∨ ∗ r−rk(T )−1 ∨ log ′ · δZ(T,ϕf )σv crk(E)−1( , ) ΩE − det σv(T )yv ∧ E ∇ ∧ Xv|∞ if T is totally positive semidefinite; when T is non-degenerate, the additional term vanishes. Example 4.4.3. We treat the orthogonal and unitary cases simultaneously here. Assume that V is anisotropic and that 0 T = (4.4.9) S with S non-degenerate, where S Symt(F ) (resp. Hert(E)) in the orthogonal (resp. unitary) cases, and t = rk T . Then any∈ x with T (x)= T is of the form x = (0,..., 0, x′) with T (x′)= S. Suppose that ϕ = ϕ′ ϕ′′ with ϕ′ S(V(A )r−t) and ϕ′′ S(V(A )t), so that f f ⊗ f f ∈ f f ∈ f Z(T, ϕ )= ϕ′ (0) Z(S, ϕ′′). Suppose furthermore that y is of the form f f · f y′ y = , (4.4.10) y′′ where y′ and y′′ are totally positive definite of rank r t and t, respectively. − It follows from Example 2.6.8 that, after descending to the Shimura variety σk (C), we have the equation of currents X g(T,y, ϕf )σk ′ ′′ ′′ r−t ′ ∨ ∗ r−t−1 = ϕ (0) g(S, y , ϕ )σ Ω ∨ log(det y ) δZ(S,ϕ′′) crk(E)−1( , ) Ω ∨ f · f k ∧ E − f σk ∧ E ∇ ∧ E ∞ h ′′ ′′ du ∨ ∗ r−t−1 + (r t) ω(S, uy , ϕ )σ . δ ′′ c ( , ) Ω ∨ . f k Z(S,ϕf )σk rk(E)−1 E − 1 − u ∧ E ∇ ∧ Z h i (4.4.11) i

Since ω(S, uy′′, ϕ′ ) and δ ′′ are cohomologous, the term appearing on the second line f Z(τ,ϕf ) above is exact, i.e.

′′′ ′′ ′′ r−t ′ ∨ ∗ r−t−1 g(T, y, ϕf )σ ϕ (0) g(S, y , ϕ )σ Ω ∨ log(det y ) δZ(S,ϕ′′) crk(E)−1( , ) Ω ∨ k ≡ f · f k ∧ E − k f σk ∧ E ∇ ∧ E h Dr−1,r−1( (C)) im∂ + im∂. (4.4.12) i ∈ Xσk This expression generalizes a similar term appearing indirectly in the work of Kudla- Rapoport-Yang [32, p. 178], which dealt with the case of a Shimura curve over Q and r = 2, r′ = 1. Finally, we note that for T = 0r and V anisotropic, a computation along the same lines o gives g (0r, y, ϕf ) = 0 and the pleasant expression ∨ ∗ r−1 g(0 , y, ϕ ) = log(det y ) ϕ (0) c ( , ) Ω ∨ . (4.4.13) r f σk − k · f · rk(E)−1 E ∇ ∧ E GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 54

Remark 4.4.4. We continue to assume V is anisotropic, and note two useful invariance properties for g(T, y, ϕf ). Set k = F in the orthogonal case and k = E in the unitary one. 0 (1) Suppose T = S for a non-degenerate matrix S Symt(F ) (resp. S Hert(E)), and θ SL (k ) is of the form ∈ ∈ ∈ r R 1 θ = r−t . (4.4.14) 1∗ t Then t g T, θ y θ, ϕf = g(T, y, ϕf ). (4.4.15) t −1 −1 (2) Suppose γ SLr(k), and let T [γ] := γ T γ . Assume that γ is chosen so that det′ T [γ] =∈ det′ T . Then t ′ g T [γ], γ y γ, ϕf = g (T, y, ϕf ) (4.4.16) where ϕ′ (x)= ϕ (x γ). f f · Given T , one can always ﬁnd an element γ SL (k) as above such that T [γ]= 0 for ∈ r S some non-degenerate matrix S. Similarly, we may choose θ as above, such that θytθ = ′ y ′′ y′′ with y of the same rank as S; thus we may always place ourselves in the setting of Example 4.4.3.

4.5. Star products on XK . In this section, we continue to treat both the orthogonal and unitary cases. Let g(T1, y1, ϕ1) and g(T2, y2, ϕ2) denote two currents attached to special cycles Z(T1, ϕ1) and Z(T2, ϕ2). Assume that T1 and T2 are non-degenerate and that Z(T1, ϕ1) and Z(T2, ϕ2) intersect properly, and consider the star product g(T , y , ϕ ) g(T , y , ϕ ) (4.5.1) 1 1 1 ∗ 2 2 2 := g(T , y , ϕ ) δ + g(T , y , ϕ ) ω(T , y , ϕ ) 1 1 1 ∧ Z(T2,ϕ2) 2 2 2 ∧ 1 1 1 ∗ in D (XK ). Theorem 4.5.1. Let ϕ = ϕ ϕ . With assumptions as above, 1 ⊗ 2 y1 g(T , y , ϕ ) g(T , y , ϕ ) g (T, ( y ) , ϕ) 1 1 1 ∗ 2 2 2 ≡ 2 T = T1 ∗ X∗ T2 ∗ ∗ in D˜ (XK ) := D (XK )/(im∂ + im∂). t Proof. Fix an embedding σk, matrices αi GLri (K) such that σk(yi)= αi αi for i = 1, 2, + ∈ · + and a component Xj = Γj D σk (C); working with Γj-invariant currents on D , the proof of Proposition 4.3.3(i)\ implies⊂ X that

−1 −1 o o g(T1, y1, ϕ1) g(T2, y2, ϕ2) = ϕ1(h x1) ϕ2(h x2) g (v1) g (v2) (4.5.2) ∗ Xj j j ∗ x ∈Ω (V[k]) 1 XT1 x ∈Ω (V[k]) 2 T2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 55

where we write v := σ (x ) α ; note that the proper intersection assumption implies i k i · i that v = (v1, v2) is regular for any non-zero term above, i.e. Dv is either empty or of codimension (r + r )rk( ). By Theorem 2.7.1, the previous line becomes 1 2 E o ϕ (h−1x )ϕ (h−1x ) g (v) ∂α(v , v ) ∂β(v , v ) 1 j 1 2 j 2 − 1 2 − 1 2 x 1∈ΩT1 (V[k]) x X 2∈ΩT2 (V[k]) o = ϕ(h−1x)g (v) ϕ (h−1x )ϕ (h−1x ) ∂α(v , v )+ ∂β(v , v ) j − 1 j 1 2 j 2 1 2 1 2 T ∗ x∈Ω (V[k]) x ∈Ω (V[k]) T = 1 T 1 T1 X∗ T2 X x X 2∈ΩT2 (V[k])

y1 −1 −1 = g (T, ( y ) , ϕ) ϕ (h x )ϕ (h x ) ∂α(v , v )+ ∂β(v , v ) 2 − 1 j 1 2 j 2 1 2 1 2 T ∗ x ∈Ω (V[k]) T = 1 1 T1 X∗ T2 x X 2∈ΩT2 (V[k]) (4.5.3) where x = (x1, x2) and α(v1, v2) and β(v1, v2) are as in Theorem 2.7.1. Again, an argument along the lines of Proposition 4.3.3(i) shows that the sum −1 −1 ϕ1(hj x1)ϕ2(hj x2)α(v1, v2), (4.5.4) x 1∈ΩT1 (V[k]) x X 2∈ΩT2 (V[k]) and its analogue with α replaced by β, converge to currents on D+, that are moreover Γj-invariant by Theorem 2.7.1. The theorem follows upon descending to Xj.

5. Local archimedean heights and derivatives of Siegel Eisenstein series Here we prove Theorem 5.3.1, our main global result relating archimedean local heights and derivatives of Siegel Eisenstein series. We review the definition of these Eisenstein series and the Siegel-Weil formula in Section 5.1. For the proof we also need to explicitly ′ determine the asymptotics of the Fourier coefficients ET (λy, Φf ,s0) as λ ; we do this in Section 5.2 and give the proof of Theorem 5.3.1 in Section 5.3. →∞ In Section 5.5 we explain how, using our results, Kudla’s conjectural arithmetic Siegel- Weil fomula can be rephrased in terms of Faltings heights of special cycles. Fix a totally real number field F of degree d, and a CM extension E, and set F, orthogonal case, R, orthogonal case k = and K = (5.0.1) (E, unitary case (C, unitary case.

Let σ1,...,σd the archimedean places of F in the orthogonal case, or the elements of a ﬁxed CM type of E in the unitary case.

We ﬁx an m-dimensional Hermitian k-vector space (V,Q) such that Vσi := V k,σi K is positive deﬁnite when i> 1 and ⊗ (p, 2), orthogonal case sig Vσ1 = (5.0.2) ((p, 1), unitary case GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 56

with p 1. From now on we assume that V is anisotropic. Finally,≥ let η : F × A× 1 the quadratic character corresponding to E and ﬁx a \ F → {± } × × × × m unitary character χ: E AE C such that χ A = η . \ → | F 5.1. Siegel Eisenstein series and the Siegel-Weil formula.

5.1.1. Let Mp2r(AF ) be the metaplectic double cover of Sp2r(AF ) and for a positive integer r, set

′ Mp2r(AF ), case 1, Gr(A)= (5.1.1) (U(r, r)(AF ), case 2, ′ Denote by Pr(A) the standard Siegel parabolic of Gr(A); then Pr(A) = Mr(A) ⋉ Nr(A), where

Mr(A)= (m(a),ǫ) a GLr(AF ), ǫ = 1 , { | ∈ ± } (5.1.2) N (A)= (n(b), 1) b Sym (A ) r { | ∈ r F } in case 1 and M (A)= m(a) a GL (A ) ,N (A)= n(b) b Her (A ) (5.1.3) r { | ∈ r E } r { | ∈ r E } in case 2. We also write (m(a), 1), for a GL (A ) in case 1, m(a)= ∈ r F m(a), for a GL (A ) in case 2, ( ∈ r E (n(b), 1), for b Sym (A ) in case 1, n(b)= ∈ r F (5.1.4) n(b), for b Her(A ) in case 2, ( ∈ E (wr, 1), case 1, wr = (wr, case 2.

1r where wr = ( −1r ). The multiplication in Mr(A) in case 1 is deﬁned by (m(a ),ǫ ) (m(a ),ǫ ) = (m(a a ),ǫ ǫ (det a , det a ) ), (5.1.5) 1 1 · 2 2 1 2 1 2 1 2 A where ( , ) denotes the Hilbert symbol of F . · · A Deﬁne a character χV of Mr(A) as follows: in case 1, set

−1 m(m−1)/2 ǫ γA(det a, ψ) , if m is odd, χV (m(a),ǫ)= det a, ( 1) det V · (5.1.6) − A · (1, if m is even, where γA denotes the Weil index, see [21], and in case 2, set

χV(m(a)) = χ(det a). (5.1.7)

We may extend χV to a character of Pr(A) by declaring it trivial on Nr(A), and deﬁne ′ Gr(A) s Ir(V,s) = Ind (χV ) (5.1.8) Pr(A) | · | GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 57

(smooth induction), where the induction is normalized so that s = 0 belongs to the unitary axis. Concretely, elements of I (V,s) are smooth functions Φ( ,s): G′ (A) C satisfying r · r → r + 1 Φ((m(a),ǫ)(n(b), 1)g′,s)= det a s+ρ χ (m(a),ǫ) Φ(g′,s), ρ = (5.1.9) | |Ak · V · 2 in case 1 and r Φ(m(a)n(b)g′,s)= det a s+ρ χ (m(a)) Φ(g′,s), ρ = (5.1.10) | |Ak · V · 2 in case 2. We say that a section Φ(s) Ir(V,s) is standard if its restriction to the standard ′ ′ ∈ ′ maximal compact Kr,A of Gr(A) is Kr,A-ﬁnite and independent of s. Let

′ Sp2r(F ), orthogonal case, Gr(F )= (5.1.11) (U(r, r)(F ), unitary case; then there is an embedding G′ (F ) G′ (A); given by simply by the diagonal embedding r → r in case 2, and the canonical splitting of the metaplectic cover Mp2r(AF ) Sp2r(AF ) over ′ → Sp2r(F ) in case 1. In the sequel we will tacitly identify Gr(F ) with its image under this embedding. Given a standard section Φ(s) I (V,s) and g′ G′ (A), the Siegel Eisenstein series ∈ r ∈ r E(g′, Φ,s)= Φ(γg′,s) (5.1.12) ′ γ∈Pr(FX)\Gr (F ) converges for Re(s) 0 and admits meromorphic continuation to s C. It admits a Fourier expansion ≫ ∈ ′ ′ E(g , Φ,s)= ET (g , Φ,s). (5.1.13) XT where T ranges over Symr(F ) in case 1 (resp. Herr(E) in case 2), and

E (g′, Φ,s)= E(n(b)g′, Φ,s) ψ( tr(T b)) dn(b), (5.1.14) T − Nr(F )Z\Nr(A)

where dn(b) denotes the Haar measure on Nr(AF ) that is self-dual with respect to the pairing (b, b′) ψ(tr(bb′)). 7→ ′ r 5.1.2. Let ω = ωψ,χ be the Weil representation of Gr(AF ) U(V(A)) on (V(AF ) ). For φ (V(A )r), g′ G′ (A ) and h U(V(A)), deﬁne the theta× series S ∈ S F ∈ r F ∈ Θ(g′, h; φ)= ω(g′, h) φ(v). (5.1.15) r v∈XV(k) The Siegel-Weil formula relates the integral of this function over U(V)(F ) U(V)(A) to the value of an Eisenstein series. Rather than discussing the formula in full generality,\ it will be convenient to recast the theta integral in the context of the Shimura varieties discussed above. For k = 1,...,d, we have the “nearby” spaces V[k], obtained by switching invariants GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 58

at σ1 and σk. It follows immediately from definitions that Ir(V[k],s) = Ir(V,s); in the sequel, we will implicitly identify these spaces without further mention. Fix such a k and a compact open subgroup K H (A ), and let ⊂ V[k] f X = X = H (Q) D(V[k]) H (A ) K. (5.1.16) V[k] V[k],K V[k] × V[k] f Since HV[k](R) acts transitively on D(V[k]), we may identify X H (Q) H (A ) K K (5.1.17) V[k] ≃ V[k] V[k] Q ∞ where K H (R) is the stablilizer of a fixed point z D(V[k]). Thus, if φ is K K- ∞ ⊂ V[k] 0 ∈ ∞ invariant, then the theta function Θ(g′, h; φ) descends to a well defined function Θ(g′, h; φ) on G′ (A) X . r × V[k] For any φ (V(A )r), the function Φ(g′)= ω(g′)φ(0) belongs to I (V,s ), where ∈ S F r 0 (m r 1)/2, orthogonal case, s0 = s0(r)= − − (5.1.18) (m r)/2, unitary case; ( − this construction defines a G′ (A)-intertwining map that we denote λ: (V(A )r) r S F → Ir(V,s0). Theorem 5.1.3 (Siegel-Weil formula). Suppose V is anisotropic and let φ (V(A)r)K∞K . ∈ S Denote by Φ Ir(V,s) the unique standard section such that Φ( ,s0(r)) = λ(φ). Let Ω be a positive G∈(R)-invariant differential form on D(V) of top degree.· Then E(g′, Φ,s) is regular (in the variable s) at s = s0(r), and κ 0 Θ(g′, ; φ)Ω= E(g′, Φ,s (r)), vol(X , Ω) · 0 V Z[XV] where

1, if s0(r) > 0, κ0 = (5.1.19) (2, if s0(r) = 0.

1 Proof. Recall the usual formulation of the Siegel-Weil formula: set HV := ResF/Q O(V) in 1 the orthogonal case and HV = HV = ResF/Q U(V) in the unitary case. The Siegel-Weil formula reads ′ ′ E(g , Φ,s0(r)) = κ0 Θ(g , h; φ) dh, (5.1.20)

1 Z 1 HV(Q)\HV(A) 1 1 where the Haar measure dh is normalized so that HV(Q) HV(A) has volume one. This is proved in [16, 18] in the unitary case, and in [28] and [45]\ for the metaplectic cases with m even and odd, respectively; a convenient reference treating all cases simultaneously is [10]. We now claim that Θ(g′, h; φ)Ω= C E(g′, Φ,s (r)) (5.1.21) · 0 ZXV GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 59

for some constant C independent of φ. To see this, note that

Θ(g′, ; φ)Ω= Θ(g′, h; φ) d′h XV · Z HV(Q)\HVZ(A)/K∞K (5.1.22) ′ ′ = vol(K∞K) Θ(g , h; φ) d h

HV(Q)Z\HV(A) for some Haar measure d′h. By (5.1.20), this establishes the claim in the unitary case. The orthogonal case follows from the fact that the action of HV(A) = GSpin(V(A)) on (V(A)r) factors through its quotient SO(V(A)), together with [24, Thm. 4.1.(ii)], which showsS that, up to multiplying by a non-zero constant, the integral over O(V(Q)) O(V(A)) in (5.1.20) can be replaced by integration over SO(V(Q)) SO(V(A)). \ To evaluate the constant C, compare the constant terms\ in the Fourier expansion on both sides of (5.1.21); the left hand side is vol(X , Ω) ω(g′)φ(0), while, again using V[k] · (5.1.20), the right hand side is C κ ω(g′)φ(0). · 0 · 5.2. Fourier coefficients of scalar weight Eisenstein series. In this section we study ′ ′ the asymptotic behaviour of the Fourier coefficients ET (g , Φ,s) as g goes to infinity, under ′ certain hypotheses on Φ. More precisely,let Kr be the standard maximal compact subgroup ′ ′ of Gr = Gr(Fv) (where v is archimedean) described in Section 2.5.1, and let m , orthogonal case 2 l =  m + k(χ) m + k(χ) (5.2.1)  , − , unitary case  2 2 as in (3.3.11). Assume that Φ = Φ Φ , with f ⊗ ∞ l l Φ∞ :=Φ Φ Ir(V(R),s), ⊗···⊗ ∈ (5.2.2) Φ = λ(ϕ ) for some Schwartz form ϕ (V(A )r). f f f ∈ S f Here λ is as in Section 5.1.1.

′ Lemma 5.2.1. With Φ as above, the Eisenstein series E(g , Φ,s) is regular at s = s0.

Proof. When s0 = 0, this follows from [30, Thm. 1.1] and [46]. Suppose s0 > 0 so that r p: let z0 D denote the ﬁxed based point as in Section 2.2.1, and consider the Schwartz form≤ ϕ ˜ (V∈r ) deﬁned by ∈ S σ1 ϕ(v, z ) Ωp−r(z ) =ϕ ˜(v)Ωp(z ), v V r. (5.2.3) 0 ∧ 0 0 ∈ ∗ ′ where Ω = c1( , ) . As remarked in Section 2.5.3, it has weight l under Kr. Since r E ∇ r l ϕ˜(0) = ( 1) , an argument as in the proof of Lemma 3.4.2 shows that λ(ϕ ˜) = ( 1) Φ (s0). − l − Thus the global element Φ = vΦ Φf is in the image of λ, and the lemma follows from Theorem 5.1.3. ⊗ ⊗ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 60

For τ = (x + iy ) Hd, let g′ = (g′ ) G′ (R) with g′ = n(x )m(α ) as in j j 1≤j≤d ∈ r τ τj ∈ r τj j j (3.3.10) and deﬁne E(ττ, Φ ,s) := (det y y )−ιm/4 E(g′ , Φ Φl ,s). (5.2.4) f 1 · · · d τ f ⊗ ∞ Then E(ττ, Φf ,s) has a Fourier expansion

E(ττ, Φf ,s)= ET (ττ, Φf ,s) T X (5.2.5) T T 2πitr(τv σv(T )) = CT (y, Φf ,s) q , q := e , XT vY|∞ where T runs over Symr(F ) (resp. Herr(E)), and C (y, Φ ,s) := (det y y )−ιm/4 E (g′ , Φ ,s) q−T . (5.2.6) T f 1 · · · d T τ f d In the rest of this section we will determine the asymptotic behaviour of ds CT (λy, Φf ,s0) as λ + . → ∞ 5.2.2. Consider ﬁrst the case det T = 0. Assuming that Φ = Φ , we have 6 v v ′ ′ ET (g , Φ,s) = WT,v(gv, Φv,s),Q (5.2.7) v Y where W (g′ , Φ ,s)= Φ (w−1n(b)g′,s) ψ( tr(Tn(b))) dn(b), Re(s) 0. (5.2.8) T,v v v v r − ≫ ZNr(Fv) ′ Since T is nonsingular, each WT,v(gv, Φv,s) admits analytic continuation to s C and we can write ∈ d E (g′, Φ,s) = E′ (g′, Φ,s ) , (5.2.9) ds T T 0 v s=s0 v X where the sum runs over places of F and we set

d E′ (g′, Φ,s ) = W (g′ , Φ ,s ) W (g′ , Φ ,s) . (5.2.10) T 0 v T,w w w 0 · ds T,v v v w6=v s=s0 Y

Lemma 5.2.3. Suppose Φ satisﬁes (5.2.2) and det T = 0. Then 6

β(T, Φf ) := CT (y, Φf ,s0(r)) and d κ(T, Φf ) := lim CT (λy, Φf ,s) s=s (r) λ→∞ ds | 0 are independent of y. Let ι = 1 in the orthogonal case and ι = 2 in the unitary one and set κ =1+ ι (r 1). Then explicit values for these quantities are as follows: 2 − (i) If T is not totally positive deﬁnite, then β(T, Φf )= κ(T, Φf ) = 0. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 61

(ii) Suppose T is totally positive deﬁnite and set WT,f (e, Φf ,s) = v<∞ WT,v(e, Φv,s) and Q d ( 2πi)rl c(T ) := − N (det T )ιs0(r). 2r(κ−1)/2 Γ (l) · F/Q r Then β(T, Φ ) = c(T ) W (e, Φ ,s (r)) f · T,f f 0 and ιd Γ′ (l) ι κ(T, Φ )= r log π r + log N det T β(T, Φ ) f 2 − Γ (l) 2 F/Q f r ′ + c(T ) WT,f (e, Φf ,s0(r)). Note that if β(T, Φ ) = 0, then the above may be rewritten more suggestively as f 6 ιd Γ′ (l) ι κ(T, Φ )= r log π r + log N det T f 2 − Γ (l) 2 F/Q r ′ WT,f (e, Φf ,s0(r)) + β(T, Φf ). WT,f (e, Φf ,s0(r))# (iii) In both cases, ∂ ∂ C (λy, Φ ,s) = O(λ−1−C ) as λ , for some C > 0. ∂λ ∂s T f |s=s0(r) →∞ Proof. Writing (5.2.7) in classical coordinates, we have

C (λy, Φ ,s)= (λy ,s) W (e, Φ ,s) (5.2.11) T f  Wσv(T ) v  · T,f f vY|∞   where σv(T )(λyv,s) is the normalized archimedean Whittaker functional as in (3.3.12); all claimsW in the lemma follow easily from Propositions 3.3.3 and 3.3.4. d 5.2.4. We now consider the asymptotic behaviour of ds CT (λy, Φf ,s0) as λ + for a general matrix T . Our approach, which follows that of [28], involves relating→ ∞ these coeﬃcients to certain Eisenstein series on GLr. Fix a matrix T and let t = rk(T ) r. A change of variables in (5.1.14) shows that, for γ SL (k) and g G′ (A) we have ≤ ∈ r ∈ r −1 t −1 −1 ET [γ](g, Φ,s)= ET (m(γ )g, Φ,s), T [γ] := γ T γ . (5.2.12) Hence, by choosing an appropriate γ, it suﬃces to consider the case when T is of the form 0 T = r−t , (5.2.13) S where S is non-degenerate of rank t when t = 0. We assume T is of this form until Proposition 5.2.12. 6 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 62

′ For integers 1 k k and any ring R, consider the embedding Mat2k(R) Mat2k′ (R) given by ≤ ≤ →

1k′−k 0k′−k A B A B   , A,B,C,D Matk(R) (5.2.14) C D 7→ 0k′−k 1k′−k ∈  C D    It is easily checked that in the orthogonal case, this map induces an embedding k′ ηk : Mp2k(A) Mp2k′ (A) (5.2.15) ′ → ′ with ηk ([1,ǫ]) = [1,ǫ], and in the unitary case an embedding ηk : U(k, k)(A) U(k′, k′)(A); k k → the same is true over Fv for any place v. For integers 0 j r, deﬁne a parabolic subgroup of GL by ≤ ≤ r

r,j = ∗ ∗ GLr. (5.2.16) P 0j,r−j ∩ ∗ Lemma 5.2.5. Suppose T is of the form (5.2.13) and t = rk(T ). For a standard section Φ(s) I (V,s), there is a decomposition ∈ r r j a ET (g, Φ,s)= BT (m( 1t )g, Φ,s) Xj=t a∈Pr−t,j−tX(k)\GLr−t(k) where

j r −1 0r−j BT (g, Φ,s) := Φ ηj (wj n(b)) g,s ψT −b dn(b). (5.2.17) n(b)∈N (A) · Z j 1j and wj = ( −1j ). Proof. This follows from the standard unfolding argument for the Fourier coeﬃcients of Eisenstein series and the Bruhat decomposition. See e.g. [28, Lemma 2.4] for the symplectic case; the proofs for the cases required here are identical. If the section Φ = Φ is factorizable, then there is a product expansion ⊗v v j j BT (g, Φ,s)= BT,v(gv, Φv,s) (5.2.18) v Y taken over the places of F , where

j r −1 0r−j BT,v(gv, Φv,s) := Φv ηj (wj n(b))gv ,s ψT,v −b dn(b). (5.2.19) N (Fv ) Z j Note that if j = r, then these factors are the usual local Whittaker functions

W (g , Φ ,s)= Φ (w−1n(b)g,s)ψ ( b) dn(b). (5.2.20) T,v v v v r T,v − ZNr(Fv) We shall relate these, as well as the remaining terms, to Whittaker functions of lower rank; we require a bit more notation. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 63

′ For positive integers k k′, pullback by ηk induces a map ≤ k ′ k′ ∗ k k (η ) : I ′ (V,s) I V,s + − (5.2.21) k k ,v → k,v 2 preserving holomorphic standard sections. Given a place v of k, define an operator Uk,k′,v(s) as follows: for Φ I ′ (V,s) and g G′ (F ), and supposing Re(s) is sufficiently large, let ∈ k ∈ k v b b ′ ′ −1 1 2 k −1 Uk,k ,v(s)Φ (g)= Φ w ′ n b∗ 0 ηk (w g),s db1 db2, (5.2.22) k 2 k k · N Z′ b1∈ k −k(kv ) b2∈Matk′−k,k(kv ) generalizing the construction in [28, Section 7]. Also, let k(s)= v k,v(s) denote the standard intertwining operator, where M ⊗ M (s): I (V,s) I (V, s) (5.2.23) Mk,v k,v → k,v − is defined for Re(s) sufficiently large by the integral

(s)Φ(g)= Φ w−1 n(b) g,s dn(b), Φ I (V,s). (5.2.24) Mk,v k ∈ k,v Z Nk(Fv) Both (s) and (s) admit meromorphic continuation to C. Mk,v Mk 0 Lemma 5.2.6. Let v be a place of F and suppose T = ( S ) as in (5.2.13). ′ ′ ′ k −k (i) If Φ Ik ,v(V,s), then Uk,k ,v(s)Φ Ik,v(V,s 2 ). (ii) Fix an∈ integer j with t j r. If∈t = 0, deﬁne− ≤ ≤ 6 r j U (s) := U s + − (ηr)∗ v t,j,v 2 ◦ j ′ which, by (i), deﬁnes a Gt(Fv)-intertwining map r + t U (s): I (V,s) I (V,σ), where σ = s + j. v r,v → t,v 2 − Then for any Φ I (V,s) and g G′ (F ), r,v ∈ r,v ∈ r v j BT,v(g, Φr,v,s)= WS,v (e, Uv(s) (r(g)Φr,v) ,σ) . (iii) If T = 0, then 6 r j (σ) U (s) = (ηj )∗ s + − (ηr)∗. Mt,v ◦ v t ◦ Mj,v 2 ◦ j (iv) If T = 0, then r j Bj (g,s, Φ ) = s + − (ηr)∗ (r(g)Φ ) (e). 0,v r,v Mj,v 2 ◦ j r,v GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 64

Proof. Consider the orthogonal case ﬁrst. Suppose b1 Symk′−k(Fv) and b2 Matk′−k,k(Fv). For n(β) N (F ), a direct computation yields ∈ ∈ ∈ k v ′ t ′ −1 b1 b2 k −1 ′ 0 −1 b1+b2β b2 b2 k −1 w ′ n t ηk w n(β) = m n β w ′ n t ηk (w ) (5.2.25) k b2 0 k · · · k · b2 0 · k 1 where m′ = m( −b2β ) M ′ (F ). Similarly, if m(α) M (F ), then 1 ∈ k v ∈ k v ′ ′ ′ −1 b1 b2 k −1 k −1 b1 b2α k −1 w ′ n t η w m(α) = η (m(α)) w ′ n t t η (w ). (5.2.26) k b2 0 k k · k · k · α b2 0 · k k Using these relations and applying an appropriate change of variables in (5.2.22) implies that for any Φ I ′ (V,s), we have the transformation formula ∈ k ,v ′ s− k +k [U ′ (s)Φ] (m(α)n(β)g) = χ (m(α)) α 2 [U ′ (s)Φ] (g) ; (5.2.27) k,k ,v V · | | · k,k ,v ′ ′ k −k thus Uk,k ,v(s)Φ Ik,v(V,s 2 ), proving (i). To prove (ii),∈ let Φ = Φ − I (V,s) and set r,v ∈ r,v Φ′(s′) := (ηr)∗ (r(g)Φ(s)) I (V,s + (r j)/2) (5.2.28) j ∈ j,v − with s′ = s + (j r)/2. Then, identifying N (F ) Sym (F ), we may write − k v ≃ k v WS,v(e, U(s)Φ,σ)

= [U(s)Φ] w−1 n(β) ψ ( β) dβ t · S − Zβ∈Symt(Fv) ′ −1 b1 b2 j −1 −1 ′ = Φ w n t η w w n(β)) ,s db1 db2 ψS( β) dβ j b2 0 t t · t · − Z Z βt∈Symt(Fv) b1∈Symj−t(Fv) b2∈Matj−t,t(Fv)

′ −1 b1 b2 ′ = Φ w n b∗ β , s db1 db2 ψS( β) dβ b1∈Symj−t(Fv) j 2 βt∈Symt(Fv) − Z Zb2∈Matj−t,t(Fv) = Φ′(w−1 n(b), s′) ψ ( β) db j · · S − Zb∈Symj (Fv) = Φ ηr w−1 n(b) g, s ψ 0 db = Bj (g, Φ,s). j j · · T −b T,v Zb∈Symj (Fv) (5.2.29)

The proofs of (iii) and (iv) are similar, as are the statements in the unitary case. Remark 5.2.7. (i) Since ( σ) (σ) is given by a meromorphic function in σ, Mt,v − ◦ Mt,v part (iii) shows that Uv(s) admits meromorphic continuation to s C. Setting U(s) = U (s) and using the meromorphic continuation of the global∈ intertwining ⊗v v operator t(σ) we also conclude that the global operator U(s) admits meromorphic continuation.M GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 65

(ii) If T = 0 and j = t or j = r, then r−t,j−t = GLr−t and so the sum over a for 6 P r ∗ these terms in Lemma 5.2.5 is trivial. Noting that U(s) = (ηt ) when j = t, and U(s)= Ut,r(s) when j = r, the corresponding summands in Lemma 5.2.5 are r t B t (g, Φ,s)= W e, [(ηr)∗ r(g)] Φ, s + − T S t ◦ 2 and r t B r(g, Φ,s)= W e, [U (s) r(g)] Φ,s − . T S t,r ◦ − 2 In particular, when T is non-degenerate we recover the expression ET (g, Φ,s) = v WT,v(gv, Φv,s). InQ particular, the lemma implies the global identity j BT (g, Φr,s)= WS (e, U(s) (r(g)Φr) ,σ) . (5.2.30) We will also need the following invariance property: suppose x Mat (A ) and let ∈ r−t,t k 1r−t x θ = GLr(Ak). (5.2.31) 0 1t ∈ Then a direct computation using (5.2.17 ) yields the transformation formula j j BT (m(θ)g, Φ,s)= BT (g, Φ,s) . (5.2.32) 5.2.8. Our next step is to relate the individual terms in Lemma 5.2.5 to Eisenstein series on GLr−t, generalizing the discussion in [28]. Consider first the orthogonal case and let ′ ′ GLr−t(A) denote the metaplectic double cover of GLr−t(AF ): as a set, GLr−t(A) = GL (A) 1 , with multiplication r−t × {± } (a ,ǫ ) (a ,ǫ ) = (a a , (det a , det a ) ǫ ǫ ) (5.2.33) 1 1 · 2 2 1 2 1 2 A 1 2 It follows from the formulas in [41, 5] that there is an embedding § ′ a (ι: GL (A) ֒ M (A), given by (a, ǫ) (m ( 1 ) , ǫ) (5.2.34 r−t → r 7→ t Abusing notation, let χ : GL′ (A) C denote the character χ (a)= χ (ι(a)), where V r−t → V V the latter χV is defined in (5.1.6). For the moment, fix an integer j with t j r; to lighten notation, we write ≤ ≤ p1 = r−t,j−t = ∗ p1 GLr−j, p2 GLj−t GLr−t. (5.2.35) P P 0j−t,r−j p2 ∈ ∈ ⊂ Let ′ denote the inverse image of (A) with respect to the projection GL′ (A) PA P r−t → GLr−t(A). Consider the (smooth normalized) induced representation

′ r+1 j−t j+1 j GLr−t(A) s+ 2 − 2 −s+ 2 I (V,s) := Ind ′ χ det(p1) det(p2) ; (5.2.36) P PA | |A | |A concretely,e I j (V,s) consists of smooth functions Ψ( ,s): GL′ (A) C such that P,v · r−t → r+1 r−1 s+ 2 −s− 2 +j e Ψ(pa,s) = χV(p) det(p1) det(p2) Ψ(a) (5.2.37) | |A | |A GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 66

a ′ p p1 ∗ ′ for all GLr−t(A) and = [( 0 p2 ),ǫ] A. ∈ ∈ P j In the unitary case, we may be more direct: let IP (V,s) denote the space of smooth functions Ψ( ,s): GLr−t(AE) C such that · →r r s+ 2 −s− 2 +j e p1 ∗ Ψ(pa,s) = χV(p) det(p1) det(p2) Ψ(a), for all p = ( p ) (AE). | |AE | |AE 2 ∈ P (5.2.38) ′ a We also write ι: GLr−t(AE) Gr(A) for the embedding a m ( 1t ). Finally, for a ﬁnite place v→of F , let 7→

K = GL ( k ) GL (k ); (5.2.39) v r−t O ,v ⊂ r−t v in the orthogonal case we identify K as a subset of GL′ (F ) via the map k [k, 1]. For e v r−t v 7→ a real place v, set Kv = O(r) or Kv = U(r), and finally define K = Kv. e j Q 5.2.9. To formulatee the connectione between BT (g, Φ,s) and Eisensteine e series on GLr−t, assume for the moment that t = rk(T ) > 0, and fix an integer j with t j r. Let l ≤ ≤ Φr = vΦr,v Ir(V,s) be a standard section such that Φr,v = Φr at each archimedean place v⊗. ∈ Suppose that g G′ (A) is of the form ∈ r ′ r ′′ g = ι(g ) ηt (g ), (5.2.40) ′′ ′ ′ ′ ′ where g Gt(A) and g GLr−t(A) or g GLr−t(AE) in the orthogonal or unitary cases, respectively.∈ As a function∈ of g′′, the expression∈ ′ ′′ U(s) (r(g)Φr) (e) = U(s) r(ι(g ))Φr (g ) (5.2.41)

defines an element of It(V,σ) by Lemma 5.2.6, and a change of variables in the definition j ′ shows that it defines an element of IP (V,s) as a function of g . In particular, the function Bj (g, Φ, s)= Bj ι(g′), r(g′′)Φ , s (5.2.42) T e T v ′ ′′ is in IP (V,s) when viewed as a function of g with g fixed. ′ It follows from multiplicity one for Kt-types in It(Vv,σ) for archimedean v that, as a ′′ l ′′ functione of g∞, the expression (5.2.41) is proportional to v|∞ Φt(gv ,σ); evaluating at g′′ = e to determine the constant of proportionality, we find that ∞ Q ′ ′′ l ′′ U(s) (r(g)Φr) (e)= Φ(g , gf ,s) Φt(gv ,σ), (5.2.43) vY|∞ e ′ ′′ ′ r ′′ ˜ ′ ′′ where Φ(g , gf ,s) = U(s) r(ι(g ) ηt (gf ))Φr (e). Note that Φ(g , gf ,s) is meromorphic in ′ ˜ ′′ j s and, as a function of g , is Kf -finite and K∞-invariant. Moreover Φ( , gf ,s) IP (V,s) e ′′ · ∈ for fixed gf and so we can write e e e Φ(˜ g′, g′′,s)= α (s) Φ˜ (g′,s)Φ′′(g′′,σ)+ + α (s) Φ˜ (g′,s)Φ′′ (g′′,σ), (5.2.44) f 1 1 1 f · · · N N N f where Φ˜ ( ,s) are K -invariant standard sections of I j (V,s), Φ′′( ,σ)= Φ′′ ( ,σ) are i · ∞ P i · ⊗v∤∞ i,v · standard sections of It(V(Af ),σ), and the coefficients αi(s) meromorphic in s. e e GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 67

Substituting (5.2.43) and (5.2.44) in (5.2.30) we conclude that, for g as above with ′′ gf = 1,

j a j ′ ′′ l B (m( 1 )g, Φ,s)= γ (s) (g , Φ˜ ,s) W (g , Φ ,σ), T t  i G i  S,v v t a∈P(k)X\GLr−t(k) 1≤Xi≤N vY|∞   (5.2.45) where j(g′, Φ˜ ,s) := Φ˜ (ag′,s) (5.2.46) G i i a∈P(k)X\GLr−t(k) ′ is an Eisenstein series on GLr−t(A) (case 1) or GLr−t(AE) (case 2), = r−t,j−t is as in (5.2.35), and P P γ (s) := α (s) W e, Φ′′ ,σ , 1 i N, (5.2.47) i i S,v i,v ≤ ≤ v∤∞ Y is meromorphic in s. We turn now to the case T = 0. If 1 j r 1, then the same argument as above ′ ′ ≤ ≤ − ′ shows that, for g GLr(A) (orthogonal case) or g GLr(AE) (unitary case), we can write ∈ ∈ Bj m(a)ι(g′), Φ,s = γ (s) j(g′, Φ˜ ,s) (5.2.48) 0 i G i a∈P (k)\GL (k) 1≤i≤N r,j X r X for some standard K˜ -invariant sections Φ˜ (s) I˜j (V,s) and meromorphic functions ∞ i ∈ P γi(s). In addition, Lemma 5.2.6 shows that 0 ′ ′ B0 (ι(g ), Φr,s)=Φr(ι(g ),s), (5.2.49) Br(ι(g′), Φ ,s)= (s)Φ(ι(g′)) = W (ι(g′), Φ ,s). 0 r M 0 r 5.2.10. We can now generalize Lemma 5.2.3 to arbitrary matrices T ; we consider the case T = 0 first. 6 0 Definition 5.2.11. Assume that Φf Ir(V(Af ),s) satisfies (5.2.2) and T = ( S ) = 0 with S non-degenerate, cf. (5.2.13). Let∈ Φ′ , Φ′′ I (V(A ),σ) be given by 6 f f ∈ t f Φ′ (σ) = (ηr)∗Φ (s) and Φ′′(σ) = (ηr)∗( ( s)Φ ( s)), f t f f t Mr − f − r ∗ where (ηt ) is defined as in (5.2.21), r(s) is the standard intertwining operator (5.2.24), r−t M and σ = s + 2 . Define constants β(T, Φf ) and κ(T, Φf ) as follows: ′ ′ (i) If s0(r) > 0 set β(T, Φf )= β(S, Φf ) and κ(T, Φf )= κ(S, Φf ), where these quantities are defined for the non-degenerate matrix S as in Lemma 5.2.3. (ii) Suppose s0(r) = 0, so that r = m 1 (resp. r = m) in the orthogonal (resp. unitary) case, and let −

r ιm Γr(ιs) − 2 ( 2 −1)−ιs ιmr/2 d(s) = 2 (2πi) ι ιs . (5.2.50) Γr 2 (s + m) Γr 2 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 68

Note that d(s) is holomorphic and non-vanishing at s = 0. Define ′ β(T, Φf ) = 2β(S, Φf ) and d′(0) κ(T, Φ ) := κ(S, Φ′ ) d β(S, Φ′ )+ d(0)dκ(S, Φ′′) . f f − · d(0) f f ⋄ Let us now define constants κ(T, Φf ) and β(T, Φf ) for an arbitrary (i.e. not necessarily block diagonal) matrix T of rank t> 0. Let γ SL (k) such that ∈ r 0 T [γ] := tγ−1T γ−1 = r−t , (5.2.51) S where S is non-degenerate and det S = det′ T := product of non-zero eigenvalues of T ; (5.2.52) such γ always exists for fixed T . Define

β(T, Φf ) := β(T [γ],m(γ)Φf ) (5.2.53) κ(T, Φf ) := κ(T [γ],m(γ)Φf ).

That β(T, Φf ) and κ(T, Φf ) are independent of γ (and hence are well-deﬁned) follows from a direct computation, or alternatively from the invariance property t −1 CT (y, Φf ,s)= CT [γ](y[ γ ],m(γ)Φf ,s), (5.2.54)

(cf. (5.2.12)) and the next proposition, which shows that the constants β(T, Φf ) and d κ(T, Φf ) determine the asymptotic behaviour of the derivative ds CT (y, Φf ,s0).

Proposition 5.2.12. Assume that Φf satisﬁes (5.2.2) and let y = (yv)v|∞ Symr(FR)≫0 ′ ∈ (resp. Her(ER)≫0). Write det A for the product of non-zero eigenvalues of a square matrix A, and let ′ d ι det σv(T ) det yv F (y) := CT (y, Φf ,s) κ(T, Φf ) β(T, Φf ) log ′ · . ds − − 2 det (σv(T )yv) s=s0(r) v|∞ X

Then for every ﬁxed y, we have ∂ lim F (λy) = 0 and F (λy)= O(λ−1−C ) (5.2.55) λ→∞ ∂λ as λ , for some C > 0 . →∞ Proof. We begin with a few preliminary reductions. Choose γ SLr(k) satisfying eqs. (5.2.51) ′ det σv(T )·det yv ∈ and (5.2.52), and note that the quantities ′ are unchanged upon simultane- det (σv (T )yv ) ously replacing T with T [γ] and y with γ y tγ. Combining this observation with (5.2.54), 0 it suﬃces to prove the proposition for T of the form T = ( S ) with S non-degenerate of rank t = rk(T ). GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 69

r j In this case, Lemma 5.2.5 gives a decomposition CT (y, Φf ,s)= j=t CT (y, Φf ,s), where j −ιm/4 j a P′ −T CT (y, Φf ,s)= det(yv) BT m( 1t )gy, Φ,s q . (5.2.56) v|∞ a∈P − − \GL − (k) Y r t,jXt r t 1r−t ∗ d Next, we may ﬁx an element θ = SLr(kR) SLr(K) such that 1t ∈ ≃ y′ θ y tθ = (5.2.57) · · y′′ is block diagonal, where y′ and y′′ are totally positive deﬁnite of rank r t and t, respec- − tively. Let θ = (θ, e, . . . ) SL (A); then, using the invariance property (5.2.32) and the ∈ r relation g′ = m(θ)g′ , we have θytθ y e j a ′ j a ′ eBT m( 1t )gθytθ, Φ,s = BT m( 1t )gy, Φ,s . (5.2.58) 0 ′ Moreover, for T = ( S ), the quantities det (σv(T )yv) are unchanged upon replacing y by t θy θ, as are the determinants det yv. ′ 0 y Thus, we may reduce to the case where T = ( S ) and y = ( y′′ ) are both block diagonal, so that ′ det σv(T ) det yv det σv(S) det yv ′ ′ · = · ′′ = det yv. (5.2.59) det σv(T )yv det σv(S) det yv

Our approach will be to show that the value at s0(r) of the derivative of the expression ′ (5.2.56), after possibly subtracting oﬀ a constant and a multiple of log(det yv), satisﬁes the condition (5.2.55). Fix an element µ GL (k )d of totally positive determinantP with y′ = µ tµ, ∈ v|∞ r−t v · and let µ˜ = ((µ, Id,... ), 1) GL′ in the orthogonal case, and µ˜ = (µ, Id,... ) Q ∈ r−t,AF ∈ GLr−t(AE) in the unitary case. Then ′ r ′ g = ι (µ˜) η g ′′ . (5.2.60) y · t y Taking into account our normalizations, cf. (3.3.12 ), we may use (5.2.45) to write

Cj (y, Φ ,s)= γ (s) j (µ˜, s, Φ˜ ) det(y′ )−ιm/4 (y′′,σ) (5.2.61) T f i G i v Wσv(S) v X vY|∞ ˜ j for some meromorphic functions γi(s) and standard sections Φi IP (V,s), where σ = r+t ∈ σ(s)= s + 2 j, Now for ﬁxed−y and a parameter λ> 0, replacing y by λy correspondse to replacing y′′ by λ y′′, and replacing µ by √λ µ. Using the transformation formulas (5.2.37) or (5.2.38) to determine the central character of the Eisenstein series j in the orthogonal or unitary case, respectively, a short computation yields G − j ιd (r+t−2j)(s−s (r))+ ιd(r t)m − ιd(j−t)s (j) j (√λµ˜, Φ˜,s) = λ 2 0 4 0 (µ˜, Φ˜,s) (5.2.62) G G GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 70

j for any section Φ.˜ Since each (µ˜, Φ˜ i,s) is itself meromorphic, it follows that for fixed y, we have G j ιd (r+t−2j)(s−s (r))−ιd(j−t)s (j) ′′ C (λy, Φ ,s)= f(y,s) λ 2 0 0 (λy ,σ) (5.2.63) T f Wσv(S) v vY|∞ for some meromorphic function f(y,s) independent of λ. A priori, we do not know whether f(y,s) has a pole at s = s0(r), though one could imagine that an analysis along the lines of [28, 45] could be used to determine its order. In any case, let j (y, Φ ) := (Cj (y, Φ ,s)) (5.2.64) A1 f A1 T f denote the coefficient of (s s (r)) in the Laurent expansion at s = s (r), so that − 0 0 r ∂ C (λy, Φ ,s) = j (λy, Φ ). (5.2.65) ∂s T f A1 f s=s0(r) j=t X j Then (λy, Φ ) can be written as a sum of terms of the form A1 f k′ k −ιd(j−t)s0(j) ∂ ′′ a(y) (log λ) λ ′ λy ,σ(s) (5.2.66) ∂sk  Wσv(S) v  v|∞ Y s=s0(r)   for some Laurent coefficients a(y) of f(y,s) at s = s (r), and integers k, k′ 0. 0 First, consider the case where S is not totally positive definite. If v is a place≥ such that ′′ σv(S) is not positive, then, by Proposition 3.3.4, the derivatives of σv(S)(λyv ,σ) in s and j W λ are all of exponential decay as λ . It follows that 1(λy, Φf ) satisfies the properties (5.2.55) for each t j r. On the→∞ other hand, for non-positiveA S the constants κ(S, Φ ) ≤ ≤ f and β(S, Φf ) are both zero by definition; this proves the proposition in this case. Next, suppose S is totally positive definite. Assume further that j>t and s (j) = 0, 0 6 which implies that ιd(j t)s0(j) 1/2; this is the exponent of λ in (5.2.66). For fixed y, Proposition 3.3.3−(i) implies− that≤− each (λy′′,σ), along with its derivatives in s, are Wσv(S) v bounded as λ . In light of (5.2.66), this implies →∞ j lim 1(λy, Φf ) = 0 (5.2.67) λ→∞ A for such j. Similarly, differentiating (5.2.66) with respect to λ and applying Proposi- tion 3.3.3(ii) yields the estimate ∂ j (λy, Φ ) = O(λ−1−C ) (5.2.68) ∂λA1 f j for some C > 0. In other words, for j>t with s0(j) = 0, the term 1(y, Φf ) satisfies the condition (5.2.55). 6 A Now suppose that S is totally positive definite and j = t; the corresponding term can be written more concretely, using Remark 5.2.7, as C t (y, Φ ,s)= det(y )−ιm/4 W e, (ηr)∗ r(g′ ) Φ,σ q−T (5.2.69) T f v S t ◦ y v|∞ Y GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 71

r−t r ∗ where σ = s + 2 and (ηt ) : Ir(V,s) It(V,σ) is the map deﬁned in (5.2.21). For → ′ an archimedean place v, note that multiplicty one for Kt,v-types immediately implies the r ∗ l l ′ relation (ηt ) Φr =Φt. More generally, for block diagonal y as above and any hv Gt,v we have ∈ r ∗ ′ l l r r ′ (η ) (r(g )Φ )(h ,σ)=Φ η (h ) ι(˜µ ) η (g ′′ ), s t yv r v r t v v t yv l r ′ =Φ ι(˜µ ) η (h g ′′ ), s r v t v yv ι (5.2.70) ′ (s+ρr) l r ′ = det(y ) 2 Φ η (h g ′′ ), s v r t v yv ι ′ (s+ρr) l ′ = det(y ) 2 Φ h g ′′ , σ . v t v yv Hence ι t ′ 2 (s−s0(r)) ′′ ′ CT (y, Φf ,s)= det yv CS(y , Φf ,σ), (5.2.71) v|∞ Y where C (y′′, Φ′ ,σ) := W (e, Φ′ ,σ) (y′′,σ) (5.2.72) S f S,f f Wσv(S) v vY|∞ and Φ′ = (ηr)∗Φ I (V(A ),σ). f t f ∈ t f Now consider the special point s = s0(r), and note that r t σ = s (r)+ − = s (t). (5.2.73) |s=s0(r) 0 2 0 ′′ ′ Since S is non-degenerate, the term CS(y , Φf ,σ) is holomorphic at s = s0(r), and therefore t the same is true for CT (y, Φf ,s). Therefore, ∂ ι ∂ Ct (y, Φ ,s) = log det(y′ ) β(S, Φ′ ) + C (y′′, Φ′ ,σ) . (5.2.74) ∂s T f 2 v f ∂σ S f s=s0(r) v|∞ σ=s0(t) X

Applying Lemma 5.2.3 to CS(...), it follows immediately that the diﬀerence ∂ ι Ct (y, s, Φ) β(S, Φ′ ) log det y′ κ(S, Φ′ ) (5.2.75) ∂s T − 2 f v − f s=s0(r) v|∞ X

satisﬁes the conditions in (5.2.55 ). Finally, it remains to consider the case j>t and s0(j) = 0. Since we are assuming s (r) 0, we must have s (j) = s (r) = 0 and hence j = r as well. On the other hand, 0 ≥ 0 0 if s0(r) > 0, then such a term does not arise and we have completed the proof of the proposition at this stage. Otherwise, by Remark 5.2.7, the corresponding term is

Cr (y, Φ ,s)= det(y )−ιm/4 W e, U (s) r(g′ ) Φ,σ q−T (5.2.76) T f v S r,t ◦ y v|∞ Y GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 72

r−t with σ = s 2 . Since s0(r) = 0, we have m = r + 1 in the orthogonal case, and m = r in the unitary− case; it follows in both cases that r t σ = − = s (t). (5.2.77) |s=s0(r)=0 − 2 − 0 To compute this term, note that the functional equation of the genus t Eisenstein series implies W (e, Φ ,σ) = W (e, (σ)Φ , σ) (5.2.78) S t S Mt t − for any Φ I (V,σ). To apply this to the present case, recall that t ∈ t r j (σ) U(s) = (ηj )∗ s + − (ηr)∗ (5.2.79) Mt ◦ t ◦ Mj 2 ◦ j as in Lemma 5.2.6(iii). Consider an archimedean place v , and deﬁne a meromorphic function d(s) by the relation |∞ (s)Φl (s) = d(s)Φl ( s). (5.2.80) Mr r,v r,v − To determine the function d(s), we may evaluate both sides at the identity, and apply [43, (1.31)]; a little algebra, using the fact s0(r) = 0, shows that d(s) is given by the formula (5.2.50). On the other hand, by Lemma 5.2.6(iii) with j = r, we ﬁnd [ (σ) U (s)] r(g′ )Φl (s) = [(ηr)∗ (s)] r(g′ )Φl (s) Mt,v ◦ r,t,v yv r,v t ◦ Mr,v yv r,v = d(s) (ηr)∗ r(g′ )Φl ( s) (5.2.81) · t yv r,v − ι ′ h (−s+ρr) ′ i l = d(s) det y 2 r(g ′′ )Φ ( σ). v yv t,v − Thus, applying the functional equation for WS(...), taking into account our normalizations, and noting that s0(r) = 0, we have r d ′ − ιs ′′ ′′ C (y, Φ ,s) = d(s) det(y ) 2 C (y , Φ , σ), (5.2.82) T f · v · S f − vY|∞ where Φ′′( σ) = (ηr)∗( (s)Φ (s)). Therefore f − t Mr f ∂ d′(0) ι Cr (y, Φ ,s) = d(0)d d log det y′ C (y′′, Φ′′,s (t)) ∂s T f  d(0) − 2 v · S f 0 s=s0(r)=0 v|∞ X  ∂  d(0)d C (y′′, Φ′′,σ) . − ∂σ S f σ=s0(t)

(5.2.83) Setting β′ = d(0)d C (y′′, Φ′′,s (t)) = d(0)d β(S, Φ′′) (5.2.84) − · S f 0 − · f and d′(0) κ′ = d(0)d d β(S, Φ′′) κ(S, Φ′′) (5.2.85) d(0) f − f GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 73

it follows, in the same manner as the previous case, that the diﬀerence

∂ ι Cr (λy, Φ ,s) β′ log det y′ κ′ (5.2.86) ∂s T f − 2 · v − s=s0(r)=0 v|∞ X satisfies the conditions of (5.2.55). Combining this fact with the previous computations for the terms with t j < r, we conclude that ≤ ∂ ι C (λy, Φ ,s) β(S, Φ′)+ β′ log det y′ κ(S, Φ′ )+ κ′ (5.2.87) ∂s T f − 2 · v − f s=0 v|∞ X satisfies (5.2.55). It remains to identify the two terms in square brackets in the preceding display as β(T, Φf ) and κ(T, Φf ), respectively. First, consider the value of CT (λy, Φ,s) at s = s0(r)= 0; by eqs. (5.2.71) and (5.2.82) and an argument analogous to the one leading to (5.2.66), we may write t j CT (λy, Φf , 0) = CTs=0 CT (λy, Φf ,s) Xj=1 = β(S, Φ′ ) β′ + O(λ−C ) (5.2.88) f − for some C > 0. On the other hand, the Eisenstein series E(g, Φ,s) is incoherent, in the sense of [30], and hence vanishes identically at s = 0 (this result is [30, Theorem 4.10] in the orthgonal case with m even; see [10, Proposition 6.2] and the references therein for ′ ′ more details in the remaining cases). In particular, CT (y, Φf , 0) = 0 and β(S, Φf ) = β , and, comparing with Definition 5.2.11, we find ′ ′ ′ ′ β(S, Φf )+ β = 2β(S, Φf )= β(T, Φf ) (5.2.89) and ′ ′ κ(S, Φf )+ κ = κ(T, Φf ) (5.2.90) as required.

It remains to consider the case T = 0. Proposition 5.2.13. Let

Φf (e), if s0(r) > 0 β(0, Φf ) := (2Φf (e), if s0(r) = 0. and 0, if s0(r) > 0 κ(0, Φ ) := d′ f d (0) Φ (e) d(0)d W ′ (e, Φ , 0), if s (r) = 0. (− · d(0) f − 0,f f 0 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 74

Then the diﬀerence d ι F (y)= C0(y, Φ ,s) log det y β(0, Φ ) κ(0, Φ ) ds f − 2 v · f − f s=s0(r) v|∞ X

satisﬁes the conditions (5.2.55).

r j Proof. Using Lemma 5.2.5 and (5.2.48), we may write C0(y, Φ,s) = j=0 C0(y, Φ,s), where P Cj (y, Φ ,s)= det(y )−ιm/4 γ (s) j (α, Φ˜ ,s), (5.2.91) 0 f v · i G i vY|∞ 1≤Xi≤N ˜ j for some meromorphic functions γi(s) and standard sections Φi IP (V,s). Applying the transformation formula (5.2.62), with t = 0, we have ∈

j d (r−2j)(s−s (r))−djs (j) j e C (λy, Φ ,s)= λ 2 0 0 C (y, Φ ,s). (5.2.92) 0 f · 0 f Thus ∂ Cj (λy, Φ ,s) is a ﬁnite sum of terms of the form ∂s 0 f |s=s0(r) log(λ)k λ−djs0(j) a(y) (5.2.93) · · j for some integer k 0 and Laurent coeﬃcient a(y) of C0(y, Φf ,s). In particular, if j≥ 1 and s (j) = 0, then ≥ 0 6 ∂ j ∂ ∂ j −1−C lim C0(λy, Φf ,s) s=s (r) =0 and C0(λy, Φf ,s) s=s (r) = O(λ ). λ→∞ ∂s | 0 ∂λ ∂s | 0 (5.2.94) Next, consider the case j = 0, so that, by (5.2.49), we have

0 −ιm/4 0 ′ C0(y, Φf ,s) = det(yv) B0 gy, Φf ,s v|∞ Y = det(y )−ιm/4 Φ(g′ ,s) (5.2.95) v · y vY|∞ ι (s−s0(r)) = det(yv) 2 Φf (e). vY|∞ Note here that Φf (e, s)=Φf (e) is independent of s. Therefore ∂ ι C0(y, Φ ,s) = log det y Φ (e). (5.2.96) ∂s 0 f |s=s0(r) 2 v · f Xv|∞ Finally, consider the case s0(j) = 0, so that j = r with s0(r)=0 and Cr(y, Φ ,s)= det(y )−ιm/4 [ (s)Φ(s)](g′ ). (5.2.97) 0 f v · M y vY|∞ GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 75

We have [ (s)Φ(s)](g′ )= [ (s)Φl (s)](g′ ) [ Φ (s)](e) M y Mv r,v yv · Mf f vY|∞ = d(s)d Φl (g′ , s) W (e, Φ , s) r,v yv − · 0,f f − (5.2.98) vY|∞ d ι (−s+ρ ) = d(s) det(y ) 2 r W (e, Φ , s) v 0,f f − vY|∞ Moreover, applying [10, Lemma 6.3] to the incoherent section Φ gives (0)Φ(0)(g′ ) = det(y )ιm/4 Φ(e) = det(y )ιm/4 Φ (e) (5.2.99) M y − v · − v · f vY|∞ vY|∞ and hence, after a little algebra we obtain d ι Cr(y, Φ ,s) = det log y Φ (e) ds 0 f 2 v f s=s0(r)=0 v|∞ X (5.2.100) d′(0) d Φ (e) d(0)dW ′ (e, Φ , 0). − · d(0) f − 0,f f The proposition follows immediately from these observations. 5.3. Archimedean height pairings. In this section, we prove our main theorem relating the integrals of the Green forms g(T, y, ϕ) constructed in Section 4 to Eisenstein series. We continue to assume V is anisotropic and rk( ) = 1. Let denote the canonical E X model of the Shimura variety XV = XV,K , for a ﬁxed open compact subgroup K H(Af ), and recall that there is a decomposition (C) = X in terms of the nearby⊂ spaces X V[k] V[k] for k = 1,...,d, see Sections 4.1 and 4.4. For each k, let ` p Vol(XV[k], Ω) = Ω (5.3.1) Z[XV[k]] where Ω = Ω = c ( , )∗ is the ﬁrst Chern form of the positive line bundle on X . E 1 E ∇ E V[k] Then Vol(XV[k], Ω) = degE (XV[k]) is a positive rational number (a positive integer if K is neat). As remarked in 2.2.2, the line bundle is a rational multiple of the canonical E bundle; in particular we have Vol(XV[k], Ω) = Vol(XV, Ω) for every k. Theorem 5.3.1. Assume that V is anisotropic and rk( ) = 1. Let r 1 and E ≥ (m r 1)/2, orthogonal case, s0(r)= − − (m r)/2, unitary case, ( − r K and assume that s0(r) 0. Given ϕf (V(Af ) ) , let Φf ( ,s) be the unique standard section of I (V(A ),s) ≥such that Φ ( ,s∈)= S λ(ϕ ) and set · r f f · 0 f Φ= Φl Φ I (V(A),s), (5.3.2) ⊗v|∞ r ⊗ f ∈ r GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 76

l where Φr is the standard weight l section given by (3.3.11), (3.1.10) and (3.2.11). For τ = x + iy Hd, let ∈ r E(ττ, Φ ,s) = (det y y )−ιm/4 E(g′ , Φ ,s) = E (ττ, Φ ,s) f 1 · · · d τ f T f XT be the corresponding Eisenstein series of scalar weight l defined in (5.2.4), and write ′ d ET (ττ, Φf ,s) = ds ET (ττ, Φf ,s). Then for any T Symr(F ) (resp. T Herr(E)) we have ∈ ∈ ( 1)rκ − 0 g(T, y, ϕ ) Ωp+1−r qT = E′ (ττ, Φ ,s (r)) κ(T, Φ ) qT , 2Vol(X , Ω) f ∧ T f 0 − f V Z[X (C)] where κ(T, Φf ) is explicit and defined in Definition 5.2.11, and κ0 = 2 if s0(r) = 0 and κ0 = 1 otherwise. Proof. Fix an archimedean embedding σ , and a component X (C). Recall that k V[k] ⊂ X the restriction of the Green form g(T, y, ϕf ) to XV[k] is given by ′ o det σk(T ) det yk r−rk(T )−1 g(T, y, ϕf ) XV = g (T, y, ϕf )σ log · δZ(T,ϕ ) Ω ∨ | [k] k − det′ (σ (T )y ) f σk ∧ E k k

(5.3.3) when T is positive semi-definite, and g(T, y, ϕ ) = go(T, y, ϕ ) otherwise (see Defi- f |XV[k] f σk nitions 4.3.5 and 4.4.2); here o o g (T, y, ϕf ) = CT g (T, y, ϕf ; ρ) (5.3.4) ρ=0 as in Propositions 4.3.4 and 4.4.1. Consider the contribution of go(T, y, ϕ ) to the integral over (C); by definition, it f σk X equals CTρ=0 I(ρ, σk), where r ( 1) κ o I(ρ, σ )= − 0 g (T, y, ϕ ; ρ) Ωp+1−r. (5.3.5) k 2Vol(X , Ω) f σk ∧ V[k] Z[XV[k]] Let us compute I(ρ, σ1). Define the archimedean Schwartz function φ =ν ˜ ϕ ϕ (Vr R)= (Vr ), (5.3.6) ⊗ + ⊗···⊗ + ∈ S ⊗F S σi 1O≤i≤d r withν ˜ (Vσ1 ) as in (3.4.2) and where ϕ+ denotes the Gaussian for the positive definite ∈r S spaces Vσk (k > 1), given by −π Q(vi,vi) ϕ+(v)= e Pi , (5.3.7) v r for = (v1, . . . , vr) Vσk . Consider the theta∈ series Θ(g′ , h; φ ϕ ) = Θ (g′ , h; φ ϕ ) (5.3.8) τ ⊗ f T τ ⊗ f XT GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 77

defined in (5.1.15), and write C (y, h; φ ϕ ) := (det y y )−ιm/4 Θ (g′ , h; φ ϕ ) q−T . (5.3.9) Θ,T ⊗ f 1 · · · d · T τ ⊗ f For z = hz D+ (h U(V )) and Re(ρ) 0, we have 0 ∈ ∈ σ1 ≫ ∞ o dt g (T, y, ϕ ; ρ) (z) Ωp+1−r(z)= C (ty, h; φ ϕ ) Ωp(z), (5.3.10) f σ1 ∧ Θ,T ⊗ f tρ+1 ∧ Z1 where the estimates in the proof of Proposition 4.3.3 justify the interchange of sum and integral. Using the Siegel-Weil formula (Theorem 5.1.3) to compute the integral over XV, we find ( 1)r ∞ dt I(ρ, σ )= − C (ty, λ(φ ϕ ),s (r)) ; (5.3.11) 1 2 E,T ⊗ f 0 tρ+1 Z1 here the relevant Eisenstein series is E(g′, λ(φ ϕ ),s)= E (g′, λ(φ ϕ ),s) (5.3.12) ⊗ f T ⊗ f XT where λ: S(V(A)r) I (V,s (r)) is as in Section 5.1.2, and we have written → r 0 C (y, λ(φ ϕ )),s) := (det y y )−ιm/4 E (g′ , λ(φ ϕ ),s) q−T . (5.3.13) E,T ⊗ f 1 · · · d · T τ ⊗ f Our next step is to relate this expression to the coefficient −T CT (y, Φf ,s) := ET (ττ, Φf ,s) q (5.3.14) of the scalar weight l Eisenstein series in the statement. Comparing archimedean compo- l r−1 ˜ nents, for i 2 we have λ(φσi ) = λ(ϕ+)=Φ (s0), while λ(φσ1 ) = λ(˜ν) = ( 1) Φ(s0) ≥ ′ d − as in (3.4.6). Thus, writing CT (y, Φf ,s0) = ds CT (y, Φf ,s) s=s0 and y = (y1,...,yd), the argument in the proof of Lemma 3.4.3 shows that |

r−1 2 d ′ ′ ′ CE,T (ty, λ(φ ϕf ),s0) = ( 1) t CT ((ty1,t y2,...,t yr)), Φf ,s0) . (5.3.15) ⊗ − ι · dt ′ t=t

Adding the contributions from all archimedean places, we conclude that

r ( 1) κ0 o p+1−r − g (T, y, ϕf ) Ω 2Vol(XK , Ω) ∧ Z[X (C)] (5.3.16) ∞ d dt = CT C′ (ty, Φ ,s ) . − ρ=0 dt T f 0 tρ Z1 This integral can be evaluated using the following lemma, whose proof is straightforward.

Lemma 5.3.2. Let f : R>0 C be a smooth function such that for some constants a, b C, → ′ −1−C∈ the function F (t) = f(t) a b log t satisﬁes limt→∞ F (t) = 0 and F (t) = O(t ) as t , for some positive− constant− C. Then →∞ ∞ dt CT f ′(t) = f(1) a. − ρ=0 tρ − Z1 GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 78

′ By Proposition 5.2.12, the function CT (ty, Φf ,s0), regarded as a function of t, satisfies the hypotheses of the lemma with ′ ι det σv(T ) det yv a = κ(T, Φf )+ β(T, Φf ) log ′ · . (5.3.17) 2 det σv(T )yv Xv|∞ To finish the proof, it suffices to show that the contributions from the second terms in (5.3.3) for the various components X (C) sum to V[k] ⊂X d ′ rk(T ) det σk(T ) det yk ( 1) κ0 p−rk(T ) ι log ′ · − δZ(T,ϕf )σ Ω det (σ (T )y ) 2Vol(XV, Ω) k ∧ k=1 k k Z[XV[k]] X ′ ? ι det σ (T ) det y = log k · k β(T, Φ ), 2 det′ (σ (T )y ) f k k k X (5.3.18) where we used the identity ΩE∨ = ΩE = Ω. Note that on account of the logarithms, both sides vanish if T is non-degenerate.− When− T is degenerate, the claim is essentially contained in [27]; we outline the argument in Lemma 5.3.4 below. When the matrix T is non-degenerate, we recover the identities (1.1.19) and (1.1.20) by using the explicit expression for κ(T, Φf ) given by Lemma 5.2.3. Combining Theorems 4.5.1 and 5.3.1, we obtain the following corollary generalizing the main result of [23] (for U(p, 1) this corollary is due to Liu [35]; recently Bruinier and Yang [14] have treated the O(p, 2) case).

Corollary 5.3.3. Assume that T1 and T2 are non-degenerate and that Z(T1, ϕf,1) and Z(T2, ϕf,2) intersect transversely. Assume also that r1 + r2 = p + 1. Then ( 1)p+1 − g(T , y , ϕ ) g(T , y , ϕ ) qT Vol(X , Ω) 1 1 f,1 ∗ 2 2 f,2 V Z[X (C)] = E′ (( τ 1 ) , λ(ϕ ϕ ), 0) , T τ 2 f,1 ⊗ f,2 ∞ T = T1 ∗ X∗ T2 It remains to establish the following lemma, which is an application of the results of [27]. 0 Lemma 5.3.4. Suppose T = S is degenerate. Then for any k = 1,...,d, rk(T ) ( 1) κ0 p−rk(T ) − δZ(T,ϕ ) Ω = β(T, Φf ), Vol(X , Ω) f σk ∧ V Z[XV[k]] with notation as in Theorem 5.3.1. Proof. Suppose ﬁrst that t = rk(T ) > 0, so that S is non-degenerate of rank t. As both sides of the desired identity are linear in ϕf , we may also assume that ϕf is of the form ϕ = ϕ′ ϕ′′ S(V(A )r−t) S(V(A )t). (5.3.19) f f ⊗ f ∈ f ⊗ f GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 79

′ ′′ ′ By construction, we have Z(T, ϕf ) = ϕf (0)Z(S, ϕf ). On the other hand, let Φ(ϕf ) and ′′ ′ ′′ Φ(ϕf ) denote the standard sections corresponding to ϕf and ϕf , respectively. Then Deﬁ- nition 5.2.11 and a direct computation using explicit formulas for the Weil representation (see e.g. [45, Proposition 2.2.5]) give β(T, Φ )= κ β (S, η∗Φ )= κ C (e, η∗Φ ,s (t))) f 0 · f 0 · S f 0 = κ ϕ′ (0) C e, Φ(ϕ′′),s (t) 0 · f · S f 0 = κ ϕ′ (0) β(S, Φ(ϕ′′ )). (5.3.20) 0 · f · f ∗ r ∗ where η = (ηt ) as in (5.2.21). To prove the lemma, say for k = 1, let

′′ ′′ ′′ ′′ S ΘKM(τ , ϕf )σ1 = ω(S, y , ϕf )σ1 q (5.3.21) XS ′′ ′′ ′′ d be the Kudla-Millson theta series of genus t, as deﬁned in (4.3.5); here τ = x + iy Ht . ′′ ′′ ∈ It is shown in [27] that ω(S, y , ϕf )σ1 is a closed form on XV whose cohomology class is ′′ [Z(S, ϕf )σ1 ]. In particular, we have

p−t ′ ′′ p−t ′ ′′ p−t δZ(T,ϕ ) Ω = ϕ (0) δZ(S,ϕ ) Ω = ϕ (0) ω(S, y , ϕf )σ1 Ω ; f σ1 ∧ f f σ1 ∧ f ∧ Z[XV] Z[XV] Z[XV] (5.3.22) this also follows from Propositions 4.3.4 and 4.4.1. To compute the latter integral, define an archimedean Schwartz function ϕ := ϕ ϕ ϕ S(Vt ) (5.3.23) ∞ ⊗ + ⊗···⊗ + ∈ σk where ϕ is the Schwartz function on Vt defined by (5.2.3O), and ϕ is the standard Gauss- e σ1 + ian. Note that e λ(ϕ ) = ( 1)t Φl (s ), (5.3.24) ∞ − ⊗v|∞ t 0 as in the proof of Lemma 5.2.1, and ω(S, y′′, Φ′′) (z) Ωp−t(z) = C (y′′, h, ϕ ϕ′′)Ωp(z), z = hz , (5.3.25) f σ1 ∧ Θ,S ∞ ⊗ f 0 where C (y′′, h, ϕ ϕ′′) is the coefficient of qS of the theta series attached to ϕ ϕ′′. Θ,S ∞ ⊗ f ∞ ⊗ f Applying the Siegel-Weil formula (Theorem 5.1.3) again, noting that s0(t) > 0 here, we conclude ( 1)t − ω(S, y′′, ϕ′′ ) Ωp−t = C y′′, Φ(ϕ ),s (t) = β(S, Φ′′). (5.3.26) Vol(X , Ω) f σ1 ∧ S f 0 f V Z[XV] Comparing this with (5.3.20) and (5.3.22) proves the lemma for k = 1, and the proof for all other values of k follows in exactly the same way. Finally, when T = 0r, the left hand side is κ0ϕf (0) = κ0Φf (e), which is by definition equal to β(0, Φf ), cf. Proposition 5.2.13. GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 80

5.4. Classes in arithmetic Chow groups. In this section, we describe how the currents g(T, y, ϕf ) arise as the archimedean parts of classes in arithmetic Chow groups lifting the cycles Z(T, ϕf ). As we will ultimately have little to say about arithmetic aspects of the theory, we shall gloss over many serious difficulties regarding suitable integral models, bad reduction, etc. A key point is a natural geometric context for the analogue (4.3.15) of Green’s equation in the degenerate case. We continue to assume that V is anisotropic, but drop the assumption rk( ) = 1. We E also assume the level structure K H(Af ) is neat. Let k = F (resp. k = E) in the⊂ orthogonal (resp. unitary) case, so that is proper X over Spec k. Suppose X is a regular integral model, proper and flat over Spec( k), with an extension of the tautological bundle that we continue to denote . Finally, forO each T and ϕ , let (T, ϕ ) denote a cycle on X extending Z(T, ϕ ) on theE generic fibre whose f Z f f codimension in X is equal to the codimension of Z(T, ϕf ) in = Xk. • r X Let CHC(X) = r CHC(X) denote the Gillet-Souléarithmetic Chow ring (with C coefficients), as in [44]. Classes in CHr (X) are represented by pairs (Z, g ), where Z is L C Z codimensiond r cycle ondX (with C-coefficients) and g is an (r 1, r 1) current on Z − − (C)= σ : k→C σ(C) that is invariantd under complex conjugation, and satisfies Green’s Xequation X ` c dd gZ + δZ(C) = [ωZ] for some smooth differential form ω on (C); we may also view g = g as a Z X Z { Z,σ}σ collection consisting of a current gZ,σ on σ(C) for each complex embedding σ. When T is non-degenerate, PropositionX 4.3.4 (and the discussion around (4.3.6)) or Proposition 4.4.1 gives rise to a class ′ ( (T, ϕ ), g(T, y, ϕ )) CHrq (X), (5.4.1) Z f f ∈ C where q′ = rk( ). E d Now consider a pair (T, ϕf ) with T Symr(F ) (resp. T Herr(E)) a degenerate matrix, and set t = rank(T ). Let ∈ ∈ ′ ∨ q c ′ ( ) CH (X) q E ∈ C denote the arithmetic Chern class attached to ∨, as in [44, Section IV]. The class of E c ′ ( ∨)r−t may be represented by ab pair ( , g d) such that the generic fibre of intersects q E Z0 0 Z0 properly with Z(T, ϕf ), and where the current g0 satisfies the equation b c r−t ∨ dd g0 + δZ0(C) = ΩE and is of logarithmic type, see [44, II.2]. On the other hand, consider the set of currents g of degree (r 1, r 1) satisfying the§ analogue − − c r−t dd g + δ Ω ∨ = [ω] (5.4.2) Z(T,ϕf )(C) ∧ E of Green’s equation, for some smooth form ω; a short computation reveals that the map g g + g δ 7→ 0 ∧ Z(T,ϕf )(C) GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 81

deﬁnes a bijection between the solutions of (5.4.2) and Green currents for the intersection (T, ϕ ) . Therefore, applying Propositions 4.3.4 and 4.4.1, we obtain a class Z f Z0 · ′ Z(T, y, ϕ ) := (T, ϕ ) , g(T, y, ϕ )+ g δ CHrq (X). f Z f ·Z0 f 0 ∧ Z(T,ϕf )(C) ∈ C ∨ r−t To see thatb this construction is independent of the choice of ( 0, g0) representingd cq′ ( ) , choose any Green current g′ for (T, ϕ ), and note that Z E Z f ′ ′ r−t b Z(T, y, ϕ ) = ( (T, ϕ ), g ) ( , g ) + (0, g(T, y, ϕ ) g Ω ∨ ) f Z f Z0 0 f − ∧ E ′ · ∨ r−t ′ r−t = ( (T, ϕf ), g ) cq′ ( ) + (0, g(T, y, ϕf ) g Ω ∨ ) (5.4.3) b Z · E − ∧ E To preserve uniformity of notation, set Z(T, y, ϕ ) = ( (T, ϕ ), g(T, y, ϕ )) when T is b f Z f f non-degenerate. For any (possibly degenerate) T , restricting the cycle Z(T, y, ϕf ) to the generic ﬁbre (and forgetting the current) yieldsb a cycle that coincides with the construction in [22]. b As an example of our construction, take T = 0r; applying the computation (4.4.13) for g(0, y, ϕf ) gives the concrete expression

∨ r ∨ ∗ r−1 Z(0 , y, ϕ ) = ϕ (0) c ′ 0, log det y c ′ ( , ) Ω ∨ . r f f · q E − k · q −1 E ∇ ∧ E k=1,...,d (5.4.4) b b 5.5. Kudla’s arithmetic height conjecture. We recast our results in the setting of Kudla’s conjectures on the arithmetic heights of the cycles Z(T, y, ϕf ) considered in the previous section. Assume that V is anisotropic and rk( ) = 1. Let b E ω := c ( ∨) CH1 (X ) (5.5.1) 1 E ∈ C denote the arithmetic class attached to ∨, or more precisely, to its integral extension as d in the preceding section, and considerb b theE generating series

p+1−r T φω(τ) := deg Z(T, y, ϕf ) ω q (5.5.2) T · b X d b where deg : CHp+1(X ) C is the arithmetic degree mapb [44, Section III]. A rough form C → of Kudla’s conjectural programme, as outlined in e.g. [25], suggests that φω(τ) is, up to a normalization,d d equal to the special derivative of an Eisenstein series. b More precisely, let Φf Ir(V(Af ),s) denote the standard section of parallel scalar weight l determined by ϕ as in∈ (5.3.2), and consider the parallel weight l Eisenstein series

(ττ, Φ ,s)= A (s)E(ττ, Φ ,s)=: C (y, Φ ,s)qT , (5.5.3) E f r f T f XT for an appropriate normalizing factor Ar(s); then one should have an identity

deg Z(T, y, ϕ ) ωp+1−r ? C ′ (y, Φ ,s (r)) (5.5.4) f · ∼ T f 0 d b b GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 82

up to correction terms involving rational multiples of log p with p in a fixed, finite set of primes that might depend on ϕ, the level structure K, and the choices of integral models. As these correction terms are expected to arise as contributions from (components of) cycles at primes of bad reduction, it is reasonable to assume that they are independent of the parameter y. Let h( (T, ϕ )) := deg ωp+1−rk(T ) (T, ϕ ) (5.5.5) Z f Z f denote the Bost-Gillet-Souléheight of (T, ϕ ) along ω, as defined in [6, Proposition 2.3.1]. d f Using [6, (2.3.3)] and (5.4.3) above, aZ brief computationb gives

p+1−r 1 b p+1−r deg Z(T, y, ϕ ) ω = h( (T, ϕ )) + g(T, y, ϕ ) Ω ∨ f · Z f 2 f ∧ E ZX (C) p+1−r d b ( 1) p+1−r b = h( (T, ϕ )) + − g(T, y, ϕ) Ω . Z f 2 ∧ E ZX (C) By Theorem 5.3.1, the conjecture (5.5.4) is equivalent to the statement p+1 ? ( 1) vol(X ) h( (T, ϕ )) A (s (r)) − V C′ (y, Φ ,s (r)) Z f ∼ r 0 − κ T f 0 0 p+1 ′ ( 1) vol(XV) + Ar(s0(r)) CT (y, Φf ,s0(r)) + − κ(T, Φf ) (5.5.6) · κ0 · p for all T and ϕf , where vol(XV) = vol(XV, ΩE ). ′ Note that only the values Ar(s0(r)) and Ar(s0(r)) appear in this expression. To pin these values down further, suppose for the moment that s0(r) > 0, and take y = λ Id, say; then Proposition 5.2.12 implies that · ιd (r rkT ) C′ (λ Id, Φ ,s (r)) = κ(T, Φ )+ · − log λ β(T, Φ )+ F (λ) (5.5.7) T · f 0 f 2 · · f for some function F satisfying lim F (λ) = 0. Similarly, C (λ Id, Φ ,s (r)) = β(T, Φ ). λ→∞ T · f 0 f Thus, choosing T and ϕf such that rk(T ) < r and β(T, Φf ) = 0, and noting that h( (T, ϕ )) is evidently independent of y, a necessary condition for6 (5.5.6) to hold is Z f p+1 ( 1) vol(XV) Ar(s0(r)) = − . (5.5.8) κ0 ′ Now for the derivative Ar(s0(r)), assume that ϕ(0) = 1, and consider the matrix T = 0. Then p+1 h( (0, ϕ )) degX ω =: vol (X ) (5.5.9) Z f ∼ ω On the other hand, by Proposition 5.2.13, b d b c κ(0, Φf ) = 0, (5.5.10) and C0(y,s0, Φ) = ϕ(0) = 1. Therefore (5.5.6) for T = 0 and s0(r) > 0 gives the further necessary condition A′ (s (r)) vol (X ). (5.5.11) r 0 ∼ ω b c GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 83

In other words, taking A (s) such that A (s (r)) = ( 1)p+1vol(X )κ−1 and A′ (s (r)) = r r 0 − V 0 r 0 volω(X ), the conjecture (5.5.4), for s0(r) > 0, takes the form b p+1 ? ( 1) vol(XV) c h( (T, ϕf )) vol ω(X ) β(T, Φf )+ − κ(T, Φf ). (5.5.12) Z ∼ · κ0 b When s0(r) = 0, note thatcCT (y, Φf , 0) = 0 and the conjecture (5.5.6) becomes p+1 ? ( 1) vol(XV) h( (T, ϕf )) − κ(T, Φf ) (5.5.13) Z ∼ κ0 · The point here is that the right hand sides of these relations involve explict constants depending only on T and ϕf . This conjecture has been verified in the case of full level Shimura curves for r = 1 [31], for r = 2 [32], and for Hilbert modular surfaces with r = 1 in [7] (these results also include contributions from places of bad reduction). A slightly weaker version of this conjecture (i.e. away from an explicit set of primes determined by T and ϕ) was proved for general orthogonal Shimura varieties over Q by Hörmann [15]. Several cases involving cycles of top arithmetic codimension supported at finite primes were also established by Kudla and Rapoport, see e.g. [20], or the discussion in [25, II] for more details. §

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Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada E-mail address: [email protected] GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 86

Department of Mathematics, University of Manitoba, 420 Machray Hall, Winnipeg, Canada E-mail address: [email protected]