Heights of CM points IGross–Zagier formula

Xinyi Yuan, Shou-wu Zhang, Wei Zhang

September 24, 2008

Contents

1 Introduction and state of main results 21.1 L-function and root numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Waldspurger’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Gross–Zagier formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Plan of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Weil representation and kernel functions 102.1 Weil representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Theta series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Shimizu lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Integral representations of L-series . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Proof of Waldspurger formula . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Vanishing of the kernel function . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Derivatives of kernel functions 213.1 Discrete series of weight two at infinite Places . . . . . . . . . . . . . . . . . 213.2 Derivatives of Whittaker Functions . . . . . . . . . . . . . . . . . . . . . . . 243.3 Degenerate Schwartz functions at non-archimedean places . . . . . . . . . . . 283.4 Non-archimedean components . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Archimedean Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.7 Holomorphic kernel function . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Shimura curves, Hecke operators, CM-points 444.1 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Hecke correspondences and generating series . . . . . . . . . . . . . . . . . . 46

1

4.3 CM-points and height series . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Arithmetic intersection pairing . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 Decomposition of the height pairing . . . . . . . . . . . . . . . . . . . . . . . 504.6 Hecke action on arithmetic Hodge classes . . . . . . . . . . . . . . . . . . . . 53

5 Local heights of CM points 555.1 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Supersingular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Superspecial case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4 Ordinary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.5 The j -part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 Pseudo-theta series and Gross–Zagier formula 736.1 Pseudo-theta series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Proof of Gross–Zagier formula . . . . . . . . . . . . . . . . . . . . . . . . . . 77

1 Introduction and state of main results

In 1984, Gross and Zagier [GZ] proved a formula that relates the Neron–Tate heights ofHeegner points to the central derivatives of some Rankin L-series under certain ramificationconditions. Since then some generalizations are given in various papers [Zh1, Zh2, Zh3].The methods of proofs of the Gross–Zagier theorem and all its extensions depend on somenewform theories. There are essential difficulties to remove all ramification assumptions inthis method. The aim of this paper is a proof of a general formula in which all ramificationcondition are removed. Such a formula is an analogue of a central value formula of Wald-spurger [Wa] and has been more or less formulated by Gross in 2002 in term of representationtheory. In the following, we want to describe statements of the main results and main ideaof proof.

1.1 L-function and root numbers

Let F be a number field with adele ring A = AF . Let π = ⊗vπv be a cuspidal automorphicrepresentation of GL2(A). Let E be a quadratic extension of F , and χ : E×\A×

E → C× be acharacter of finite order. We assume that

χ|A× · ωπ = 1

where ωπ is the central character of π.Denote by L(s, π, χ) the Rankin-Selberg L-function. Denote by ε(1

2, πv, χv) = ±1 the

local root number at each place v of F , and denote

Σ = v : ε(1/2, πv, χv) 6= χvηv(−1),

2

where ηv is the quadratic character associated to the extension Ev/Fv. Then Σ is a finite setand the global root number is given by

ε(1/2, π, χ) =∏v

ε(1/2, πv, χv) = (−1)#Σ.

The following is a description of root numbers in terms of linear functionals:

Proposition 1.1.1 (Saito–Tunnell [Tu, Sa]). Let v be place of F and Bv a quaterniondivision algebra over Fv. Let π′v be the Jacquet–Langlands correspondence of πv on B×

v if πvis square integrable. Fix embeddings E×

v ⊂ GL2(Fv) and E×v ⊂ B×

v as algebraic subgroups.Then v ∈ Σ if and only if

HomE×v(πv ⊗ χv,C) = 0.

Moreoverdim HomE×v

(πv ⊗ χv,C) + HomE×v(π′v ⊗ χv,C) = 1.

Here the second space is treated as 0 if π′v is not square integrable.

Arithmetic properties of the L-function depends heavily on the parity of #Σ. When #Σis even, the central value L(1

2, π, χ) is related to certain period integral. Explicit formulae

have been given by Gross, Waldspurger and S. Zhang. We will recall the treatment ofWaldspurger [Wa] in next section.

When #Σ is odd then L(12, π, χ) = 0. Under the assumption that E/F is a CM-extension,

that πv is discrete of weight 2 for all infinite place v, and that χ is of finite order, thenthe central derivative L′(1

2, π, χ) is related to the height pairings of some CM divisors on

certain Shimura curves. Explicit formulae have been obtained by Gross-Zagier and one ofthe authors under some unramified assumptions. The goal of this paper is to get a generalexplicit formula in this odd case without any unramified assumption.

1.2 Waldspurger’s formula

Assume that the order of Σ is even in this subsection. We introduce the following notations:

1. B is the unique quaternion algebra over F with ramification set Σ;

2. B× is viewed as an algebraic group over F ;

3. T = E× is a torus of G for a fixed embedding E ⊂ B;

4. π′ = ⊗vπ′v is the Jacquet–Langlands correspondence of π on B×(A).

Define a period integral `(·, χ) : π′ → C by

`(f, χ) =

∫Z(A)T (F )\T (A)

f(t)χ(t)dt, f ∈ π′.

Here the integral uses the Tamagawa measure.

3

Assume that ωπ is unitary. Then π′ is unitary with Petersson inner product 〈·, ·〉 usingTamagawa measure which has volume 2 on A×B×\B×(A). Fix any non-trivial Hermitianform 〈·, ·〉v on π′v so that their product gives 〈·, ·〉. Waldspurger proved the following formulawhen ωπ is trivial:

Theorem 1.2.1 (Walspurger when ωπ = 1). Assume that f = ⊗vfv ∈ π′ is decomposableand nonzero. Then

|`(f, χ)|2 =ζF (2)L(1

2, π, χ)

2 L(1, π, ad)

∏v≤∞

α(fv, χv),

where

α(fv, χv) =L(1, ηv)L(1, πv, ad)

ζv(2)L(12, πv, χv)

∫F×v \E×v

〈π′v(t)fv, fv〉vχv(t)dt.

Moreover, α(fv, χv) is nonzero and equal to 1 for all but finitely many places v.

We interpret the formula as a result on bilinear functionals. As π′ is unitary, the contra-gredient π′ is equal to π′. Thus we have two bilinear functionals on π′ ⊗ π′. The first oneis

`(f1, f2) = `(f1, χ)`(f2, χ−1) =

∫(Z(A)T (F )\T (A))2

f1(t)f2(t)χ(t1)χ(t2)dt1dt2, f1 ∈ π′, f2 ∈ π′.

And the second one is the product of local linear functionals:

α(f1, f2) =∏v

α(f1v, f2v)

α(f1v, f2v) =L(1, ηv)L(1, πv, ad)

ζv(2)L(12, πv, χv)

∫F×v \E×v

〈π′v(t)f1,v, f2,v〉vχv(t)dt.

It is easy to see that both pairings are bilinear and (χ−1, χ)-equivariant under the ac-tion of T (A) × T (A). But we know such functionals are unique up to scalar multiples bythe uniqueness theorem of the local linear functionals of Saito–Tunnell (Proposition 1.1.1).Therefore, these two functionals must be proportional. Theorem 1.1 says that their ratio isrecognized as a combination of special values of L-functions.

1.3 Gross–Zagier formula

Now assume that Σ is odd. We further assume that

1. F is totally real and E is totally imaginary.

2. πv is discrete of weight 2 at all infinite places v of F .

3. χ is a character of finite order.

4

In this case, the set Σ must contain all infinite places. We have a totally definite quaternionalgebra B over A with ramification set Σ which does not have a model over F . For each opencompact subgroup U of B×

f := (B ⊗A Af )×, we have a Shimura curve XU whose complex

points at a real place τ of F can be described as

XU,τ (C) = B(τ)×\H ± × B×f /U∞U

where B(τ) is a quaternion algebra over F with ramification set Σ\τ, Bf is identified withB(τ)Af

as an Af -algebra, and B(τ)× acts on H ± through an isomorphism B(τ)τ 'M2(R).Fix an embedding EA−→B, then we have a set CU ⊂ XU(Eab) of CM-points by E which

has a (non-canonical) identification:

CU ' E×\E×(Af )/U ∩ E×(Af ) (1.3.1)

with Galois action of Gal(Eab/E) given by left multiplication of E×(Af ) and the class fieldtheory. More precisely, at a complex place τE of E over τ of F , CτE(C) in XU,τ (C) isrepresented by (z0, t) with t ∈ E×(Af ) and z0 ∈ H ± is the unique point fixed by E× suchthat the action of E× on the tangent space TH ±,z0 is given by inclusion τE.

Assume that U ∩ E×(Af ) is included into kerχ. Define a divisor of degree zero

YU =∑

t∈E×\E×(Af )/U∩E×(Af )

χ(t)([t]− ξt)

where [t] denote the CM-point in CU via identification (1.3.1) and ξt is the Hodge class ofdegree 1 in the fiber of XU containing [z0, t]. The divisor YU up to a multiple by a root ofunity does not depend on the choice of identification (1.3.1).

Let π′f denote the Jacquet–Langlands correspondence of πf on B×f . Let f ∈ π′f , f ∈ π′f .

We then have a Hecke operator Tf⊗ ef attached to a φ ∈ C∞0 (B×

f ) which has image f ⊗ f inπ′f ⊗ π′f via the projection

C∞0 (B×

f )−→π′f ⊗ π′f

and image 0 in the projections σ′f⊗σ′f for σ′f the Jacquet–Langlands correspondences of finiteparts of all other cuspidal representations σ of GL2(A) with discrete components of weight2 at all archimedean places. It is easy to shown that the Neron–Tate height 〈Y,Tf⊗ efY 〉NTdoes not depend on the choice of the lifting φ.

Theorem 1.3.1. Assume that f = ⊗fv ∈ π′f and f = ⊗fv ∈ π′f are decomposable. Then

〈Y,Tf⊗ efY 〉NT =ζF (2)L′(1/2, π, χ)

4L(1, π, ad)

∏v<∞

α(fv, fv).

Remark. For new form f , some partial results have been proved in [Zh1, Zh2, Zh3] withmore precise formula under some unramified assumptions.

5

1.4 Applications

Let π and χ satisfy the same condition as in §1.3. Then we have an abelian variety A definedover E such that

L(s, A) =∏σ

L(s− 1

2, πσ, χσ)

where (πσ, χσ) are conjugates of (π, χ) for automorphisms σ of C in the sense that the Heckeigenvalues of πσ and χσ are σ-conjugates of those of π and χ. Let Q[π, χ] denote the subfieldgenerated by Heck eigenvalues of π and values of χ. Then A has a multiplication by an orderin Q[π, χ]. Replace A by an isogenous one, we may assume that A has multiplication byZ[π, χ].

Theorem 1.4.1 (Tian–Zhang). Under the assumption above, we have:

1. If ords=1/2L(s, π, χ) = 1, then the Mordell-Weil group A(E) as a Z[π, χ]-module hasrank 1 and the Shafarevich-Tate group X(A) is finite.

2. If ords=1/2L(s, π, χ) = 0 and A does not have CM-type, then the Mordell-Weil groupA(E) is finite. Furthermore, if χ is trivial, and A is geometrically simple, then theShafarevich-Tate group X(A) is also finite.

Remark. For π as above, there is an abelian variety Aπ defined over F with L-series givenby

L(s, Aπ) =∏σ

L(s− 1/2, πσ).

We may assume that Aπ has multiplications by the ring Z[π] of the Hecke eigenvalues in π.The variety A up to an isogeny can be obtained by the algebraic tensor product:

A := Aπ ⊗Z[π] Z[π, χ].

In terms of algebraic points,

A(E) = Aπ(E)⊗Z[π] Z[π, χ]

with usual Galois action by Gal(E/E) on A(E) and by character χ on Z[π, χ].

Example. The theorem applies to all modular F -elliptic curves: the elliptic curve A over aGalois extension of F such that Aσ is isogenous to A for all σ ∈ Gal(F /F ).

1.5 Plan of proof

The ideal of proof of Theorem 1.3.1 which is still based on Gross–Zagier’s paper [GZ] is toshow that the kernel functions representing the the central derivative of L-series is equal tothe generating series of height pairings of CM-points. The new idea here is to construct thekernel function and generating series systematically using Weil representations. This is avariation of an idea used by Waldspurger in the central value formula [Wa].

6

First of all, for given Σ, we will have a quaternion algebra B over A with ramificationset Σ. The reduced norm on B defines an orthogonal space V with group GO of orthogonalsimilitudes. Then we have a Weil representation of GL2(A)×GO(A) on the space S (V×A×).For each φ ∈ S (V × A×) we will construct an automorphic form I(s, g, φ) as a mixedEisenstein and theta series to represent the L-series L(s, π, χ).

If Σ is even, then B = BA for some quaternion algebra B over F . Then by Siegel–Weilformulae, I(0, g, χ, φ) is equal to a period integral θ(g, φ, χ) of a theta series associate to φ.This is essentially the Waldspurger formula.

If Σ is odd, I(0, g, χ, φ) = 0 and I ′(0, g, χ, φ) will represent L′(1/2, π, χ). In the casewhere E/F is a CM-quadratic extension, where π has weight 2 at each archimedean place,and where χ is a finite character, we will write I ′(0, g, χ) as a sum I ′(0, g, χ, φ)(v) indexedover places v of F and a finite sum of Eisenstein series and their derivations defined in §4.4.4in our previous paper [Zh1].

The role of theta series in Waldspurger’s work is played by the following generating seriesof heights of CM-points:

Zφ(g, χ) = 〈Zφ(g)YU , YU〉

where Zφ(g) is a generating series of Hecke operators on XU defined by Kudla extending theclassic formula

∑n Tnq

n. The modularity of this generating series is proved in our previouspaper [YZZ]. Using Arakelov theory, we may decompose Zφ(g, χ) into a sum of pairingsinvolving the Hodge class ξ’s and pairings only involving CM-points. The pairings involvesξ’s give a linear combination of Eisenstein series and their derivations. The pairing overCM-points can be further decomposed into a sum Zφ(g, χ)(v) indexed over places of F .

For good v, one can show that I ′(0, g, χ)(v) is essentially Zφ(g, χ)(v). For bad v’s, wecan show that they are essential equal to a pseudo-theta series associate to quadratic spaceover F . Since both I ′(0, g, χ, φ) and Zφ(g, χ) are both automorphic, some density argumentcan be used to show that their difference is essentially a linear combination of theta seriesand Eisenstein series. Now the main theorem follows from the result of Saito and Tunnel onlinear forms, Proposition 1.1.1.

Acknowledgement

This research has been supported by some grants from the National Science Foundation andChinese Academy of Sciences.

1.6 Notation and terminology

Local fields and global fields

We normalize the absolute values, additive characters, and measures following Tate’s thesis.Let k be a local field of characteristic zero.

• Normalize the absolute value | · | on k as follows:

It is the usual one if k = R.

7

It is the square of the usual one if k = C.

It takes the uniformizer to the inverse of the cardinality of the residue field if k isnon-archimedean.

• Normalize the additive character ψ : k → C× as follows:

If k = R, then ψ(x) = e2πix.

If k = C, then ψ(x) = e4πiRe(x).

If k is non-archimedean, then it is a finite extension of Qp for some prime p. Takeψ = ψQp trk/Qp . Here the additive character ψQp of Qp is defined by ψQp(x) = e−2πiι(x),where ι : Qp/Zp → Q/Z is the natural embedding.

• We take the Haar measure dx on k to be self-dual with respect to ψ. More precisely,

If k = R, then dx is the usual Lebesgue measure.

If k = C, then dx is twice of the usual Lebesgue measure.

If k is non-archimedean, then vol(Ok) = |dk|12 . Here Ok is the ring of integers and

dk ∈ k is the different of k over Qp.

• We take the Haar measure d×x on k× as follows:

d×x = ζk(1)|x|−1v dx.

Recall that ζk(s) = (1 − N−sv )−1 if v is non-archimedean with residue field with Nv-

elements, and ζR(s) = π−s/2Γ(s/2), ζC(s) = 2(2π)−sΓ(s). With this normalization, ifk is non-archimedean, then vol(O×

k ) = vol(Ok).

Now go back to the totally real field F . For each place v, we choose | · |v, ψv, dxv, d×xvas above. By tensor products, they induce global | · |, ψ, dx, d×x.

Quadratic extensions

As for the CM field E, we view V1 = (E, q) as a two-dimensional vector space over F , whichuniquely determines self-dual measures dx on AE and Ev for each place v of F . We definethe measures on the corresponding multiplicative groups by the same setting as above.

Let E1 = y ∈ E× : q(y) = 1 act on E by multiplication. It induces an isomorphismSO(V1) ' E1 of algebraic groups over F . We also have SO(V1) = E×/F× given by

E×/F× → E1, t 7→ t/t.

It is an isomorphism by Hilbert Theorem 90.We have the following exact sequence

1 → E1 → E× q→ q(E×) → 1.

8

For all places v, we endow E1v the measure such that the quotient measure over q(E×

v ) is thesubgroup measure of F×

v . On the other hand, we endow E×v /F

×v with the quotient measure.

It turns out that these two measures induce the same one on SO(V1).Let v be a non-archimedean place of F , denote by dv ∈ OFv the local different of Fv,

and by Dv ∈ OFv the local discriminant of Ev/Fv. Then vol(OFv) = vol(O×Fv

) = |dv|12v and

vol(OEv) = vol(O×Ev

) = |Dv|12 |dv|v. We set dv and Dv to be 1 when v is archimedean.

If v is a non-archimedean place inert in E, then

vol(E1v) =

vol(O×Ev

)

vol(q(O×Ev

))=

vol(O×Ev

)

vol(O×Fv

)= |Dv|

12 |dv|

12 = |dv|

12 .

If v is a non-archimedean place ramified in E, then

vol(E1v) =

vol(O×Ev

)

vol(q(O×Ev

))= 2

vol(O×Ev

)

vol(O×Fv

)= 2|Dv|

12 |dv|

12 .

If v is archimedean, then vol(E1v) = 2.

Notation on GL2

We introduce the matrix notation:

m(a) =

(a

a−1

), d(a) =

(1

a

), d∗(a) =

(a

1

)n(b) =

(1 b

1

), kθ =

(cos θ sin θ− sin θ cos θ

), w =

(1

−1

).

We denote by P ⊂ GL2 and B ⊂ SL2 the subgroups of upper triangular matrices. And Nbe the standard unipotent subgroup.

For any place v, the character δv : P (Fv) → R× defined by

δv :

(a b

d

)7→∣∣∣ad

∣∣∣ 12which extends to a function δv : GL2(Fv) → R× by Iwasawa decomposition.

If v is a real place, we define a function λv : GL2(Fv) → C by λv(g) = eiθ if

h =

(cc

)(a b

1

)(cos θ sin θ− sin θ cos θ

)in the form of Iwasawa decomposition.

For g ∈ GL2(A), we define δ(g) =∏

v δv(gv) and λ∞(g) =∏

v|∞ λv(gv).

9

2 Weil representation and kernel functions

In this section, we will review the theory of Weil representation and its applications to in-tegral representations of Rankin–Selberg L-series L(s, π, χ) and to a proof of Waldspurger’sthe central value formula. We will mostly follow Waldspurger’s treatment with some modi-fications including the construction of incoherent Eisenstein series from Weil representation.

We will start with the classical theory of Weil representation of O(F )×SL2(F ) on S (V )for an orthogonal space V over a local field F and its extension to GO(F ) × GL2(F ) onS (V, F ) by Waldspurger. We then define theta function, state Siegel–Weil formulae, anddefine normalized local Shimizu lifting. The main result of this section is an integral formulafor L-series L(s, π, χ) using a kernel function I(s, g, φ, χ). This kernel function is a mixedEisenstein and theta series attached to each φ ∈ S (V × A×) for V an orthogonal spaceobtained form a quaternion algebra over A. The Waldspurger formula is a direct consequenceof the Siegel–Weil formula. We conclude the section by proving the vanishing of the kernelfunction at s = 0 using information on Whittaker function of incoherent Eisenstein series.

2.1 Weil representations

Let us start with some basic set up on Weil representation. We follow closely from Wald-spurger’s paper [Wa] but we will work on a slightly bigger space S (V, F×).

Non-archimedean case

Let F be a non-archimedean local field and (V, q) a quadratic space over F . Let O = O(V, q)

denote the orthogonal group of (V, q) and SL2 the double cover of SL2. Then for any non-

trivial character ψ of F , the group SL2(F ) × O(F ) has an action r on the space S (V ) oflocally constant with compact support as follows:

• r(h)φ(x) = φ(h−1x), h ∈ O(F );

• r(m(a))φ(x) = χV (a)φ(ax)|a|dimV/2, a ∈ F×;

• r(n(b))φ(x) = φ(x)ψ(bq(x)), b ∈ F ;

• r(w)φ = γφ(x), γ8 = 1.

Archimedean case

When F ' R, we may define an analogous representation of the pair (G ,K ) of the Lie

algebra G and a maximal compact subgroup K = SO2(R)×K 0 of SL2(R)× O(R). Moreprecisely, the maximal compact subgroup K 0 stabilizes a unique orthogonal decompositionV = V + + V − such that the restrictions of q on V ± are positive and negative definiterespectively. Then we take S (V ) as to be the space of functions of the form

P (x)e−2π(q(x+)−q(x−)), x = x+ + x−, x± ∈ V ±

10

where P is a polynomial function on V . The action of (G ,K ) on S (V ) can be deducedformally from the same formulae as above. Notice that the space S (V ) depends on thechoice of the maximal compact group K 0 of O(V ) and ψ.

Extension to GL2

Assume that dimV is even, we want to extend this action to an action r of GL2(F )×GO(F )or more precisely their (G ,K ) analogue in real case. If F is non-archimedean, let S (V, F×)be the space of smooth functions φ(x, u) on V ×F× which is in S (V ) for any fixed u. Thenthe Weil representation is defined by the following formulae:

• r(h)φ(x, u) = φ(h−1x, ν(h)u), h ∈ GO(F );

• r(g)φ(x, u) = ru(g)φ(x, u), g ∈ SL2(F );

• r(d(a))φ(x, u) = φ(x, a−1u)|a|− dimV/4, a ∈ F×

where in the second formula φ(x, u) is a function of x, and ru is the Weil representation onV with new norm uq.

Remark. In Waldspurger’s paper [Wa], he only considers the Weil representation on the thespace S (V × F×) consisting of locally constant functions with compact support if F isnon-archimedean, and linear combinations of functions of the form

h(u)P (x)e−2π|u|(q(x+)−q(x−))

with h(u) ∈ C∞0 (R) if F is archimedean.

2.2 Theta series

Now we assume that F is a totally real number field and (V, q) is an orthogonal space overF . Then we can define a Weil representation r on S (VA) (which actually depends on ψ)

of SL2(A) × O(VA). When dimV is even, we can define an action r of GL2(A) × GO(VA)on S (VA,A×) which is the restricted tensor product of S (Vv, Fv) with base element ascharacteristic function of VOFv

×O×Fv

.Notice that the representation r depends only on the quadratic space (VA, q) over A.

Thus we may define representations directly for a pair (V, q) of a free A-module V withnon-degenerate quadratic form q.

If (V, q) is a base change of an orthogonal space over F , then we call this Weil represen-tation is coherent; otherwise it is called incoherent.

Let F be a number field, and (V, q) a quadratic space over F . Fix a nontrivial additivecharacter ψ of F\A. Then for any φ ∈ S (VA), we can form a theta series as a function on

SL2(F )\SL2(A)×O(F )\O(A):

θφ(g, h) =∑x∈V

r(g, h)φ(x), (g, h) ∈ SL2(A)×O(A).

11

Similarly, when V has even dimension we can define theta series for φ ∈ S (VA × A×) as anautomorphic form on GL2(F )\GL2(A)×GO(F )\GO(A):

θφ(g, h) =∑

(x,u)∈V×F×r(g, h)φ(x, u).

Siegel–Weil formulae

Still in the global case. Let V be an orthogonal space over F . For φ ∈ S (VA), s ∈ C, wehave a section

g 7→ δ(g)sr(g)φ(0)

in IndSL2B (χV |·|dimV/2+s) where δ is the modulo function which assigns g to |a|s in the Iwasawa

decomposition g = m(a)n(b)k. Thus we can form an Eisenstein series

E(s, g, φ) =∑

γ∈P (F )\SL2(F )

δ(γg)sr(γg)φ(0).

This Then we have Siegel–Weil formula in the following cases:

A. (V, q) = (E,NE/F ) with E a quadratic extension of F , and

B. V is the space of the trace free elements in a quaternion algebra over F and q is thethe restriction of the reduced norm.

Theorem 2.2.1 (Siegel–Weil formula). Let φ ∈ S (VA). Then∫SO(F )\SO(A)

θφ(g, h)dh = mE(0, g, φ)

where dh is the Tamagawa measure on SO(A) which has values 2L(1, χ) and 2 on SO(F )\SO(A)for dimV = 2 and 3 respectively, and m = L(1, η) and 2 respectively.

Notice that the Eisenstein series can be defined for any quadratic space (V, q) withrational detV . But its value vanishes on s = 0. Kudla has proposed a connection betweenthe derivative of Eisenstein series and arithmetic intersection of certain arithmetic cycles onShimura varieties.

2.3 Shimizu lifting

Let F be a local field and B a quaternion algebra over F . Write V = B as an orthogonalspace with quadratic form q defined by the reduced norm on B. Let B× ×B× act on V by

x 7→ h1xh−12 , x ∈ B, hi ∈ B×.

Then we have an exact sequence:

1−→F×−→(B× ×B×) o 1, ι−→GO(F )−→1.

12

Here ι acts on V = B by the canonical involution, and on B××B× by (h1, h2) 7→ (hι−12 , hι−1

1 ),and here F× is embedded into the middle group by x 7→ (x, x) o 1. The theta lifting of anyrepresentation π of GL2(F ) on GO is induced by representation JL(π)⊗ JL(π) on B××B×,JL(π) is the Jacquet–Langlands lifting of π. Recall that JL(π) 6= 0 only if B = M2(F ) or πis discrete, and that if JL(π) 6= 0,

dim HomGL2(F )×B××B×(S (V × F×)⊗ π, JL(π)⊗ JL(π)) = 1.

This space has a normalized form θ as follows: for any ϕ ∈ π realized as W−1(g) in aWhittaker model for the additive character ψ−1, φ ∈ S (V × F×), and for F the canonicalform on JL(π)⊗ JL(π),

Fθ(φ⊗ ϕ) =ζ(2)

L(1, π, ad)

∫N(F )\GL2(F )

W−1(g)r(g)φ(1, 1)dg (2.3.1)

The constant before the integral is used to normalize the form so that F (θϕφ) = 1 when everything is unramified.

Let F be a number field and B a quaternion algebra over F . Then the Shimuzu liftingcan be realized as a global theta lifting:

θ(φ⊗ ϕ)(h) =ζ(2)

2L(1, π, ad)

∫GL2(F )\GL2(A)

ϕ(g)θφ(g, h)dg, h ∈ B×A ×B×

A . (2.3.2)

Here is the relation between global theta lifting and normalized local theta lifting:

Proposition 2.3.1. We have a decomposition θ = ⊗θv in

HomGL2(A)×B×A ×B×A(S (VA × A×)⊗ π, JL(π)⊗ JL(π)).

Proof. It suffices to show that for decomposable φ = ⊗φv and ϕ = ⊗ϕv,

Fθ(φ⊗ ϕ) =∏

Fθv(φv ⊗ ϕv).

The inner product between JL(π) and JL(π) is given by integration on the diagonal (A×B×\B×A )2.

Let V = V0⊕V1 correspond to the decomposition B = B0⊕F with B0 the subspace of tracefree elements. Then the diagonal can be identify with SO′ = SO(V0) thus we have globally

Fθ(φ⊗ ϕ) =

∫SO′(F )\SO′(A)

dh

∫GL2(F )\GL2(A)

θφ(g, h)ϕ(g)dg.

To compute this integral, we interchange the order of integrals and apply the Siegel–Weilformula:∫

SO′(F )\SO′(A)

θφ(g, h)dh = 2

∑γ∈P (F )\GL2(F )

δ(γg)s∑

(x,u)∈V1×F

r(γg)φ(x, u)

s=0

.

13

Thus we have

Fθ(φ⊗ ϕ) =2

∫GL2(F )\GL2(A)

ϕ(g)∑

γ∈P (F )\GL2(F )

∑(x,u)∈V1×F

r(γg)φ(x, u)dg

=2

∫P (F )\GL2(A)

ϕ(g)∑

(x,u)∈V1×F×r(g)φ(x, u)dg

The sum here can be written as a sum of two parts I1 + I2 invariant under P (F ) where I1 isthe sum over x 6= 0 and I2 over x = 0. It is easy to see that I2 is invariant under N(A) whichcontributes 0 to the integral as ϕ is cuspidal. The sum I1 is a single orbit over diagonalgroup. Thus we have

2

∫N(F )\GL2(A)

ϕ(g)∑

(x,u)∈V1×F

r(g)φ(1, 1)dg

=2

∫N(A)\GL2(A)

W−1(g)r(g)φ(1, 1)dg.

2.4 Integral representations of L-series

In the following we want to describe an integral representation of the L-series L(s, π, χ). LetF be a number field with ring of adeles A. Let B be a quaternion algebra with ramificationset Σ. Fix an embedding EA−→B. We have an orthogonal decomposition

B = EA + EAj, j2 ∈ A×.

Write V for the orthogonal space B with reduced norm q, and V1 = EA and V2 = EAj assubspaces of VA. Then V1 is coherent and is the base change of F -space V1 := E, and V2 iscoherent if and only if Σ is even.

For φ ∈ S (V× A×), we can form a mixed Eisenstein–Theta series:

I(s, g, φ) =∑

γ∈P (F )\GL2(F )

δ(γg)s∑

(x1,u)∈V1×F×r(γg)φ(x1, u)

I(s, g, φ, χ) =

∫T (F )\T (A)

χ(t)I(s, g, r(t, 1)φ)dt.

For ϕ ∈ π, we want to compute the integral

P (s, ϕ, φ, χ) =

∫Z(A)GL2(F )\GL2(A)

ϕ(g)I(s, g, φ, χ)dg.

14

Proposition 2.4.1 (Waldspurger). If φ = ⊗φv and ϕ = ⊗ϕv are decomposable, then

P (s, ϕ, φ, χ) =∏v

Pv(s, ϕv, φv, χv)

where

Pv(s, ϕv, φv, χv) =

∫Z(Fv)\T (Fv)

χ(t)dt

∫N(Fv)\GL2(Fv)

δ(g)sW−1,v(g)r(g)φv(t−1, q(t))dg.

Proof. Bring the definition formula of I(s, g, φ, χ) to obtain an expression for P (s, ϕ, φ, χ):∫Z(A)P (F )\GL2(A)

ϕ(g)δ(g)s∫T (F )\T (A)

χ(t)∑

(x,u)∈V1×F×r(g, (t, 1))φ(x, u)dtdg.

We decompose the first integral as a double integral:∫Z(A)P (F )\GL2(A)

dg =

∫Z(A)N(A)P (F )\GL2(A)

∫N(F )\N(A)

dndg

and perform the integral on N(F )\N(A) to obtain:∫Z(A)N(A)P (F )\GL2(A)

δ(g)sdg

∫T (F )\T (A)

χ(t)∑

(x,u)∈V1×F×Wq(x)−1u(g)r(g, (t, 1))φ(x, u)dt.

Here as ϕ is cuspidal, the term x = 0 has no contribution to the integral. In this way, wemay change variable (x, u) 7→ (x, q(x−1)u) to obtain the following expression of the sum:∑

(x,u)∈E××F×W−u(g)r(g, (t, 1))φ(x, q(x−1)u)

=∑

(x,u)∈E××F×W−u(g)r(g, (tx, 1))φ(1, u)).

The sum over x ∈ E× collapses with quotient T (F ) = E×. Thus the integral becomes∫Z(A)N(A)P (F )\GL2(A)

δ(g)sdg

∫T (A)

χ(t)∑u∈F×

W−u(g)r(g, (t, 1))φ(1, u)dt.

The expression does not change if we make the substitution (g, au) 7→ (gd(a)−1, u). Thus wehave ∫

Z(A)N(A)P (F )\GL2(A)

δ(g)sdg

∫T (A)

χ(t)∑u∈F×

W−u(d(u−1)g)r(d(u−1)g, (t, 1))φ(1, u)dt.

The sum over u ∈ F× collapses with quotient P (F ), thus we obtain the following expression:

P (s, ϕ, φ, χ) =

∫Z(A)N(A)\GL2(A)

δ(g)sdg

∫T (A)

χ(t)W−1(g)r(g)φ(t−1, q(t))dt.

15

We may decompose inside integral as ∫Z(A)\T (A)

∫Z(A)

and move the integral∫Z(A)\T (A)

to the outside. Then we use the fact that ωπ · χ|A× = 1 to

obtain

P (s, ϕ, φ, χ) =

∫Z(A)\T (A)

χ(t)dt

∫N(A)\GL2(A)

δ(g)sW−1(g)r(g)φ(t−1, q(t))dg.

When everything is unramified, Waldspurger has computed these integrals:

Pv(s, ϕv, φv, χv) =L((s+ 1)/2, πv, χv)

L(s+ 1, ηv).

Thus we may define a normalized integral P 0v by

Pv(s, ϕv, φv, χv) =L((s+ 1)/2, πv, χv)

L(s+ 1, ηv)P 0v (s, ϕv, φv, χv).

This normalized local integral P 0v will be regular at s = 0 and equal to

L(1, ηv)L(1, πv, ad)

ζv(2)L(1/2, πv, χv)

∫Z(Fv)\T (Fv)

χv(t)F (JL(π)(t)θ(φv ⊗ ϕv)) dt.

We may write this as αv(θ(φv ⊗ ϕv)) with

αv ∈ Hom(JL(π)⊗ JL(π),C)

given by the integration of matrix coefficients:

αv(ϕ⊗ ϕ) =L(1, ηv)L(1, πv, ad)

ζv(2)L(1/2, πv, χv)

∫Z(Fv)\T (Fv)

χv(t)(π(t)ϕ, ϕ)dt. (2.4.1)

And we define an element α := ⊗vαv in Hom(JL(π)⊗ JL(π),C).We now take value or derivative at s = 0 to obtain

Proposition 2.4.2.

P (0, ϕ, φ, χ) =L(1/2, π, χ)

L(1, χE)

∏v

αv(θ(φv ⊗ ϕv)).

If Σ is odd, then L(1/2, π, χ) = 0, and

P ′(0, ϕ, φ, χ) =L′(1/2, π, χ)

2L(1, χE)α(θ(φ⊗ ϕ)).

16

Remark. Let AΣ(GL2, χ) denote the direct sum of cusp forms π on GL2(A) such thatΣ(π, χ) = Σ. If Σ is even, let I (g, φ, χ) be the projection of I(0, g, φ, χ) on AΣ(GL2, χ

−1).If Σ is odd, let I ′(g, φ, χ) denote the projection of I ′(0, g, φ, χ) on AΣ(GL2, χ

−1). Then haveshown that I (g, φ, χ) and I ′(g, φ, χ) represent the functionals

ϕ 7→ L(1/2, π, χ)

L(1, χE)α(θ(φ⊗ ϕ)) if Σ is even

ϕ 7→ L′(1/2, π, χ)

2L(1, χE)α(θ(φ⊗ ϕ)) if Σ is odd

on π respectively.

2.5 Proof of Waldspurger formula

Assume that Σ is even and we recall Waldspurger’s proof of his central value formula. Nowthe space V = V (A) is coming from a global V = B over F . For φ ∈ S (V×A×) and ϕ ∈ π,we want to compute the double period integral of θ(φ⊗ ϕ):∫

(T (F )Z(A)\T (A))2θ(φ⊗ ϕ)(t1, t2)χ(t1)χ

−1(t2)dt1dt2.

Using definition, this equals

ζ(2)

2L(1, π, ad)

∫Z(A)GL2(F )\GL2(A)

ϕ(g)θ(g, φ, χ)dg

where

θ(g, φ, χ) =

∫Z∆(A)T (F )2\T (A)2

θφ(g, (t1, t2))χ(t1t−12 )dt1dt2,

where Z∆ is the image of the diagonal embedding F×−→(B×)2. We change variable t1 = tt2to get a double integral

θ(g, φ, χ) =

∫T (F )\T (A)

χ(t)dt

∫Z(F)T (F )\T (A)

θφ(g, (tt2, t2))dt2,

Notice that the diagonal the diagonal embedding Z\T can be realized as SO(Ej) in thedecomposition V = B = E + Ej. Thus we can apply Siegel Weil formula 2.2.1 to obtain

θ(g, φ, χ) = L(1, χ)I(0, g, φ, χ).

Here recallI(s, g, φ) =

∑γ∈P (F )\GL2(F )

δ(γg)s∑

(x,ψ)∈E×F×r(γg, (t, 1)φ)(x, u).

17

Combining with Proposition 2.4.2, we have∫(T (F )Z(A)\T (A))2

θ(φ⊗ ϕ)(t1, t2)χ(t1)χ−1(t2)dt1dt2 =

ζ(2)L(1/2, π, χ)

2L(1, π, ad)α(θ(φ⊗ ϕ)).

This is certainly an identity as functionals on JL(π)⊗ JL(π):

`χ ⊗ `χ−1 =ζ(2)L(1/2, π, χ)

2L(1, π, ad)· α. (2.5.1)

2.6 Vanishing of the kernel function

We want to compute the Fourier expansion of I(s, g, φ, χ) and I(s, g, φ) in the case that Σis odd, and prove that they vanish at s = 0. We will mainly work on φ ∈ S (V× A×).

By writing φ as a linear combination, we may reduce the computation to the decompos-able case: φ(x1 + x2, u) = φ1(x1, u)φ2(x2, u) for xi ∈ Vi. Then we have

I(s, g, φ) =∑u∈F×

θ(g, u, φ1)E(s, g, u, φ2)

and

I(s, g, φ, χ) =

∫T (F )\T (A)

χ(t)I(s, g, r(t, 1)φ)dt

whereθ(g, u, φ1) =

∑x∈E

r(g)φ1(x, u),

E(s, g, u, φ2) =∑

γ∈P (F )\GL2(F )

δ(γg)sr(γg)φ2(0, u).

It suffices to study the behavior of Eisenstein series at s = 0. The work of Kudla-Rallisin more general setting shows that the incoherent Eisenstein series E(s, g, u, φ2) vanishes ats = 0 by local reasons. For readers’ convenience, we will show this fact by detailed analysisof the local Whittaker functions. We will omit φ1 and φ2 in the notation.

We start with the Fourier expansion of Eisenstein series:

E(s, g, u) = E0(s, g, u) +∑a∈F×

Ea(s, g, u)

where the a-th Fourier coefficient is

Ea(s, g, u) =

∫F\A

E

(s,

(1 b

1

)g, u

)ψ(−ab)db, a ∈ F.

An easy calculation gives the following result.

18

Lemma 2.6.1.

E0(s, g, u) = δ(g)sr2(g)φ2(0, u)−∫

Aδ(wn(b)g)s

∫V2

r2(g)φ2(x2, u)ψ(buq(x2))dux2db,

Ea(s, g, u) = −∫

Aδ(wn(b)g)s

∫V2

r2(g)φ2(x2, u)ψ(b(uq(x2)− a))dux2db, a ∈ F×.

Here the measure dux2 on V2 is the self-dual Haar measure with respect to uq.

Proof. It is a standard result that the constant term is given by

E0(s, g, u) = δ(g)sr2(g)φ2(0, u) +

∫N(A)

δ(wng)sr2(wng)φ2(0, u)dn.

By definition, we have

r2(wn(b)g)φ2(0, u) = γ(V2)

∫V2

r2(n(b)g)φ2(x2, u)dux2

= γ(V2)

∫V2

r2(g)φ2(x2, u)ψ(buq(x2))dux2.

Here γ(V2) is the Weil index of the quadratic space (V2, uq), which is apparently independentof u ∈ F×. By the orthogonal decomposition V = V1 ⊕ V2, we have γ(V) = γ(V1)γ(V2).Here γ(V1) = 1 since V1 = E(Af ) is coherent, and γ(V) =

∏v γ(Vv, q) = (−1)#Σ = −1 since

Σ is assumed to be odd. It follows that γ(V2) = −1. So we get the result for E0(s, g, u).The formula for Ea(s, g, u) is computed similarly.

Notation. We introduce the following notations:

Wa(s, g, u) =

∫Aδ(wn(b)g)s

∫V2

r2(g)φ2(x2, u)ψ(b(uq(x2)− a))dux2db,

Wa,v(s, g, u) =

∫Fv

δ(wn(b)g)s∫

V2,v

r2(g)φ2,v(x2, u)ψv(b(uq(x2)− a))dux2db,

W 0,v(s, g, u) =

L(s+ 1, ηv)

L(s, ηv)W0,v(s, g, u).

Here the normalizing factorL(s+ 1, ηv)

L(s, ηv)has a zero at s = 0 when Ev is split, and is equal

to π−1 at s = 0 when v is archimedean.Now we list the values of these local Whittaker functions when s = 0. They are essentially

the Siegel-Weil formula in the coherent case. But in the incoherent case, they will lead tothe vanishing of our kernel function at s = 0.

Proposition 2.6.2. (1) In the sense of analytic continuation for s ∈ C,

W0(0, g, u) =r2(g)φ2(0, u),

W 0,v(0, g, u) =|Dv|

12 |dv|

12 r2(g)φ2,v(0, u).

19

(2) Assume a ∈ F×v .

(a) If au−1 is not represented by (V2,v, q2,v), then Wa,v(0, g, u) = 0.

(b) Assume that there exists ξ ∈ V2,v satisfying q(ξ) = au−1. Then

Wa,v(0, g, u) =1

L(1, ηv)

∫E1

v

r2(g)φ2,v(ξτ, u)dτ.

Proof. It is easy to reduce to the case u = 1. By Siegel-Weil theorem, we know that all theequalities hold up to constants independent of g and φv. Hence we only need to check theconstants here. Many cases are in the literature (see [KRY] for example).

Proposition 2.6.3. E(0, g, u) = 0, and thus I(0, g, φ) = 0.

Proof. It is a direct corollary of Proposition 2.6.2. The Eisenstein series has Fourier expan-sion

E(0, g, u) =∑a∈F

Ea(0, g, u).

The vanishing of the constant term easily follows Proposition 2.6.2. For a ∈ F×,

Ea(0, g, u) = −∏v

Wa,v(0, g, u)

is nonzero only if au−1 is represented by (Ev, q2,v) at each v.In fact, Bv is split if and only if there exists a non-zero element ξ ∈ V2,v (equivalently,

for all nonzero ξ ∈ V2,v) such that −q(ξ) ∈ q(E×v ). In another word, the following identities

hold:ε(Bv) = ηv(−q(ξ)),

where ε(Bv) is the Hasse invariant of Bv and ηv is the quadratic character induced by Ev.Now the existence of ξ at v such that q2,v(ξ) = au−1 is equivalent to

ε(Bv) = ηv(−au−1).

If this is true for all v, then ∏v

ε(Bv) =∏v

ηv(−au−1) = 1.

It contradicts to the incoherence condition that the order of Σ is odd.

20

3 Derivatives of kernel functions

In this section, we want to study the derivative of the kernel function for L-series when Σ isodd and π has discrete components of weight 2 at archimedean places. The main content ofthis section is various local formulae.

In §3.1, we will extend the result in the last section to functions in φ ∈ S (V×A×)GO(F∞),the maximal quotient of S (V×A×) with trivial action by GO(F∞). In §3.2, we decomposethe derivative I ′(0, g) of the kernel function into a sum of finitely many usual theta series,their derivations, and infinite many local terms I ′(0, g)(v) indexed by places of F . Eachlocal term is a period integral of some kernel function K (v)(g, (t1, t2)). In §3.3, to simplifythe kernel function, we introduce a class of “degenerate” Schwartz functions and provesa non-vanishing result concerning them. We will give some precise formulae for the kernelfunctions in §3.4 and 3.5. In the last two section we will define and compute the holomorphicprojection of I ′(0, g).

3.1 Discrete series of weight two at infinite Places

Discrete series of weight two at infinity

If V is positive over an F ' R of even dimension 2d, the subspace S (V, F )GO(F ) of functionsinvariant under GO(V ) will be an representation of GL2(F ) isomorphic to the discrete seriesof weight d. In fact this space consists of functions on V × F× of the form:

(P1(|u|q(x)) + sgn(u)P2(|u|q(x)))e−2π|u|q(x)

with polynomials Pi on R. The space S (V, F )GO(F ) can be a quotient of S (V ×R×) (resp.S (V × R×)O(F )) via integration over GO(F ):

φ 7→∫

GO(F )

r(h)φdh, (resp.

∫Z(F )

r(h)φdh).

Thus we have an equality

S (V, F×)GO(F ) = S (V × F×)GO(F ) = S (V × F×)O(F )Z(F ).

where the second (resp. third) the space denote the maximal quotient of S (V × F×) (resp.S (V × F×)O(F ) with trivial action of GO(F ) (resp. Z(F )).

The the standard Schwartz function φ ∈ S (V,R×) is the Gaussian

φ0(x, u) =1

2(1 + sgn(u))e−2|u|q(x)

Then one verifies thatr(g)φ(x, u) = W

(d)uq(x)(g)

21

where W(d)a (g) is the standard Whittaker function of weight d for character e2πiax:

W (d)a (g) =

|y0|

d2 ediθ if a = 0

|y0|d2 e2πia(x0+iy0)ediθ if ay0 > 0

0 if ay0 < 0

for any a ∈ R and

g =

(z0

z0

)(y0 x0

1

)(cos θ sin θ− sin θ cos θ

)∈ GL2(R)

in the form of Iwasawa decomposition.We may extend the result in the last section to functions in

S (V× A×)GO(F∞) := S (V∞, F×∞)GO(F∞) ⊗S (Vf × A×

f ).

Theta series

First let us consider the definition of theta series. Let V be a positive definite quadratic spaceover a totally real field F . Let φ ∈ S (V (A)×A×)GO(F∞). There is an open compact subgroupK ⊂ GO(V )(Af ) such that φf is invariant under the action of K by Weil representation.Denote µK = F× ∩K. Then µK is a subgroup of the unit group O×

F , and thus is a finitelygenerated abelian group. Our theta series is of the following form:

θφ(g, h) =∑

(x,u)∈µK\(V×F×)

r(g, h)φ(x, u),

The definition here depends on the choice of K. We may normalize this by adding a factor[O×

F : µK ]−1 in front of sum. But we will not use this normalization.By choosing a different fundamental domain, we can write

θφ(g, h) = ε−1K

∑u∈µ2

K\F×

∑x∈V

r(g, h)φ(x, u).

Here εK = |1,−1∩K| ∈ 1, 2. In particular, εK = 1 for K small enough. The summationover u is well-defined since φ(x, u) = r(α)φ(x, u) = φ(αx, α−2u) for any α ∈ µK . It is anautomorphic form on GL2(A)×GO(V )(A), provided the absolute convergence.

To show the convergence, we claim that the summation over u is actually a finitesum depending on (g, h). For fixed (g, h), there is a compact subset K ′ ⊂ A×

f such thatr(g, h)φf (x, u) 6= 0 only if u ∈ K ′. Thus the summation is taken over u ∈ µ2

K\(F× ∩ K ′),which is a finite set. In fact, it is a result of finiteness of µK\(F× ∩K ′) and µ2

K\µK . Theformer follows from the injection µK\(F× ∩ K ′) → (A×

f ∩ K)\K ′ and compactness of K ′.

And µ2K\µK is finite because µK ⊂ O×

F is a finitely generated abelian group.

22

Alternatively, we may construct theta series for above φ ∈ S (V (A),A×) by some function

φ = φ∞ ⊗ φf ∈ S (V× A×)O(F∞) such that∫Z(F∞)

r(h)φdh = φ0.

In this case,

θφ(g, h) =

∫Z(F∞)/µK

θeφ(g, zh)dz.

Kernel functions

For φ ∈ S (V× A×)GO(F∞), we may define the mixed Eisenstein-theta series as

I(s, g, φ) =∑

γ∈P (F )\GL2(F )

δ(γg)s∑

(x1,u)∈µF \V1×F×r(γg)φ(x1, u).

If χ has trivial component at infinity, then we may define

I(s, g, φ, χ) = vol(KZ)

∫T (F )\T (A)/Z(F∞)KZ

χ(t)I(s, g, r(t, 1)φ)dt.

It is clear that I(s, g, φ) is a finite linear combination of I(s, g, φ, χ). The definitions here donot depend on the choice of K. Moreover if

φ = φ∞ ⊗ φf ∈ S (V× A×)O(F∞)

such that

φ∞ =

∫Z(F∞)

r(z)φ∞dz

then

I(s, g, φ) =

∫Z(F×∞)/µF

I(s, g, r(z)φ)dz

I(s, g, φ, χ) = I(s, g, φ, χ).

Indeed, in the definition of I(s, g, φ, χ) in section 2.4, we may decompose the inte-gral over T (F )\T (A) into double integrals over T (F )\T (A)/Z(F∞)KZ and integrals overZ(F∞)KZT (F )/T (F ) to obtain

I(s, g, φ, χ) =

∫T (F )\T (A)/Z(F∞)KZ

χ(t)dt

∫T (F )\T (F )Z(F∞)KZ

I(s, g, r(tz, 1)φ)dz.

The inside integral domain can be identified with

T (F )\T (F )Z(F∞)KZ ' µZ\Z(F∞)KZ

23

with has a fundamental domain Z(F∞)/µK ×KZ . Thus the second integral can be writtenas

vol(KZ)

∫Z(F∞)/µK

I(s, g, r(tz, 1)φ)dz.

For this last integral, we write the sum over V1×F× in the definition of I(s, g, φ) as a doublesum over µZ\V1×F× and a sum of µZ . The first sum commutes with integral over Z(F∞)/µKwhile the second the sum collapse with quotient Z(F∞)/µF to get an simple integral over

Z(F∞) which changes φ to φ.

3.2 Derivatives of Whittaker Functions

We now compute the derivative of the kernel function I(s, g, φ) for φ ∈ S (V × A×)GO(F∞).It suffices to do this for φ of the form φ = φ1 ⊗ φ2 with φi ∈ S (Vi × A×)GO(Vi∞). In thiscase,

I(s, g) =∑

u∈µ2K\F×

θ(g, u, φ1)E(s, g, u, φ2)

where KZ is an open subset of Z(Af ) not including such that φf is invariant under KZ . Itamounts to compute the derivative of the Eisenstein series E(s, g, u, φ2). We may furtherassume that both φi have standard components φi∞ defined in the last subsection. As beforewe will suppress the dependence on φ.

Let us start with the Fourier expansion:

E(s, g, u) = δ(g)sr2(g)φ2(0, u)−∑a∈F

Wa(s, g, u).

Denote by F (v) the set of a ∈ F× that is represented by (E(Av), uqv2) but not by (Ev, uq2,v).Then F (v) 6= ∅ only if E is non-split at v. By Proposition 2.6.2, Wa,v(0, g, u) = 0 for anya ∈ F (v). Then taking the derivative yields

W ′a(0, g, u) = W ′

a,v(0, g, u)Wva (0, g, u).

It follows that

E ′(0, g, u) = log δ(g)r2(g)φ2(0, u)−W ′0(0, g, u)−

∑v nonsplit

∑a∈F (v)

W ′a,v(0, g, u)W

va (0, g, u).

Notation. For any non-split place v, denote the v-part by

E ′(0, g, u)(v) :=∑a∈F (v)

W ′a,v(0, g, u)W

va (0, g, u).

I ′(0, g)(v) :=∑

u∈µ2K\F×

θ(g, u)E ′(0, g, u)(v).

24

Then we have a decomposition

I ′(0, g) =−∑

v nonsplit

I ′(0, g)(v)−∑

u∈µ2K\F×

W ′0(0, g, u)θ(g, u)+

+ log δ(g)∑

u∈µ2K\F×

r2(g)φ2(0, u)θ(g, u).

We first take care of I ′(0, g)(v) for any fixed non-split v. Denote by B = B(v) the nearbyquaternion algebra. Then we have a splitting B = E + Ej. Let V = (B, q) be the corre-sponding quadratic space with the reduced norm q, and V = V1 + V2 be the correspondingorthogonal decomposition. We identify the quadratic spaces V2,w = V2,w unless w = v. Itfollows that for a ∈ F×, a ∈ F (v) if and only if a is represented by (V2, uq).

Proposition 3.2.1. For non-split v,

I ′(0, g)(v) = 2

∫Z(A)T (F )\T (A)

K (v)φ (g, (t, t))dt,

where the integral

∫employs the Haar measure of total volume one, and

K (v)φ (g, (t1, t2)) = K (v)

r(t1,t2)φ(g) =∑

u∈µ2K\F×

∑y∈V−V1

kr(t1,t2)φv(g, y, u)r(g, (t1, t2))φv(y, u).

Here for any y = y1 + y2 ∈ Vv − V1v,

kφv(g, y, u) =L(1, ηv)

vol(E1v)r(g)φ1,v(y1, u)W

′uq(y2),v(0, g, u).

Proof. By Proposition 2.6.2,

E ′(0, g, u)(v) =∑

y2∈E1\(V2−0)

W ′uq(y2),v(0, g, u)W

vuq(y2)(0, g, u)

=1

Lv(1, η)

∑y2∈E1\(V2−0)

W ′uq(y2),v(0, g, u)

∫E1(Av)

r(g)φv2(y2τ, u)dτ

=1

vol(E1v)L

v(1, η)

∑y2∈E1\(V2−0)

∫E1(A)

W ′uq(y2τ),v

(0, g, u)r(g)φv2(y2τ, u)dτ

=1

vol(E1v)L

v(1, η)

∫E1\E1(A)

∑y2∈V2−0

W ′uq(y2τ),v

(0, g, u)r(g)φv2(y2τ, u)dτ.

Therefore, we have the following expression for I ′(0, g)(v):

I ′(0, g)(v) =1

vol(E1v)L

v(1, η)

∑u∈µ2

K\F×

∑y1∈V1

r(g)φ1(y1, u)·

·∫E1\E1(A)

∑y2∈V2−0

W ′uq(y2τ),v

(0, g, u)r(g)φv2(y2τ, u)dτ

25

Move two sums inside integral to obtain:∫A×E×\A×E

∑u∈µ2

K\F×

∑y=y1+y2∈V

y2 6=0

r(g)φ1,v(y1, u)W′uq(y2t−1 t),v(0, g, u)r(g)φ

v(t−1yt, u)dt

By definition of kφv and K (v)φ , we have

I ′(0, g)(v) =1

Lv(1, η)

∫Z(A)T (F )\T (A)

∑u∈µ2

K\F×

∑x∈V−V1

kφv(g, t−1yt, u)r(g)φv(t−1yt, u)dt

=1

L(1, η)

∫Z(A)T (F )\T (A)

K (v)φ (g, (t, t))dt.

Since vol(Z(A)T (F )\T (A)) = 2L(1, η), we get the result. Here we have used the relationkφv(g, t

−1yt, u) = kr(t,t)φv(g, y, u) in the lemma below.

Lemma 3.2.2. The function kφv(g, y, u) behaves like Weil representation under the actionof P (Fv) and E×

v × E×v . Namely,

kφv(m(a)g, y, u) = |a|2kφv(g, ay, u), a ∈ F×v

kφv(n(b)g, y, u) = ψ(buq(y))kφv(g, y, u), b ∈ Fvkφv(d(c)g, y, u) = |c|−1kφv(g, y, c

−1u), c ∈ F×v

kr(t1,t2)φv(g, y, u) = kφv(g, t−11 yt2, q(t1t

−12 )u), (t1, t2) ∈ E×

v × E×v

Proof. These identities follow from the definition of Weil representation and some transfor-mation of integrals. We will only verify the first identity. By definition, we can compute thetransformation of m(a) on Whittaker function directly:

Wa0,v(s,m(a)g, u) =

∫Fv

δ(wn(b)m(a)g)sr2(wn(b)m(a)g)φ2,v(0, u)ψv(−a0b)db

=

∫Fv

δ(m(a−1)wn(ba−2)g)sr2(m(a−1)wn(ba−2)g)φ2,v(0, u)ψv(−a0b)db

= |a|−s∫Fv

δ(wn(ba−2)g)sr2(wn(ba−2)g)φ2,v(0, u)|a|−1ηv(a)ψv(−a0b)db

= |a|−s−1ηv(a)

∫Fv

δ(wn(b)g)sr2(wn(b)g)φ2,v(0, u)ψv(−a0a2b)|a|2db

= |a|−s+1ηv(a)Wa2a0,v(s, g, u).

It follows thatW ′a0,v

(0,m(a)g, u) = |a|ηv(a)W ′a2a0,v

(0, g, u).

This implies the result by combining

r(m(a)g)φ1,v(y1, u) = |a|ηv(a)φ1,v(ay1, u).

26

Now we rewrite the second and the third summations, which is less complicated than thefirst one.

Proposition 3.2.3.

I ′(0, g) =−∑

v nonsplit

I ′(0, g)(v) + 2 log δ(g)∑

u∈µ2K\F×

∑y∈V1

r(g)φ(y, u)

− c0∑

u∈µ2K\F×

∑y∈V1

r(g)φ(y, u)−∑v

∑u∈µ2

K\F×

∑y∈V1

cφv(g, y, u)r(gv)φv(y, u),

where

c0 =d

ds|s=0

(log

L(s, η)

L(s+ 1, η)

)is a constant, and

cφv(g, y, u) = r1(g)φ1,v(y, u)W

0,v′(0, g, u)

|Dv|12 |dv|

12

+ log δ(gv)r(g)φv(y, u).

Moreover, cφv = 0 identically for all archimedean places v and almost all non-archimedeanplaces v.

Proof. Take derivative on

W0(s, g, u) =L(s, η)

L(s+ 1, η)W

0 (s, g, u) =L(s, η)

L(s+ 1, η)

∏v

W 0,v(s, g, u).

By Proposition 2.6.2,

W ′0(0, g, u) =

d

ds|s=0

(L(s, η)

L(s+ 1, η)

)W

0 (0, g, u) +L(0, η)

L(1, η)

∑v

W 0,v

′(0, g, u)∏v′ 6=v

W 0,v′(0, g, u)

=d

ds|s=0

(log

L(s, η)

L(s+ 1, η)

)r(g)φ(0, u) +

∑v

W 0,v

′(0, g, u)

|Dv|12 |dv|

12

r(gv)φv(0, u).

It gives the desired equality. The map

δ(g)sr(g)φ2(0, u) 7→ W0(s, g, u, φ2), I(s, η) → I(−s, η)

is an intertwining operator. For archimedean places v and almost all non-archimedean placesv, one has the local component

W 0,v(s, g, u, φ2,v) = δ(g)−sr(g)φ2(0, u).

And thus we have cφv = 0. See [KRY].

27

3.3 Degenerate Schwartz functions at non-archimedean places

In this subsection we introduce a class of “degenerate” Schwartz functions at a non-archimedeanplace. It is generally very difficult to obtain an explicit formula of I ′(0, g)(v) for a ramifiedfinite prime v. When we choose a degenerate Schwartz function at v, the function I ′(0, g)(v)turns out to be easier to control as we will see in the next subsection.

Define

S 0(Bv × F×v ) =

φv ∈ S (Bv × F×

v ) : φv|Bvsing∪Ev×F×v ) = 0, if Ev/Fv is nonsplit;

φv ∈ S (Bv × F×v ) : φv|Bv

sing×F×v = 0, if Ev/Fv is split.

Here Bvsing = x ∈ Bv : q(x) = 0.

By compactness, for any φv ∈ S 0(Bv × F×v ), there exists a constant C > 0 such that

φv(x, F×v ) = 0 if |v(q(x))| > C. The following result will be useful when we extend our final

results to general φv.

Proposition 3.3.1. Let v be a non-archimedean place. Then for any nonzero GL2(Fv)-equivariant homomorphism S (Bv×F×

v ) → πv, the image of S 0(Bv×F×v ) in πv is nonzero.

We will prove a more general result. For simplicity, let F be a non-archimedean localfield and let (V, q) be a non-degenerate quadratic space over F of even dimension. Thenwe have the Weil representation of GL2(F ) on S (V × F×), the space of Bruhat-Schwartzfunctions on V × F×. Let α : S (V × F×)−→σ be a surjective morphism to an irreducibleand admissible representation of GL2(F ). We will prove the following result which obviouslyimplies Proposition 3.3.1.

Proposition 3.3.2. Let W be a proper subspace of V of even dimension. Assume that σis not one dimensional, and that in case W 6= 0, W is non-degenerate and its orthogonalcomplement W ′ is anisotropic. Then there is a function φ ∈ S (V ×F×) such that α(φ) 6= 0and that the support supp(φ) of φ contains only elements (x, u) such that q(x) 6= 0 and

W (x) := W + Fx

is non-degenerate of dimension dimW + 1.

Let us start with the following Proposition which allows us to modify any test functionto a function with support at points (x, u) ∈ V × F× with components x of nonzero normq(x) 6= 0.

Proposition 3.3.3. Let φ ∈ S (V × F×) be an element with nonzero image in σ. Then

there is a function φ ∈ S (V × F×) supported on

supp(φ) ⊂ supp(φ) ∩(Vq 6=0 × F×) .

The key to prove this proposition is the following lemma. It is well-known but we give aproof for readers’ convenience.

28

Lemma 3.3.4. Let π be an irreducible and admissible representation of GL2(F ) with dim >

1. Then the group N(F ) of matrixes n(t) :=

(1 t0 1

)does not have any non-zero invariant

on π.

Proof. If there exists such an invariant vector v 6= 0, then by the smoothness, v is also fixed bysome compact open subgroup. In particular, v is fixed by some element γ ∈ SL2(F )−B(F ).Then v is in fact SL2-invariant since SL2(F ) is generated by N and any one element γ inSL2(F )−B(F ). Then such π must be one dimensional. Contradiction!

Proof of Proposition 3.3.3. Applying the lemma above, we obtain elements t ∈ F such that

σ(n(t))α(φ)− α(φ) 6= 0.

The left hand side is equal to α(φ) with

φ = n(t)φ− φ.

By definition, we haveφ(x, u) = (ψ(tuq(x))− 1)φ(x).

Thus such a φ has support

supp(φ) ⊂ supp(φ) ∩(Vq 6=0 × F×) .

Proof of Proposition 3.3.2. Choose any φ such that α(φ) 6= 0. If W = 0, then we apply

Proposition 3.3.3 to obtain a function φ satisfying the proposition. Thus we assume thatW 6= 0. Since W ′ is anisotropic, we have an orthogonal decomposition V = W ⊕W ′, andan identification S (V ×F×) = S (W ×F×)⊗S (F×) S (W ′×F×). The action of GL2(F ) isgiven by actions on S (W × F×) and S (W ′ × F×) respectively. We may assume that φ isa pure tensor:

φ = f ⊗ f ′, f ∈ S (W × F×), f ′ ∈ S (W ′ × F×).

Since W ′ is anisotropic, S (W ′q 6=0 × F×) is a subspace in S (W ′ × F×) with quotient

S (F×). The quotient map is given by evaluation at (0, u). Thus

S (W ′ × F×) = S (W ′q 6=0 × F×) + wS (W ′

q 6=0 × F×)

as w acts as the Fourier transform up to a scale multiple. In this way, we may write

f ′ = f ′1 + wf ′2, f ′i ∈ S (W ′q 6=0 × F×).

Then we have decomposition

φ = φ1 + wφ2. φ1 := f ⊗ f ′1, φ2 = w−1f ⊗ f ′2.

One of α(φi) 6= 0. Thus we may replace φ by this φi to conclude that the support of φconsists of points x = (w,w′) with W (x) = W ⊕ Fw′ which is non-degenerate. ApplyingProposition 3.3.3 again, we may further assume that φ has support on Vq 6=0 × F×.

29

3.4 Non-archimedean components

Assume that v is a non-archimedean place non-split in E. Resume the notations in the lastsection. We now compute the local kernel function:

kφv(g, y, u) =L(1, ηv)

vol(E1v)r(g)φ1,v(y1, u)W

′uq(y2),v(0, g, u), y = y1 + y2 ∈ Vv − V1v.

Recall that vol(OFv) = |dv|12v and vol(OEv) = |Dv|

12 |dv|v under the self-dual measures.

Here dv ∈ OFv is the local different of Fv, and Dv ∈ OFv is the local discriminant of Ev/Fv.

Proposition 3.4.1. Assume a ∈ F (v) for a finite prime v.

(1) If v - 2 or E is unramified at v, then for g = 1,

W ′a,v(0, 1, u) = |dv|

12 logNv

v(adv)∑n=0

Nnv

∫Dn

φ2,v(x2, u)dux2,

and thus

kφv(1, y, u) =L(1, ηv)

vol(E1v)|dv|

12 logNv φ1,v(y1, u)

v(uq(y2)dv)∑n=0

Nnv

∫Dn

φ2,v(x2, u)dux2.

Here the domainDn = x2 ∈ V2,v : uq(x2) ∈ pnvd−1

v is an OEv-lattice of V2,v, and the measure dux2 is the self-dual measure of (V2,v, uq)

which gives vol(OEvx2) = |Dv|12 |dvuq(x2)| for any x2 ∈ V2,v.

(2) In all cases including v | 2, if φv ∈ S 0(Vv×F×v ), then kφv(1, y, u) extends to a Schwartz

function of (y, u) ∈ Vv × F×v .

Proof. It suffices to verify the formulae for Whittaker functions under the condition thatu = 1. The general case is obtained by replacing q by uq. We will drop the index u tosimplify the notation. Recall

Wa,v(s, 1) =

∫Fv

δ(wn(b))s∫

V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db.

By

δ(wn(b)) =

1 if b ∈ OFv ,|b|−1 otherwise,

we will split the integral over Fv into the sum of an integral over OFv and an integral overFv −OFv . Then

Wa,v(s, 1) =

∫OFv

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db

+

∫Fv−OFv

|b|−s∫

V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db

30

The second integral can be decomposed as

∞∑n=1

∫p−n

v −p−n+1v

N−nsv

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db

=∞∑n=1

∫p−n

v

N−nsv

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db

−∞∑n=1

∫p−(n−1)v

N−nsv

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db.

Combine with the first integral to obtain

Wa,v(s, 1) =∞∑n=0

∫p−n

v

N−nsv

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db

−∞∑n=0

∫p−n

v

N−(n+1)sv

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db

=(1−N−sv )

∞∑n=0

N−nsv

∫p−n

v

∫V2,v

φ2,v(x2)ψv(b(q(x2)− a))dx2db.

As for the last double integral, change the order of the integration. The integral on b isnonzero if and only if q(x2)− a ∈ pnvd−1

v . Here dv is the local different of F over Q, and alsothe conductor of ψv. Denote

Dn(a) = x2 ∈ V2,v : q(x2)− a ∈ pnvd−1v .

Then we have

Wa,v(s, 1) = (1−N−sv )

∞∑n=0

N−nsv vol(p−nv )

∫Dn(a)

φ2,v(x2)dx2

= |dv|12 (1−N−s

v )∞∑n=0

N−ns+nv

∫Dn(a)

φ2,v(x2)dx2.

We will see that Dn(a) is empty for n large enough by the condition that a ∈ F (v). Thenthe sum above have only finitely many terms, and thus Wa,v(0, 1) = 0 just by the vanishingof the factor 1−N−s

v . Hence,

W ′a,v(0, 1) = |dv|

12 logNv

∞∑n=0

Nnv

∫Dn(a)

φ2,v(x2)dx2.

We first show (1). We claim that v(q(x2) − a) = minv(a), v(q(x2)) for any x2 ∈ V2,v.Actually, if this is not true, we will have v(a−1q(x2)) = 0 and v(1 − a−1q(x2)) > 0. Then

31

a−1q(x2) ∈ 1 + pv ⊂ q(E×v ), and thus a is represented by (V2,v, q), which contradicts to the

assumption that a ∈ F (v). This simple fact implies that

Dn(a) = x2 ∈ V2,v : q(x2) ∈ pnvd−1v , a ∈ pnvd−1

v .

We see that Dn(a) is empty if v(a) < n− v(dv). Otherwise, Dn(a) = Dn. It proves (1).Now we consider (2). If v satisfies the assumption of (1), then by the results we know

that W ′a,v(0, 1) = 0 if v(a) is too small. On ther other hand, we have φ2,v(Dn, u) = 0 if n is

large enough by the vanishing property of the Schwartz function. It follows that W ′a,v(0, 1)

is constant when v(a) is large enough. By both facts, we can obtain the result.If v divides 2 and ramifies in E, then Dn(a) has no simple expression as above since we

don’t have v(q(x2)− a) = minv(a), v(q(x2)) any more. However, we have

v(q(x2)− a) ≤ minv(a), v(q(x2))+ v(8)

by a similar argument: if this were not true, then v(a−1q(x2)) = 0 and v(1−a−1q(x2)) > v(8).It follows that a−1q(x2) ∈ 1 + 8pv ⊂ q(E×

v ), and thus a is represented by (Ev, q), whichcontradicts the assumption that a ∈ F (v).

Therefore, Dn(a) is non-empty only if v(a) > −v(8), and

Dn(a) ⊂ x2 ∈ V2,v : 8q(x2) ∈ pnvd−1v .

It is easy to obtain (2) by a similar argument.

Corollary 3.4.2. Assume that v is a finite prime nonsplit in E such that:

• Bv is split at v;

• v - 2 if Ev/Fv is ramified;

• the local different dv is trivial (which makes the additive character ψv unramified);

• φv is the characteristic function of OBv ×O×Fv

.

Then kφv(1, y, u) 6= 0 only if (y, u) ∈ OBv × O×Fv

. Here Bv the quaternion over Fv that isnon-isomorphic to Bv. Under this condition, we have

kφv(1, y, u) =v(q(y2)) + 1

2logNv.

Proof. Note that φ1,v(y1, u) is a factor of kφv(1, y, u). To make it nonzero, we need to assume(y1, u) ∈ OEv ×O×

Fv. Then

kφv(1, y, u) =L(1, ηv)

vol(E1v)

logNv

v(q(y2))∑n=0

Nnv

∫Dn

φ2,v(x2, u)dqx2 =L(1, ηv)

vol(E1v)

logNv

v(q(y2))∑n=0

Nnv vol(Dn)

32

withDn = x2 ∈ V2,v : q(x2) ∈ pnv.

It is nonzero only if v(q(y2)) ≥ 0, which we now assume.Recall V2,v = Evjv. Since Bv is a matrix algebra, we can assume that q(jv) ∈ O×

Fv. Then

Dn = x2jv : x2 ∈ Ev, q(x2) ∈ pnv.

If E is unramified at v, we have Dn = p[n+1

2]

v OEv jv and vol(Dn) = N−2[n+1

2]

v . ThenNnv vol(Dn) = 1 for even n, and Nn

v vol(Dn) = N−1v for odd n. Note that v(q(y2)) is odd since

Bv is nonsplit. Hence,

kφv(1, y, u) = L(1, ηv) logNvv(q(y2)) + 1

2(1 +N−1

v ) =v(q(y2)) + 1

2logNv.

If E is ramified at v, then Dn = pn2v OEv jv and vol(Dn) = N−n

v |Dv|12 . By vol(E1

v) = 2|Dv|12 ,

we still get the result.

3.5 Archimedean Places

For an archimedean place v, the quaternion algebra Bv is isomorphic to the Hamiltonianquaternion. We will compute kφv(g, y, u) for standard φv of weight 2 introduced in Section3.1. The computation here is done by [KRY].

The result involves the exponential integral Ei defined by

Ei(z) =

∫ z

−∞

et

tdt, z ∈ C.

Another expression is

Ei(z) = γ + log(−z) +

∫ z

0

et − 1

tdt,

where γ is the Euler constant. It follows that it has a logarithmic singularity near 0. Thisfact is useful when we compare the result here with the archimedean local height, since weknow that Green’s functions have a logarithmic singularity.

Proposition 3.5.1.

kφv(g, y, u) =

−1

2Ei(4πuq(y2)y0) |y0|e2πiuq(y)(x0+iy0)e2iθ if uy0 > 0

0 if uy0 < 0

for any

g =

(z0

z0

)(y0 x0

1

)(cos θ sin θ− sin θ cos θ

)∈ GL2(Fv)

in the form of the Iwasawa decomposition.

33

Proof. It suffices to show the formula in the case g = 1. The general case is obtained byProposition 3.2.2 and the fact that r(kθ)φv = e2iθφv.

Now we show that

kφv(1, y, u) =

−1

2Ei(4πuq(y2))e

−2πuq(y) if u > 0;

0 if u < 0.

It amounts to show that, for any a ∈ F (v),

W ′a,v(0, 1, u) =

−πe−2πaEi(4πa) if u > 0;

0 if u < 0.

Assume u > 0 since the case u < 0 is trivial. Then a < 0 by the condition a ∈ F (v).We compute explicitly as follows. First of all, it is easy to check δ(wn(b)) = 1/(1 + b2). Itfollows that

Wa,v(s, 1, u) =

∫Fv

δ(wn(b))s∫V2v

φ2,v(x2, u)ψv(b(uq(x2)− a))dux2db

=

∫R

(1

1 + b2

) s2∫

Ce−2πuq(x2)+2πib(uq(x2)−a)dux2db

Take an isometry of quadratic space (V2v, uq) ' (C, | · |2) to obtain∫R

(1

1 + b2

) s2∫

Ce−2π|x2|2e2πib(|x2|2−a)dx2db

=

∫R

(1

1 + b2

) s2∫

Ce−2π(1−ib)|x2|2e−2πiabdx2db =

∫R

(1

1 + b2

) s2 1

1− ibe−2πiabdb

=

∫R(1 + ib)−

s2 (1− ib)−

s2−1e−2πiabdb.

By the computation in [KRY], page 19,

d

ds|s=0

∫R(1 + ib)−

s2 (1− ib)−

s2−1e−2πiabdb = −πe−2πaEi(4πa).

ThusW ′a,v(0, 1, u) = −πe−2πaEi(4πa).

3.6 Holomorphic projection

In this subsection we consider the general theory of holomorphic projection which we willapply to the form I ′(0, g, χ) in the next subsection. Denote by A (GL2(A), ω) the space of

34

automorphic forms of central character ω, by A0(GL2(A), ω) the subspace of cusp forms,

and by A (2)0 (GL2(A), ω) the subspace of holomorphic cusp forms of parallel weight two.

The usual Petersson inner product is just

(f1, f2) =

∫Z(A)GL2(F )\GL2(A)

f1(g)f2(g)dg, f1, f2 ∈ A (GL2(A), ω).

Denote by Pr′ : A (GL2(A), ω) → A (2)0 (GL2(A), ω) the orthogonal projection. Namely, for

any f ∈ A (GL2(A), ω), the image Pr′(f) is the unique form in A (2)0 (GL2(A), ω) such that

(Pr′(f), ϕ) = (f, ϕ), ∀ϕ ∈ A (2)0 (GL2(A), ω).

We simply call Pr′(f) the holomorphic projection of f . Apparently Pr′(f) = 0 if f is anEisenstein series.

For any automorphic form f for GL2(A) we define a Whittaker function

fψ,s(g) = (4π)degFW (2)(g∞)

∫Z(F∞)N(F∞)\GL2(F∞)

δ(g)sfψ(gfh)W (2)(h)dh.

Here W (2) is the standard holomorphic Whittaker function of weight two at infinity, and fψdenotes the Whittaker function of f . As s → 0, the limit of the integral is holomorphic ats = 0.

Proposition 3.6.1. Let f ∈ A (GL2(A), ω) be a form with asymptotic behavior

f

((a 00 1

)g

)= O(|a|1−ε)

as |a| → ∞ for some ε > 0. Then the holomorphic projection Pr′(f) has Whittaker function

Pr′(f)ψ(gfg∞) = lims→0

fψ,s(g).

Proof. For any Whittaker functionW of GL2(A) with decompositionW (g) = W (2)(g∞)Wf (gf )such that W (2)(g∞) is standard holomorphic of weight 2 and that Wf (gf ) compactly sup-ported modulo Z(Af )N(Af ), the Poincare series is defined as follows:

ϕW (g) := lims→0+

∑γ∈Z(F )N(F )\G(F )

W (γg)δ(γg)s,

where

δ(g) = |a∞/d∞|12 , g =

(a b0 d

)k, k ∈ U

where U is the standard maximal compact subgroup of GL2(A). Assume that W and f havethe same central character. Since f has asymptotic behavior as in the proposition, their

35

inner product can be computed as follows:

(f, ϕW ) =

∫Z(A)GL2(F )\GL2(A)

f(g)ϕW (g)dg

= lims−→0

∫Z(A)N(F )\GL2(A)

f(g)W (g)δ(g)sdg

= lims−→0

∫Z(A)N(A)\GL2(A)

fψ(g)W (g)δ(g)sdg. (3.6.1)

We may apply this formula to Pr′(f) which has the same inner product with ϕW as f .Write

Pr′(f)ψ(g) = W (2)(g∞)Pr′(f)ψ(gf ).

Then the above integral is a product of integrals over finite places and integrals at infiniteplaces: ∫

Z(R)N(R)\GL2(R)

|W (2)(g)|2dg =

∫ ∞

0

y2e−4πydy/y2 = (4π)−1.

In other words, we have

(f, ϕW ) = (4π)−g∫Z(Af )N(Af )\GL2(Af )

Pr′(f)ψ(gf )W (gf )dgf . (3.6.2)

As W can be any Whittaker function with compact support modulo Z(Af )N(Af ), the com-bination of above formulae give the proposition.

We introduce an operator Pr formally defined on the function space of N(F )\GL2(A).For any function f : N(F )\GL2(A) → C, denote as above

fψ,s(g) = (4π)degFW (2)(g∞)

∫Z(F∞)N(F∞)\GL2(F∞)

δ(g)sfψ(gfh)W (2)(h)dh

if it has meromorphic continuation around s = 0. Here fψ denotes the first Fourier coefficientof f . Denote

Pr(f)ψ(gfg∞) = lims→0fψ,s(g),

where lims→0 denotes the constant term of the Laurent expansion at s = 0. Finally, we write

Pr(f)(g) =∑a∈F×

Pr(f)ψ(d∗(a)g).

The the above result is just Pr(f) = Pr′(f). In general, Pr(f) is not automorphic whenf is automorphic but fails the growth condition of Proposition 3.6.1.

36

Now we want to consider the holomorphic projection of I ′(0, g, φ) for φ with standardinfinite component at infinity. First, let us compute the asymptotic behavior of I(s, g, χ)near a cusp for small s. This function is a finite linear combination of I(s, g, χ)’s:

I(s, g, χ) =

∫T (F )Z(F∞)\T (A)

I(s, g, r(t, 1)φ)χ(t)dt

with central character χ|A×F . Thus we can define and compute the holomorphic projectionusing Proposition 3.6.1.

Proposition 3.6.2. If χ does not factor through norm NE/F : A×E−→A×, then I(s, g, χ) has

an asymptotic O(|a|1−ε). Otherwise, let χ be induced from two different characters µi onF×\A×:

χ = µ1 NE/F = µ2 NE/F .

Then there is an Eisenstein series J(s, g, χ) for the characters(a ∗0 b

)7→ |a/b|1/2±sµ−1

i (ab)

such that I(s, g, χ)− J(s, g, χ) has asymptotic behavior O(|a|1−ε).

Proof. Recall that I(s, g, χ) can be rewritten as an integration of I(s, g):

I(s, g, χ) =

∫T (F )Z(F∞)\T (A)

I(s, g, r(t, 1)φ)χ(t)dt

and thatI(s, g) =

∑u∈µ2

K\F×θ(g, u)E(s, g, u).

It suffices to get an asymptotic behavior for the theta series and the Eisenstein series for left

action by diagonal elements

(a 00 b

). In their explicit Fourier expansions, the non-constant

term exponentially decays in as |a/b| → ∞. The constant terms have asymptotical behavior:

θ0

((a 00 b

)g, u

)= |a/b|1/2θ0(g, a

−1b−1u) = r(g)φ(0, a−1b−1u)|a/b|1/2

E0

(s,

(a 00 b

)g, u

)= |a/b|1/2+sf(g, a−1b−1u) + |a/b|1/2−sf(g, a−1b−1u)

where

f(g, u) = δ(g)sr2(g)φ2(0, u), f(g, u) = −∫N(A)

δ(wmg)sr2(wng)φ2(0, u).

37

It follows that I(s, g) has the following asymptotic behavior modulo a function with expo-nential decay,

I0

(s,

(a 00 b

)g

)= |a/b|1+s

∑µ2

K\F×θ0f(g, a−1b−1u) + |a/b|1−s

∑µ2

K\F×θ0f(g, a−1b−1u).

Similarly I(s, g, χ) has an asymptotic behavior modulo a function with exponential decay,

I0

(s,

(a 00 b

)g, χ

)= |a/b|1+sI+

0 (s, g, ab, χ) + |a/b|1−sI−0 (s, g, ab, χ)

where

I+0 (s, g, α, χ) =

∫T (F )\T (A)

∑u∈µ2

K\F×θ0f(g, q(t)uα)χ(t)dt

I−0 (s, g, α, χ) =

∫T (F )\T (A)

∑u∈µ2

K\F×θ0f(g, q(t)uα)χ(t)dt.

It is clear that these two function vanish if χ does not factor through the the norm NE/F .If χ factors through the norm, then χ is induced from exactly two characters µi on F×\A×

F :χ(t) = µ1(NE/F t) = µ2(NE/F t). Then the above integral can be written as integrals overF×\A×:

I+0 (s, g, α, χ) = µ1(α)−1I+

0 (s, g, µ1) + µ2(α)−1I+0 (s, g, µ2)

I−0 (s, g, α, χ) = µ1(α)−1I−0 (s, g, µ1) + µ2(α)−1I−0 (s, g, µ2)

where

I+(s, g, µi) =1

2

∫F×\A×

∑u∈µ2

K\F×θ0f(g, zu)µi(z)dz

I−(s, g, µi) =1

2

∫F×\A×

∑u∈µ2

K\F×θ0f(g, zu)µi(z)dz.

In this way, we see that I0(s, g, χ) is a linear combination of principal series induced fromcharacters on the upper triangular groups:(

a ∗0 b

)7→ |a/b|1/2±s · µ−1

i (ab).

Let J(s, g, χ) be the Eisenstein series formed by I0(s, g, χ):

J(s, g, χ) =∑

γ∈P (F )\GL2(F )

I0(s, γg, χ).

The constant term of J(s, g, χ) has two terms I0(s, g, χ) and I0(s, g, χ) which is a linearcombination of series in the characters:(

a ∗0 b

)7→ |a/b|−1/2±s · µ−1

i (ab).

In this way, the proof of the proposition is complete.

38

We now are ready to apply Proposition 3.6.1 to the form I ′(0, g, χ) − J ′(0, g, χ). Moreprecisely,

Pr′(I ′(0, g, χ)) = Pr′(I ′(0, g, χ)− J ′(0, g, χ)) = Pr(I ′(0, g, χ))−Pr(J ′(0, g, χ))

where in the right-hand side the operator Pr is defined after Proposition 3.6.1.In the following we want to study Pr(J ′(0, g, χ)), which is computed from the limit

of J ′ψ,s′(0, g, χ) as s′ → 0. Writing J(s, g, χ) as a linear combination of Eisenstein seriesJ±(s, g, µi) defined by functions in I±(s, g, µi), we need only treat the Fourier coefficientsJ±ψ,s′(s, g, µi). From the proof of Proposition 3.6.2, we see that their Whittaker functionshave a decomposition

J±ψ (s, g, µi) = W (2)(±s, g∞)J±ψ (s, gf , µi)

where W (2)(s, g∞) at each archimedean place has weight 2, trivial central character and value

y1+s at g =

(y 00 1

). In this way

J±ψ,s′(s, g, µi) = W (2)(g∞)J±ψ (s, gf , µi)Γ(1 + s+ s′)

(4π)s+s′.

Take the limit as s′ → 0, and the derivative at s = 0 to obtain

Pr(J ′(0, g, χ))ψ := lims′−→0

J ′ψ,s′(0, g, χ) = W (2)(g∞)d

ds|s=0J

±ψ (s, gf , µi)

Γ(1 + s)

(4π)s.

It is clear that it is a sum of Fourier coefficients of Eisenstein series and their derivations(§4.4.4 in [Zh1]). In summary, we have shown:

Proposition 3.6.3. The difference Pr′(I ′(0, g, χ))−Pr(I ′(0, g, χ)) is a finite sum of Eisen-stein series and their derivations.

3.7 Holomorphic kernel function

We now apply holomorphic projection formula to I ′(0, g) which has the following expressionby Proposition 3.2.3,

I ′(0, g) =−∑

v nonsplit

I ′(0, g)(v) + 2 log δ(g)∑

u∈µ2K\F×,y∈E

r(g)φ(y, u)

− c0∑

u∈µ2K\F×,y∈E

r(g)φ(y, u)−∑v

∑u∈µ2

K\F×,y∈E

cφv(g, y, u)r(gv)φv(y, u).

For a function on f(g) on GL2(A) which is invariant under N(F ) on the left, we let f ∗(g)denote the sum of non-constant coefficients in the Fourier coefficients under left translationby N(F )\N(A).

39

Proposition 3.7.1. Modulo a finite sum of Eisenstein series and their derivation,

Pr(I ′(0, g)) =−∑v|∞

I ′(0, g)(v)−∑

v-∞ nonsplit

I ′(0, g)(v)

+ 2∑

(y,u)∈µK\E××F×(log δf (gf ) +

1

2log |uq(y)|f )r(g)φ(y, u)

−∑v

∑u∈µ2

K\F×,y∈E×cφv(g, y, u)r(g

v)φv(y, u)− c1∑

u∈µ2K\F×,y∈E×

r(g)φ(y, u).

Here for an archimedean v,

I ′(0, g)(v) = 2

∫Z(A)T (F )\T (A)

K (v)φ (g, (t, t))dt,

K (v)φ (g, (t1, t2)) =

∑a∈F×

lims→0

∑y∈µK\(B(v)×+−E×)

r(g, (t1, t2))φ(y)a kv,s(y),

kv,s(y) =Γ(s+ 1)

2(4π)s

∫ ∞

1

1

t(1− ξv(y)t)s+1dt,

c1 =d

ds|s=0

(log

L(s, η)

L(s+ 1, η)

)+

1

2(γ + log 4π)[F : Q].

Other terms are the same as in Proposition 3.2.3.

Proof. We will consider the image under Pr of each term on the right-hand side. We will seethat just a few terms are changed since the operator Pr only changes the non-holomorphicterms.

First let’s look at I ′(0, g)(v), which is always the main term. By Proposition 3.2.1,

I ′(0, g)(v) = 2

∫Z(A)T (F )\T (A)

K (v)φ (g, (t, t))dt.

with

K (v)φ (g, (t1, t2)) = K (v)

r(t1,t2)φ(g) =∑

u∈µ2K\F×

∑y∈B(v)−E

kr(t1,t2)φv(g, y, u)r(g, (t1, t2))φv(y, u).

Note that the integral above is just a finite sum.We have a simple rule

r(n(b)g, (t1, t2))φv(y, u) = ψ(uq(y)b) r(g, (t1, t2))φ

v(y, u),

and its analoguekr(t1,t2)φv(n(b)g, y, u) = ψ(uq(y)b)kr(t1,t2)φv(g, y, u)

40

showed in Proposition 3.2.2. By these rules it is easy to see that the first Fourier coefficientis given by

K (v)φ (g, (t1, t2))ψ =

∑(y,u)∈µK\((B(v)−E)×F×)1

kr(t1,t2)φv(gv, yv, uv)r(g, (t1, t2))φv(y, u).

If v is non-archimedean, all the infinite components are already holomorphic of weighttwo. So the operator Pr doesn’t change K (v)

φ (g, (t1, t2))ψ at all. Thus

Pr(K (v)φ (g, (t1, t2))) =

∑a∈F×

K (v)φ (d∗(a)g, (t1, t2))ψ

=∑a∈F×

∑(y,u)∈µK\((B(v)−E)×F×)1

kr(t1,t2)φv(d∗(a)gv, yv, uv)r(d

∗(a)g, (t1, t2))φv(y, u)

=∑a∈F×

∑(y,u)∈µK\((B(v)−E)×F×)1

kr(t1,t2)φv(gv, ayv, a−1uv)r(g, (t1, t2))φ

v(ay, a−1u)

=∑a∈F×

∑(y,u)∈µK\((B(v)−E)×F×)a

kr(t1,t2)φv(gv, yv, uv)r(g, (t1, t2))φv(y, u)

=K (v)φ (g, (t1, t2)).

Here the third equality follows from Proposition 3.2.2 that kr(t1,t2)φv transforms accord-ing to the Weil representation under upper triangular matrices. We conclude that Prdoesn’t change K (v)

φ (g, (t1, t2)) if v is non-archimedean, and thus we have Pr(I ′(0, g)(v)) =I ′(0, g)(v).

Now we look at the case that v is archimedean. The only difference is that we need toreplace kφv(g, y, u) by some kφv ,s(g, y, u), and then take a “quasi-limit” lim. It suffices toconsider the case that uq(y) = 1, it is given by

kφv ,s(g, y, u) = 4πW (2)(gv)

∫Fv,+

ys0e−2πy0kφv(d

∗(y0), y, u)dy0

y0

.

Then kφv ,s(g, y, u) 6= 0 only if u > 0, since kφv(d∗(y0), y, u) 6= 0 only if u > 0.

Assume that u > 0, which is equivalent to q(y) > 0 since we assume uq(y) = 1 for themoment. By Proposition 3.5.1,∫

Fv,+

ys0e−2πy0kφv(d

∗(y0), y, u)dy0

y0

= −1

2

∫Fv,+

ys0e−2πy0 Ei(4πuq(y2)y0) y0e

−2πy0dy0

y0

=− 1

2

∫ ∞

0

ys+10 e−4πy0 Ei(−4παy0)

dy0

y0

(α = −uq(y2) = −q(y2)

q(y)> 0)

=1

2

∫ ∞

0

ys+10 e−4πy0

∫ ∞

1

t−1e−4παy0tdtdy0

y0

=1

2

∫ ∞

1

t−1

∫ ∞

0

ys+10 e−4π(1+αt)y0

dy0

y0

dt

=Γ(s+ 1)

2(4π)s+1

∫ ∞

1

1

t(1 + αt)s+1dt.

41

Hence,

kφv ,s(g, y, u) = W (2)(gv)Γ(s+ 1)

2(4π)s

∫ ∞

1

1

t(1− q(y2)q(y)

t)s+1dt = W (2)(gv)kv,s(y).

This matches the result in the proposition. Since kv,s(y) is invariant under the multiplicationaction of F× on y, it is easy to get

Pr(K (v)φ (g, (t1, t2))) = K (v)

φ (g, (t1, t2)), Pr(I ′(0, g)(v)) = I ′(0, g)(v).

Similarly, we have

Pr

∑u∈µ2

K\F×,y∈E

r(g)φ(y, u)

=∑

u∈µ2K\F×,y∈E×

r(g)φ(y, u),

Pr

∑u∈µ2

K\F×,y∈E

cφv(g, y, u)r(gv)φv(y, u)

=∑

u∈µ2K\F×,y∈E×

cφv(g, y, u)r(gv)φv(y, u).

It remains to take care of

log δ(g)∑

u∈µ2K\F×,y∈E

r(g)φ(y, u).

Its first Fourier coefficient is just

log δ(g)∑

(y,u)∈µK\(E××F×)1

r(g)φ(y, u)

=∑

(y,u)∈µK\(E××F×)1

log δ(gf )r(g)φ(y, u) +∑

(y,u)∈µK\(E××F×)1

r(g)φf (y, u) · log δ(g∞)W (2)(g∞).

Then Pr doesn’t change the first sum of the right-hand side, but changes log δ(g∞)W (2)(g∞)in the second sum to some multiple c2 W

(2)(g∞) = c2 r(g)φ∞(y, u), where c2 is some constant.Hence we see that

Pr

log δ(g)∑

u∈µ2K\F×,y∈E

r(g)φ(y, u)

∗

=∑a∈F×

∑(y,u)∈µK\(E××F×)1

log δ(d∗(a)gf )r(d∗(a)g)φ(y, u)

+ c2∑a∈F×

∑(y,u)∈µK\(E××F×)1

r(d∗(a)g)φ(y, u)

=∑

(y,u)∈µK\E××F×(log δ(gf ) + log |uq(y)|

12 )r(g)φ(y, u) + c2

∑(y,u)∈µK\E××F×

r(g)φ(y, u).

42

As for the constant, we have

c2[F : Q]

= 4π lims→0

∫Fv,+

yse−2πy log |y|12 e−2πy dy

y= 2π

∫ ∞

0

e−4πy log ydy = −1

2(γ + log 4π).

43

4 Shimura curves, Hecke operators, CM-points

In this section, we will review the theory of Shimura curves, CM-points, generating seriesof Hecke operators, and a preliminary decomposition of height pairing of CM-points usingArakelov theory.

In §4.1, we will describe a projective system of Shimura curves XU for a totally defi-nite incoherent quaternion algebra B over a totally real field F indexed by compact opensubgroups U of B×

f . In §4.2, we will define a generating function Zφ(g) with coefficients inPic(XU ×XU) for each φ ∈ S (V×A×), and will show that it is automorphic in g ∈ GL2(A).This series is an extension of Kudla’s generating series for Shimura varieties of orthogonaltype [Ku1]. In this case, the modularity of its cohomology class is proved by Kudla-Millson[KM1, KM2, KM3]. The modularity as Chow cycles are proved in our previous work [YZZ].In §4.3, for E an imaginary quadratic extension of F , we define a set of CM-points by E bijec-tive to E×\B×

f . For two CM-points represented by βi ∈ B×f , we define a function Zφ(g, β1, β2)

for g ∈ GL2(A) by means of height pairing. It can be viewed as a function in g ∈ GL2(A) and(β1, β2) ∈ GO(A) compatible with Weil representation on S (V×A×)GO(F∞). This functionis automorphic for g and left invariant under the diagonal action of A×

E on (β1, β2). In §4.4and §4.5, using Arakelov theory, we will decompose Zφ(g, t1, t2) into a finite sum of the form

〈Zφ(g)ξ, η〉 where ξ is the arithmetic Hodge class, a theta series, and an infinite sum of localpairings:

〈Z∗φt1, t2〉 =

∑v

jv(Z∗φ(g)t1, t2) logNv + i(Z∗

φt1, t2).

In §4.6, we show that 〈Zφ(g)ξ, η〉 is equal to a finite sum of Eisenstein series and theirderivations in the sense of §4.4.4 in our previous paper [Zh1]. The computation of the localpairing is the main content in the next section.

4.1 Shimura curves

In the following, we will review the theory of Shimura curves following our previous paper[Zh2]. Let F be a totally real number field. Let B be a quaternion algebra over A withodd ramification set Σ including all archimedean places. Then for each open subset U of B×

f

we have a Shimura curve XU . The curve is not connected; its set of connected componentsis can be parameterized by F×

+ \A×f /q(U). For each archimedean place τ of F , the set of

complex points at τ is given as follows:

XU,τ (C) = B(τ)×+\H × B×f /U ∪ Cusps

where B(τ)×+ is the group of totally positive elements in a quaternion algebra B(τ) over Fwith ramification set Σ \ τ with an action on H ± by some fixed isomorphisms

B(τ)⊗τ R = M2(R)

B(τ)⊗ Af ' Bf ,

44

and where Cusp is the set of cusps which is non-empty only when F = Q and Bf = M2(Af ).For two open compact subgroups U1 ⊂ U2 of B×

f , one has a canonical morphism πU1,U2 :XU1−→XU2 which satisfies the composition property. Thus we have a projective system Xof curves XU . For any x ∈ Bf , we also have isomorphism Tx : XU−→Xx−1Ux which inducesan automorphism on the projective system X and compatible with multiplication on B×

f :Txy = Tx ·Ty. All of these morphisms on XU ’s has obvious description on complex manifoldsXU,τ (C). The induced actions are the obvious one on the sets of connected components aftertaking norm of Ui and x.

An important tool to study Shimura curves is to use modular interpretation. For a fixedarchimedean place τ , the space H ± parameterizes Hodge structures on V0 := B(τ) whichhas type (−1, 0) + (0,−1) (resp (0, 0)) on V0 ⊗τ R (resp. V0 ⊗σ R for other archimedeanplaces σ 6= τ). The non-cuspidal part of XU,τ (C) parameterizes Hodge structure and levelstructures on a B(τ)-module V of rank 1.

Due to the appearance of type (0, 0), the curveXU does not parameterize abelian varietiesunless F = Q. To get a modular interpretation, we use an auxiliary imaginary quadraticextension K over F with complex embeddings σK : K−→C for each archimedean placesσ of F other than τ . These σK ’s induce a Hodge structure on K which has type (0, 0) onK⊗τ R and type (−1, 0)+(0,−1) on K⊗σR for all σ 6= τ . Now the tensor product of Hodgestructures on VK := V ⊗F K is of type (−1, 0)+(0,−1). In this way, XU parameterizes someabelian varieties with homology group H1 isomorphic to VK . The construction makes XU acurve over the reflex field for σK ’s:

K] = Q

(∑σ 6=τ

σ(x), x ∈ K

).

See our paper [Zh1] for a construction following Carayol in the case K = F (√d) with d ∈ Q

where σK(√d) is chosen independent of σ.

The curve XU has a canonical integral model XU over OF . At each finite place v ofF , the base change XU,v over OFv will parameterizes some p-divisible group with some levelstructure. Locally at a geometric point, XU,v is the universal deformation of the representedp-divisible group.

Hodge class and Neron–Tate height pairing

The curve XU has a Hodge class LU ∈ Pic(XU) ⊗ Q which is compatible with pull-backmorphism and such that LU ' ωXU

when U is sufficiently small.We also define a class ξU ∈ Pic(XU)⊗Q which has degree 1 on each connected component

and is proportional to LU . One can use ξ to define an projection Div(XU)−→Div0(XU) bysending D to D−deg(D)ξU where degD is the degree function on the connected componentsXi of XU : degD(Xi) := deg(D|Xi

). The class degD · ξU has restriction deg(D|Xi)ξXi

onXi.

The Jacobian variety JU of XU is defined to be the variety over F to represent linebundles on XU (or base change ) of degree 0 on every connected components Xi. Over an

45

extension of F , JU is the product of Jacobian varieties Ji of the connected components Xi.The Neron–Tate height 〈·, ·〉 on JU(F ) is defined using the Poincare divisor on Ji × Ji.

4.2 Hecke correspondences and generating series

We want to define some correspondences on XU , i.e., some divisor classes on XU ×XU . Theprojective system of surfaces XU ×XU has an action by B×

f × B×f . Let K denote the open

compact subgroup K = U × U .

Hecke operators

For any double coset UxU of U\B×f /U , we have a Hecke correspondence

Z(x)U ∈ Z1(XU ×XU)

defined as the image of the morphism

(πU∩xUx−1,U , πU∩x−1Ux,U Tx) : ZU∩xUx−1−→X2U .

In terms of complex points at a place of F as above, the Hecke correspondence Z(x)Utakes

(z, g)−→∑i

(z, gxi)

for points on XU,τ (C) represented by (z, g) ∈ H ± × Bf where xi are representatives ofUxU/U .

Hodge class

On MK := XU ×XU , one has a Hodge bundle LK ∈ Pic(MK)⊗Q defined as

LK =1

2(p∗1LU + p∗2LU).

Generating Function

Let V denote the orthogonal space B with quadratic form q. Let S (V× A×)GO(V∞) denotethe space S (V∞, F

×∞)GO(F∞) ⊗ S (VAf

× A×f ) which is isomorphic to the maximal quotient

of S (V× A×) via integration on GO(F∞).For any (x, u) ∈ V×A×, let us define a cycle Z(x, u)K on XU ×XU as follows. This cycle

is non-vanishing only if q(x)u ∈ F× or x = 0. If q(x)u ∈ F×, then we define Z(x, u)K to bethe Hecke operator UxU defined in the last subsection. If x = 0, then we define Z(x, u)K tobe the push-forward of the Hodge class on the subvariety Mα which is union of connectedcomponents Xα ×Xβ with α, β ∈ F×\A×/F×

∞,+ν(U) such that u = αβ mod F×+ q(U)F×

∞,+.

46

For φ ∈ S (V×A×)GO(F∞) which is invariant under K ·GO(F∞), we can form a generatingseries

Zφ(g) =∑

(x,u)∈(K·GO(F∞))\V×A×r(g)φ(x, u)Z(x, u)K .

It is easy to see that this definition is compatible with pull-back maps in Chow groups in theprojection MK1−→MK2 with Ki = Ui × Ui and U1 ⊂ U2. Thus it defines an element in thedirect limit Ch1(M)Q := limK Ch1(MK). For any h ∈ (B×

f )2, let ρ(h) denote the pull-back

morphism on Ch1(M) by right translation of h. Then it is easy to verify

Zr(h)φ = ρ(h)Zφ.

Proposition 4.2.1. The series Zφ is absolutely convergent and defines an automorphic formfor GL2(A).

We will reduce the proposition to the modularity proved in [YZZ].Let M1

K be the subvariety of the union of connected components Xα × Xβ with α, β ∈F×

+ \A×f /ν(U) such that α ∈ F×

+ ν(U). Then M1K is a Shimura variety of orthogonal type

associated to the subgroup GSpin(V) of B× × B× of pairs (g1, g2) with same norm. For anyelement φ1 ∈ S (V)O(F∞) then one can define the generating series

Zφ1(g) =∑

x∈K1·O(F∞)\V

r(g)φ1(x)Z(x)K1

where K1 = K∩GSpin(Vf ) and Z(x)K1 is non-zero only if q(x) ∈ F× or x = 0. If q(x) ∈ F×,then Z(x)K1 is defined as Heck operator UxU ; if x = 0, then Z(x)K1 is defined as c1(L ∨

K1).In [YZZ], we have shown that Zφ1(g) is absolutely convergent and defines an automorphicform on SL2(A).

For any h ∈ GO(V ), let ih denote the composition of the embedding M1−→M and thetranslation Th on M . Then we can show that MK is covered by i(M1

Kh) for h runs through a

set of elements in GO(V ) = B×f ×B×

f whose similitudes represents the cosets F×+ \A×

f /ν(K),

and Kh = GSpin(V ) ∩ hKh−1. Thus it is clear that

Zφ =∑h

ih∗i∗hZφ.

Let us compute i∗hZφ. By definition,

i∗hZφ(g) =∑

(x,u)∈K·GO(F∞)\V×A×r(g)φ(x, u)i∗Z(x, u)K

If x = 0, then i∗hZ(x, u) 6= 0 only if u ∈ q(h)F×+ ν(K) in which case it is given as Z(0)Kh ;

if q(x)u ∈ F×, then i∗Z(x, u)K 6= 0 only if ν(h)q(x) ∈ F×+ ν(K) in which case it is given by

Z(y)Kh , where y ∈ Kx with norm in F×+ . In this way the sum above becomes

i∗hZr(g)φ =∑

u∈µ′K\F×+

∑x∈KhO(F∞)\V

r(g, h)φ(x, u)Z(y)Kh =∑

u∈µ′K\F×+

Zr(g,h)φ(·,u).

47

where µ′K = F×+ ∩ ν(K). We have thus shown that Zφ is a finite sum of generating series for

M1. It follows that Zφ(g) is invariant under left translation by elements in SL2(F ) and isabsolutely convergent. By definition, it is clear that Zφ is also invariant under left translation

by elements of the form d(a) =

(1 00 a

). Thus Zφ is invariant under left translation by

GL2(F ).

4.3 CM-points and height series

Let E be an imaginary quadratic extension of F with an embedding E(Af ) ⊂ Bf . ThenXU(F ) has a set of CM-points defined over Eab, the maximal abelian extension of E. Theset CMU is stable under action of Gal(Eab/F ) and Hecke operators. More precisely, we havea projective system of bijections

CMU ' E×\B×f /U (4.3.1)

which is compatible with action of Hecke operators and such that the Galois action is givenas follows: the action of Gal(Eab/E) acts by right multiplication of elements of E×(Af ) viathe reciprocity law in class field theory:

E×\E×(Af )−→Gal(Eab/E).

Such a bijection is unique up to left multiplication by elements in E×(Af ).If τ is a real place of F , then the set CU can be described as a subset of

XU,τ (C) = B(τ)×\H ± × B×f /U ∪ cusps

as the set of points

CMU = B(τ)×\B(τ)×z0 × B×f /U ∪ cusps ' E×\B×

f /U

where E is embedded into B(τ) compatible with isomorphism B(τ)Af= Bf and z0 ∈ H is

unique fixed point of E×.For any φ ∈ S (BA × A×)U×UGO(F∞), we define the Neron–Tate height pairing

Zφ(g, β1, β2) ∈ 〈Zφ(g)([β1]U − deg([β1]U)ξ), [β2]U − deg([β2]U)ξ〉U , β1, β2 ∈ B×f .

Using the projection formula of height pairing, we see that this definition does not dependon the choice of U . Also this definition does not depend on the choice of the bijection 4.3.1since the height pairing is invariant under Galois action. It follows that

Zφ(g, tβ1, tβ2) = Zφ(g, β1, β2).

In this way, we may view this as a function on GO(V) through projection

GO(V) = ∆(A)\B× × B×−→GO(Vf ) = ∆(Af )\B×f × B×

f

48

Using the projection formula and the formula Zr(h)φ = ρ(h)Zφ(g) where ρ(h) is the pull-back morphism of right translation by h ∈ GO(V), one can show that the resulting functionZφ(g, h) is equivariant under Weil representation:

Zr(g1,h1)φ(g2, h2) = Zφ(g2g1, h2h1), gi ∈ GL2(A), hi ∈ GO(V).

This function is automorphic for the first variable and invariant for the second variable underleft diagonal multiplication by A×

E.Let T denote E× as an algebraic group over F . Any β ∈ B×

f gives a CM-point in CMU

which is denoted by [β]U or just β if U is clear. We are particularly interested in the casethat β ∈ T (Af ), i.e. CM-points that are in the image in XU of the zero-dimensional Shimuravariety

CU = T (F )\T (Af )/UT

associated to T = E×. Here UT = U ∩ T (Af ). Notice that the set CU does not depend onthe choice of bijection 4.3.1. For a finite character χ of T (F )\T (A), we can define

Zφ(g, χ) =

∫(T (F )\T (Af ))2

Zφ(g, t1, t2)χ(t1t−12 )dt1dt2.

4.4 Arithmetic intersection pairing

The Hodge bundle LU can be extended into a metrized line bundle on XU . More precisely,at an archimedean place, the metric can defined by using Hodge structure or equivalentlynormalized such that its pull-back on Ω1

H takes the form:

‖f(z)dz‖ = 4π · Imz · |f(z)|.

At a finite place, we may take an extension LU,v by using the fact that LU is twice ofthe cotangent bundle of the divisible groups. The resulting bundle on XU with metrics at

archimedean places is denoted by LU . Also we have an arithmetic class ξU induced from

LU .The Hodge index theorem on the Jacobian Ji of geometric connected components Xi of

X implies a Hodge index theorem on XU : for any two divisors D1 and D2 of degree 0 onevery connected component. Then their Neron–Tate height pairing is given by the followingformula:

〈D1, D2〉 = −D1 · D2

where Di are some “flat” arithmetic divisors extending Di. Here, D is flat in the sense thatits curvature at archimedean places and its intersection with vertical divisors are all 0.

For the computation in the next section, we will consider the group Pic(XU)ξ of arithmetic

line bundles L such that c1(L ) − degD · c1(ξ) is flat, we call such that line bundle ξ-

admissible. Moreover for any divisor D, there is a unique extension D which is ξ-admissiblesuch that the vertical component in every fiber of XU has zero intersection with ξ. Sucha construction is totally local and compatible with pull-back morphism for the projections

49

XU1−→XU2 and Hecke operators. In particular, we have a well defined Green’s functiongv(x, y) on Xv(Fv) such that for any two distinct points x, y ∈ XU(K),

x · y = − 1

[K : F ]

∑v

∑σ:K−→Fv

gv(σ(x), σ(y)).

On XU , one has a decomposition

gv(x, y) = iv(x, y) + jv(x, y)

where iv(x, y) is the intersection of the Zariski closures of x and y and jv(x, y) is a locallyconstant function on XU,v(Fv)×XU,v(Fv).

4.5 Decomposition of the height pairing

Our goal in the geometric side is to compute

Zφ(g, t1, t2) = 〈Zφ(g)(t1 − deg(t1)ξ), t2 − deg(t2)ξ〉, t1, t2 ∈ CU .

Once ξ is extended to an arithmetic class ξ, we get the decomposition:

〈Zφ(g)t1, t2〉 − 〈Zφ(g)t1, deg(t2)ξ〉 − 〈Zφ(g) deg(t1)ξ, t2〉+ 〈Zφ(g) deg(t1)ξ, deg(t2)ξ〉.

We will take care of the last three terms, but let us first consider 〈Zφ(g)t1, t2〉 which isthe main term. We start with some general discussion on decomposition of intersection ofCM-points.

Decomposition of the height pairing

Now we want to write the height pairing into a sum of local terms in the case that β1 6= β2 ∈B×f , and get a similar decomposition even in the case β1 = β2 with an“error term” i0(β, β).

We can always write〈β1, β2〉 = i(β1, β2) + j(β1, β2)

as in [Zh2].We first assume that β1 6= β2. Then we have decompositions

i(β1, β2) =∑v∈ΣF

iv(β1, β2) logNv, j(β1, β2) =∑v∈ΣF

jv(β1, β2) logNv

with

iv(β1, β2) =1

#ΣEv

∑w∈ΣEv

iw(β1, β2), jv(β1, β2) =1

#ΣEv

∑w∈ΣEv

jw(β1, β2).

Here ΣEv denotes the set of places of E lying over v.

50

To compute these local pairings, we will use:

iw(β1, β2) =1

[L : E]

∑σ:L→Ew

iw(βσ1 , βσ2 ) =

∫T (F )\T (Af )

iw(tβ1, tβ2)dt,

jw(β1, β2) =1

[L : E]

∑σ:L→Ew

jw(βσ1 , βσ2 ) =

∫T (F )\T (Af )

jw(tβ1, tβ2)dt.

Here L is any Galois extension of E that contains the residue fields of β1, β2, and the integralon T (F )\T (Af ) takes the Haar measure with total volume one.

The pairing jw is zero identically if w is archimedean or XU,E has good reduction at w.Furthermore, all the decompositions above for j still makes sense even if β1 = β2.

Now we consider the self-intersection i(β, β) for any β ∈ CMU . In next section, we willhave an extended definition of iw(β, β) for each place w of E. Then formally, we still defineiw(β, β) and iv(β, β) by the above averaging formulas. The difference

i0(β, β) := i(β, β)−∑v∈ΣF

iv(β, β) logNv

is not zero any more. We will see that it is essentially the Faltings height of β by an arithmeticadjunction formula. But for the Gross-Zagier formula, it is enough to use the property thati0(β, β) depends only on the Galois orbit of β.

Decomposition of the kernel Function

Rewrite Zφ(g) according to a = q(x)u ∈ F to obtain

Zφ(g) = Zφ,0(g) +∑a∈F×

∑x∈K\B×f

r(g)φ(x)aZ(x)U

whereZφ,0(g) =

∑u∈ν(K)F×∞\A×

r(g)φ(0, u)[L −1u ],

andφ(x)a = φ(x, q(x)−1a).

The Hecke operator Z(x)U corresponds to the double coset UxU , i.e.,

Z(x)U t1 =∑j

t1αj, if UxU =∐j

αjU.

It follows that

Zφ(g)[t1] = Zφ,0(g)[t1] +∑a∈F×

∑x∈K\B×f

r(g)φ(x)aZ(x)U [t1]

= Zφ,0(g)[t1] +∑a∈F×

∑x∈B×f /U

r(g)φ(x)a[t1x].

51

Since [t1x] = [t2] in CMU if and only if x ∈ t−11 t2T (F )U , the part of self-intersection

comes from

Rφ(g)[t1] =∑a∈F×

∑x∈t−1

1 t2T (F )U/U

r(g)φ(x)a[t1x]

=∑a∈F×

∑y∈E×/(E×∩U)

r(g)φ(t−11 yt2)a[t2]

=1

[E× ∩ U : µU ]

∑a∈F×

∑y∈E×/µU

r(g, (t1, t2))φ(y)a[t2].

The index [E× ∩ U : µU ] = 1 when U is small enough. In the following, for simplicity wealways assume that E× ∩ U = µU .

Now we consider the decomposition of the global height 〈Zφ(g)t1, t2〉. We first write

〈Zφ(g)t1, t2〉 = 〈Zφ,0(g)t1, t2〉+ 〈Z∗φ(g)t1, t2〉

= 〈Zφ,0(g)t1, t2〉+∑v

jv(Z∗φ(g)t1, t2) logNv + i(Z∗

φ(g)t1, t2),

where

Z∗φ(g) = Zφ(g)− Zφ,0(g) =

∑a∈F×

∑x∈K\B×f

r(g)φ(x)aZ(x)U ,

Z∗φ(g)[t1] =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)a[t1x].

Then

i(Z∗φ(g)t1, t2)

=∑a∈F×

∑x∈B×f /U

r(g)φ(x)a i(t1x, t2)

=∑a∈F×

∑y∈E×/µU

r(g, (t1, t2))φ(y)ai0(t2, t2) +∑a∈F×

∑x∈B×f /U

r(g)φ(x)a∑v

iv(t1x, t2) logNv

=∑v

iv(Z∗φ(g)t1, t2) logNv + i0(1, 1)

∑a∈F×

∑y∈E×/µU

r(g, (t1, t2))φ(y)a.

Here we denote

iv(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)a∑v

iv(t1x, t2).

And we have i0(t2, t2) = i0(1, 1) since [1] and [t] are Galois conjugate CM points.

52

4.6 Hecke action on arithmetic Hodge classes

In this subsection, we want to discuss the action of Hecke operator Zφ on the arithmeticclasses. More precisely, we want to study the pairing

〈Zφξf , η〉

where f is a function on the set of connected components and η is an arithmetic divisorclass. Recall using the usual uniformization at an archimedean place, the set of connectedcomponents can be indexed by a group

F×+ \A×

f /ν(U).

The function on this set is generated by characters ω. Thus it suffices to treat the case wheref = ω is a character.

The action of Zφ on the geometric cycle ξω is given by the action on ω, the function onthe set of connected components. By a result of Kudla [Ku1], the ratio of Z∗

φ(ξω) over ξω isgiven by an Eisenstein series E(g, φ, ω) with central character ω−2:

Z∗φ(g)ξω = E∗(g, φ, ω)ξω.

The Whittaker function of this Eisenstein series is given by

Eψ(g, φ, ω) =∑

x∈KGO(F∞)\V×r(g)φ(x, q(x)−1)Z(x, ω)K

where

Z(x, ω)K =

∫Z(x)K

ω(h)dh.

It follows that the action on the arithmetic cycle ξ will have formula

Z∗φ(g)ξω = E∗(g, φ, ω)ξω + V

where V is a vertical cycle. In the following we want to discuss in more details the Whittakercoefficient function of Zφ(g).

Assume that φ = ⊗φv. By definition, the Whittaker function of Zφ(g) is given by

Zφ,ψ(g) =∏v

Zφv ,ψv(gv)

whereZv(gv) =

∑x∈Kv\V×v

r(gv)φv(xv, q(xv)−1)Z(xv)Kv .

Here Kv = GO(Fv) for v | ∞. Notice that for almost all v, Zv(gv) is a trivial operator.Indeed, for almost all v, gv ∈ GL2(Ov), φv is the characteristic function of OBv × O×

v where

53

OBv is a maximal order of Bv, and Kv = O×Bv× O×

Bv. Thus the sum Zv(gv) has only one

term Z(1)Kv which is an identity operator. For each v, Zv(gv) is an operator which is etaleoutside v.

Let S be a finite subset of F including all places in Σ and places where Uv is not maximal.Then the Shimura curve XU has good reduction outside S. Using the same argument as inour previous paper [Zh1] we can show the following identity:

Zv(gv)ξω = Zv(g, ω)ξω +D(gv)

where D(gv) is a constant multiple of the divisor ωXv supported on the special fiber of Xover v whose connected components is also indexed by F×

+ \A×f . The composition of two

operator at places u 6= v outside S is given as

Zu(gu)Zv(gv)ξω − Zu(gu, ωu)Zv(gv, ωv)ξω = Zu(gu, ωu)D(gv) + Zv(gv, ωv)D(gu).

We have shown thatg−→D(g) := ZS(g)ξω − ZS(g, ωS)ξω

behaves like derivative for g ∈ GL2(AS): for any two coprime g1, g2 ∈ GL2(AS) in the sensethat for all v /∈ S, either g1 or g2 is in GL2(Ov) one has

D(g1g2) = ZS(g1, ωS)D(g2) + ZS(g2, ω

S)D(g1).

Thus D(g) is a derivation for ZS(g, ωS) as defined in §4.4.4 in our previous paper [Zh1].Let us apply this to the function

g−→〈Zφ(g)ξ, η〉.

This function may not be automorphic but invariant under P (F ). Thus it has a Whittakerfunction

〈Zφ,ψ(g)ξ, η〉.

Let us fix gS ∈ GL2(AS). Then we obtain a function on gS ∈ GL2(AS):

gS−→〈Zφ,ψ(gSgS)ωξ, η〉 = 〈ZS(gS)ξ, Zt

S(gS)η〉

where ZtS(gS) is the transpose of ZS(gS). We have just shown that this is a derivation

function for the Whittaker function of the Eisenstein series of E(g, φ, ω). Applying this tothe decomposition of Zφ(g, t1, t2) we obtain the following:

Proposition 4.6.1. For fixed φ, there is an S such that for any t1, t2, the Whittaker functionof the difference

Zφ(g, t1, t2)− 〈Zφ(g)t1, t2〉

is a finite linear combination of the Whittaker functions of the Eisenstein series E(g, φi, ωi)and their derivations where φi are pure tensors and ωi are finite idele class characters.

54

5 Local heights of CM points

In this section, we compute the local heights of CM-points

〈Z∗φt1, t2〉 = i(Z∗

φt1, t2) + j(Z∗φt1, t2)

and compare it with the local analytic kernel function K (v)φ appeared in the decomposition

of I ′(0, g) representing series L′(1/2, π, χ). We will follow the work of Gross–Zagier and itsextension in our previous paper [Zh2].

In §5.1, we study the case of archimedean place. We obtain an explicit formula whichagrees with the analytic kernel. In §5.2, we study the supersingular case, namely the finiteplace not in Σ and inert in E. In the unramified case, the formula is also explicit and agreewith corresponding analytic kernels. In the ramified case, under the condition that φv issupported in certain non-degenerate locus of Bv × F×

v , we can show that the i-part of thelocal intersection is given by certain pseudo-theta series defined in the next section. Thesame result holds at a superspecial place v in §5.3 (namely, a finite place in Σ). In §5.4, wetreat ordinary case, namely finite places split in E. Again the formula in the unramified caseis explicit and agrees with analytic kernels. The i-part of the ramified case will contributepseudo-theta series. In §5.5, we will show that the j-part of local intersection contributepseudo-theta series for φv with non-generate supports.

5.1 Archimedean places

In this subsection we want to describe local heights of CM points at any archimedean placev. Fix an identification B(Af ) = Bf . We will use the uniformization

XU,v(C) = B×+\H ×B×(Af )/U

where B = B(v). We follow the treatment of Gross-Zagier [GZ]. See also [Zh2].

Basic formula

For any two points z1, z2 ∈ H , the hyperbolic cosine of the hyperbolic distance betweenthem is given by

d(z1, z2) = 1 +|z1 − z2|2

2Im(z1)Im(z2).

It is invariant under the action of GL2(R). For any s ∈ C with Re(s) > 0, denote

ms(z1, z2) = Qs(d(z1, z2)),

where

Qs(t) =

∫ ∞

0

(t+

√t2 − 1 coshu

)−1−sdu

55

is the Legendre function of the second kind. Note that

Q0(1 + 2λ) =1

2log(1 +

1

λ), λ > 0.

We see that m0(z1, z2) has the right logarithmic singularity.For any two distinct points of

XU,v(C) = B×+\H ×B×(Af )/U

represented by (z1, β1), (z2, β2) ∈ H ×B×(Af ), we denote

gs((z1, β1), (z2, β2)) =∑

γ∈µU\B×+

ms(z1, γz2) 1U(β−11 γβ2).

It is easy to see that the sum is independent of the choice of the representatives (z1, β1), (z2, β2),and hence defines a pairing on XU,v(C). Then the local height is given by

iv((z1, β1), (z2, β2)) = lims→0 gs((z1, β1), (z2, β2)).

Here lims→0 denotes the constant term at s = 0 of gs((z1, β1), (z2, β2)), which converges forRe(s) > 0 and has meromorphic continuation to s = 0 with a simple pole.

The definition above uses adelic language, but it is not hard to convert it to the classicallanguage. We first observe that gs((z1, β1), (z2, β2)) 6= 0 only if there is a γ0 ∈ B×

+ such thatβ1 ∈ γ0β2U , which just means that (z1, β1), (z2, β2) are in the same connected component.Assuming this, then (z2, β2) = (z′2, β1) where z′2 = γ0z2. we have

gs((z1, β1), (z2, β2)) = gs((z1, β1), (z′2, β1)) =

∑γ∈µU\B×+

ms(z1, γz′2) 1U(β−1

1 γβ1) =∑

γ∈µU\Γ

ms(z1, γz′2).

Here we denote Γ = B×+ ∩ β1Uβ

−11 . The connected component of these two points is exactly

B×+\H ×B×

+β1U/U ≈ Γ\H , (z, bβ1U) 7→ b−1z.

The stabilizer of H in Γ is exactly Γ ∩ F× = µU . Now we see that the formula is the sameas those in [GZ] and [Zh2].

Next we consider the special case of CM points. For any γ ∈ B×v,+ − E×

v , we have

1 +|z0 − γz0|2

2Im(z0)Im(γz0)= 1− 2ξ(γ).

Thus it is convenient to denote

ms(γ) = Qs(1− 2ξ(γ)), γ ∈ B×v − E×

v .

56

For any two distinct CM points β1, β2 ∈ CMU , we obtain

gs(β1, β2) =∑

γ∈µU\B×+

ms(γ) 1U(β−11 γβ2),

andiv(β1, β2) = lims→0gs(β1, β2).

Note that ms(γ) is not defined for γ ∈ E×. The summation above makes sense becauseβ1 6= β2 implies that 1U(β−1

1 γβ2) = 0 for all γ ∈ E×. But this is not true for self-intersections.In general, for any β1, β2 ∈ CMU , we introduce formally

gs(β1, β2) =∑

γ∈µU\(B×+−E×)

ms(γ) 1U(β−11 γβ2),

andiv(β1, β2) = lims→0gs(β1, β2).

Then this definition is the same as the before if β1 6= β2. In the case that β1 = β2, it willoccur from the arithmetic adjunction formula.

Kernel function

In this section we compute the local height

iv(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)aiv(t1x, t2).

for any archimedean place v of F .

The goal is to show that it is equal to K (v)φ (g, (t1, t2)) obtained in Proposition 3.7.1.

Proposition 5.1.1. For any t1, t2 ∈ CU ,

iv(Z∗φ(g)t1, t2) = M (v)

φ (g, (t1, t2)) =∑a∈F×

lims→0

∑y∈µU\(B×+−E×)

r(g, (t1, t2))φ(y)ams(y).

Therefore, M (v)φ (g, (t1, t2)) = K (v)

φ (g, (t1, t2)).

Proof. By the above formula,

iv(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)a lims→0

∑γ∈µU\(B×+−E×)

ms(γ) 1U(x−1t−11 γt2)

=∑a∈F×

lims→0

∑γ∈µU\(B×+−E×)

r(g)φ(t−11 γt2)ams(γ)

=∑a∈F×

lims→0

∑γ∈µU\(B×+−E×)

r(g, (t1, t2))φ(γ)ams(γ).

Here the second equality is obtained by replacing x by t−11 γt2.

57

We want to compare the above result with the holomorphic projection

K (v)φ (g, (t1, t2)) =

∑a∈F×

lims→0

∑y∈µU\(B×+−E×)

r(g, (t1, t2))φ(y)a kv,s(y)

computed in Proposition 3.7.1.It amounts to compare

ms(y) = Qs(1− 2ξ(y))

with

kv,s(y) =Γ(s+ 1)

2(4π)s

∫ ∞

1

1

t(1− ξv(y)t)s+1dt.

By the result of Gross-Zagier,∫ ∞

1

1

t(1− ξt)s+1dt = 2Qs(1− 2ξ) +O(|ξ|−s−2), ξ → −∞,

and the error term vanishes at s = 0. We conclude that

K (v)φ (g, (t1, t2)) = iv(Z

∗φ(g)t1, t2).

5.2 Supersingular case

Let v be a finite prime of F non-split in E but split in B. We consider the local height atthe unique place of E lying over v. We will use the local multiplicity functions treated inZhang [Zh2]. For more details, we refer to that paper.

Multiplicity function

Assume that U is decomposable at v:

U = Uv · U v.

Let B = B(v) be the nearby quaternion algebra over F . Make an identification B(Av) = Bv.Then the set of supersingular points on XK over v is parameterized by

SSU = B×\(F×v / det(Uv))× (Bv×

f /U v).

Then we have a natural isomorphism

v : CMU = E×\B×f /U → B×\(B× ×E× B×

v /Uv)× Bv×f /U v

sending β to (1, βv, βv). The reduction map CMU → SSU is given by taking norm on the

first factor:q : B× ×E× B×

v → F×v , (b, β) 7→ q(b)q(β).

58

The intersection pairing is given by a multiplicity function m on

Hv := B×v ×E×v

B×v /Uv.

More precisely, the intersection of two points (b1, β1), (b2, β2) ∈ Hv is given by

gv((b1, β1), (b2, β2)) = m(b−11 b2, β

−11 β2).

The multiplicity function m is defined everywhere in Hv except at the image of (1, 1). Itsatisfies the property

m(b, β) = m(b−1, β−1).

Lemma 5.2.1. For any two distinct CM-points β1 ∈ CMU and t2 ∈ CU , their local heightis given by

iv(β1, t2) =∑

γ∈µU\B×m(γt2v, β

−11v )1Uv((βv1)

−1γtv2).

Proof. Like the archimedean case, we compute the height by pulling back to Hv × Bv×f .

The height is the sum over γ ∈ µU\B× of the intersection of (1, β1v, βv1) with γ(1, t2v, t

v2) =

(γ, t2v, γtv2) = (γt2v, 1, γt

v2) on Hv × Bv×

f .

Analogous to the archimedean case, the summation is well-defined for all β1 6= t2. Butif β1 = t2, then the summation is not defined at γ ∈ E× such that β−1

1 γt2 ∈ U . Hence, weextend the definition as

iv(β1, t2)

=∑

γ∈µU\(B×−E×∩β1Ut−12 )

m(γt2v, β−11v )1Uv((βv1)

−1γtv2)

=∑

γ∈µU\(B×−E×)

m(γt2v, β−11v )1Uv((βv1)

−1γtv2) +∑

γ∈µU\(E×−β1Ut−12 )

m(γt2v, β−11v )1Uv((βv1)

−1γtv2)

=∑

γ∈µU\(B×−E×)

m(γt2v, β−11v )1Uv((βv1)

−1γtv2) +∑

γ∈µU\(E×−β1Uvt−12 )

m(γt2v, β−11v )1Uv((βv1)

−1γtv2).

The definition is the equal to the previous one if β1 6= t2.

The kernel function

Now we compute

iv(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)aiv(t1x, t2).

59

By the above formula,

iv(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)a∑

γ∈µU\(B×−E×)

m(γt2, x−1t−1

1 )1Uv(x−1t−11 γt2)

+∑a∈F×

∑x∈B×f /U

r(g)φ(x)a∑

γ∈µU\(E×−t1xUvt−12 )

m(γt2, x−1t−1

1 )1Uv(x−1t−11 γt2).

In the first triple sum, we replace xv by t−11 γt2 and thus get∑

a∈F×

∑γ∈µU\(B×−E×)

r(g)φv(t−11 γt2)a

∑xv∈B×v /Uv

r(g)φv(xv)am(t−11 γt2, x

−1)

=∑

u∈µ2U\F×

∑γ∈B−E

r(g, (t1, t2))φv(γ, u)

∑xv∈B×v /Uv

r(g)φv(xv, uq(γ)/q(xv))m(t−11 γt2, x

−1).

In the second triple sum, we still replace xv by t−11 γt2. The condition γ /∈ t1xUvt

−12 is

equivalent to xv /∈ t−11 γt2Uv. We get∑

a∈F×

∑γ∈µU\E×

r(g)φv(t−11 γt2)a

∑xv∈(B×v −t−1

1 γt2Uv)/Uv

r(g)φv(xv)am(t−11 γt2, x

−1)

=∑

u∈µ2U\F×

∑γ∈E×

r(g, (t1, t2))φv(γ, u)

∑xv∈(B×v −t−1

1 γt2Uv)/Uv

r(g)φv(xv, uq(γ)/q(xv))m(t−11 γt2, x

−1).

For convenience, we introduce:

Notation.

mφv(y, u) =

∫B×vm(y, x−1)φv(x, uq(y)/q(x))dx

=

∫B×vm(y−1, x)φv(x, uq(y)/q(x))dx, (y, u) ∈ (Bv − Ev)× F×

v

nφv(y, u) =

∫B×v −yUv

m(y, x−1)φv(x, uq(y)/q(x))dx

=

∫B×v −Uv

m(1, x)φv(yx, u/q(x))dx, (y, u) ∈ E×v × F×

v

By this notation, we obtain:

Proposition 5.2.2.

iv(Z∗φ(g)t1, t2) = M (v)

φ (g, (t1, t2)) + N (v)φ (g, (t1, t2))

where

M (v)φ (g, (t1, t2)) =

∑u∈µ2

U\F×

∑y∈B−E

r(g, (t1, t2))φv(y, u) mr(g,(t1,t2))φv(y, u),

N (v)φ (g, (t1, t2)) =

∑u∈µ2

U\F×

∑y∈E×

r(g, (t1, t2))φv(y, u) r(t1, t2)nr(g)φv(y, u).

60

Here we use the convention

r(t1, t2)nr(g)φv(y, u) = nr(g)φv(t−11 yt2, q(t1t

−12 )u).

Note that in the above series, we write the dependence on (t1, t2) in different manners for mφv

and nφv . This is because mφv(y, u) translates well under the action of P (Fv)×(E×v ×E×

v ), butnφv(y, u) only translates well under the action of P (Fv). We should compare the followingresult with Lemma 3.2.2 for kφv(g, y, u).

Lemma 5.2.3. (1) The function mr(g,(t1,t2))φv(y, u) behaves like Weil representation underthe action of P (Fv)× (E×

v × E×v ) on (y, u). Namely,

mr(g,(t1,t2))φv(y, u) = r(g, (t1, t2))mφv(y, u), (g, (t1, t2)) ∈ P (Fv)× (E×v × E×

v ).

More precisely, it means that

mr(m(a))φv(y, u) = |a|2mφv(ay, u), a ∈ F×v

mr(n(b))φv(y, u) = ψ(buq(y))mφv(y, u), b ∈ Fvmr(d(c))φv(y, u) = |c|−1mφv(y, c

−1u), c ∈ F×v

mr(t1,t2)φv(g, y, u) = mφv(t−11 yt2, q(t1t

−12 )u), (t1, t2) ∈ E×

v × E×v

(2) The function nr(g)φv(y, u) behaves like Weil representation under the action of P (Fv)on (y, u), i.e.,

nr(m(a))φv(y, u) = |a|2nφv(ay, u), a ∈ F×v

nr(n(b))φv(y, u) = ψ(buq(y))nφv(y, u), b ∈ Fvnr(d(c))φv(y, u) = |c|−1nφv(y, c

−1u), c ∈ F×v .

Proof. They follow from basic properties of the multiplicity function m(x, y). We only verifythe first identity.

mr(m(a))φv(y, u) =

∫GL2(Fv)

m(y−1, x)r(gv)φv(ax, uq(y)/q(x))|a|2dx

= |a|2∫

GL2(Fv)

m(y−1, a−1x)r(gv)φv(x, uq(y)/q(a−1x))dx

= |a|2∫

GL2(Fv)

m((ay)−1, x)r(gv)φv(x, uq(ay)/q(x))dx

= |a|2mφv(ay, u).

61

Unramified Case

Fixing an isomorphism Bv = M2(Fv). In this subsection we compute mφv(y, u) and nφv(y, u)in the following unramified case:

1. φv is the characteristic function of M2(OFv)×O×Fv

;

2. Uv is the maximal compact subgroup GL2(OFv).

By [Zh2], there is a decomposition

GL2(Fv) =∞∐c=0

E×v hcGL2(OFv), hc =

(1 00 $c

)(5.2.1)

We may assume that β = hc. For any b ∈ Bv we set

ξ(b) =q(b)

q(b)=q(b2)

q(b)

for the orthogonal decomposition b = b1 + b2. The following result is Lemma 5.5.2 in [Zh2].There is a small mistake in the original statement. Here is the corrected one.

Lemma 5.2.4. The multiplicity function m(b, β) 6= 0 only if q(b)q(β) ∈ O×Fv

. In this case,assume that β ∈ E×

v hcGL2(OFv). Then

m(b, β) =

12(ordvξ(b) + 1) if c = 0;

N1−cv (Nv + 1)−1 if c > 0, Ev/Fv is unramified;

12N−cv if c > 0, Ev/Fv is ramified.

Proposition 5.2.5. (1) The function mφv(y, u) 6= 0 only if (y, u) ∈ OBv × O×Fv

. In thiscase,

mφv(y, u) =1

2(ordvq(y2) + 1).

(2) The function nφv(y, u) 6= 0 only if (y, u) ∈ OEv ×O×Fv

. In this case,

nφv(y, u) =1

2ordvq(y).

Proof. We will use Lemma 5.2.4. Recall that

mφv(y, u) =∑

x∈GL2(Fv)/Uv

m(y−1, x)φv(x, uq(y)/q(x)).

Note thatm(y−1, x) 6= 0 only if ord(q(x)/q(y)) = 0. Under this condition, φv(x, uq(y)/q(x)) 6=0 if and only if u ∈ O×

Fvand x ∈M2(OFv). It follows that mφv(y, u) 6= 0 only if u ∈ O×

Fvand

n = ord(q(y)) ≥ 0. Assuming these two conditions, we have

mφv(y, u) =∑

x∈M2(OFv )n/Uv

m(y−1, x),

62

where M2(OFv)n denotes the set of integral matrices whose determinants have valuation n.Now we use decomposition (2.3), we obtain

mφv(y, u) =∞∑c=0

m(y−1, hc)vol(E×v hcGL2(OFv) ∩M2(OFv)n).

We first consider the case that Ev/Fv is unramified. The set in the right hand side isnon-empty only if n− c is even and non-negative. In this case it is given by

$(n−c)/2O×EvhcUv.

The volume of this set is 1 if c = 0 and N c−1v (Nv + 1) if c > 0 by the computation of [Zh2,

p. 101]. It follows that, for c > 0 with 2 | (n− c),

m(y−1, hc)vol(E×v hcGL2(OFv) ∩M2(OFv)n) = 1.

If n is even,

mφv(y, u) =1

2(ordvξ(y) + 1) +

n

2=

1

2(ordvq(y2) + 1).

If n is odd, mφv(y, u) =1

2(n+ 1). It is easy to see that ordvq(y1) is even and ordvq(y2) is

odd. Then n = ordvq(y2), since n = ordvq(y) = minordvq(y1), ordvq(y2) is odd. We stillget mφv(y, u) = 1

2(ordvq(y2) + 1) in this case.

Now assume that Ev/Fv is ramified. Then the condition that 2 | (n− c) is unnecessary,and vol(E×

v hcGL2(OFv) ∩M2(OFv)n) = N cv . Thus

mφv(y, u) =1

2(ordvξ(y) + 1) + n · 1

2=

1

2(ordvq(y2) + 1).

As for nφv , still write n = ordvq(y), and we have

nφv(y, u) =∞∑c=1

m(y−1, hc)vol(E×v hcGL2(OFv) ∩M2(OFv)n).

If Ev/Fv is unramified, n is always even, and we have mφv(y, u) =n

2. The ramified case is

similar.

We immediately see that in the unramified case, mφv matches the analytic kernel kφv

computed in Proposition 3.4.2. As for nφv(y, u), its counterpart coming from PrI ′(0, g) is

exactly log δ(gv) +1

2log |uq(y)|v in Proposition 3.7.1. The good news is that the extra term

log δ(gv) cancels the action of g. In summary, we have the following result:

Proposition 5.2.6. (1) Let v be unramified as above. Then

nr(g)φv(y, u) logNv = −(log δ(gv) +1

2log |uq(y)|v) r(g)φv(y, u).

63

(2) Let v be unramified as above with further conditions:

• v - 2 if Ev/Fv is ramified;

• the local different dv is trivial.

Thenkr(t1,t2)φv(g, y, u) = mr(g,(t1,t2))φv(y, u) logNv,

and thusK (v)φ (g, (t1, t2)) = M (v)

φ (g, (t1, t2)) logNv.

Proof. We first consider (2). The case (g, t1, t2) = (1, 1, 1) follows from the above resultand Corollary 3.4.2. It is also true for g ∈ GL2(OFv) since it is easy to see that such gwill not change the local kernel functions for standard φv. For the general case, apply theaction of P (Fv) and E×

v ×E×v . The equality follows from Proposition 3.2.2 and Lemma 5.2.3.

Similarly, by Lemma 5.2.3 we can show (1).

General case

Now we consider general Uv. By Proposition 5.2.5 for the unramified case, we know thatmφv may have logarithmic singularity around the boundary Ev × F×

v , and nφv may havelogarithmic singularity around the boundary 0 × F×

v . These singularities are caused byself-intersections in the computation of local multiplicity. However, we will see that there isno singularity if φv ∈ S 0(Bv × F×

v ).

Proposition 5.2.7. Assume that φv ∈ S 0(Bv × F×v ) is invariant under the right action of

Uv. Then:

(1) The function mφv(y, u) can be extended to a Schwartz function for (y, u) ∈ Bv × F×v .

(2) The function nφv(y, u) can be extended to a Schwartz function for (y, u) ∈ Ev × F×v .

Proof. We only consider mφv . The proof for nφv is similar.By the choice of φv, there is a constant C > 0 such that φv(x, u) 6= 0 only if −C <

v(q(x)) < C and −C < v(u) < C. Recall that

mφv(y, u) =

∫B×vm(y−1, x)φv(x, uq(y)/q(x))dx.

In order that m(y−1, x) 6= 0, we have to make q(y)/q(x) ∈ q(Uv) and thus v(q(y)) =v(q(x)). It follows that φv(x, uq(y)/q(x)) 6= 0 only if −C < v(u) < C. The same is true formφv(y, u) by looking at the integral above. Then it is easy to see that mφv(y, u) is Schwartzfor u ∈ F×

v .On the other hand, mφv(y, u) 6= 0 only if −C < v(q(y)) < C, since φv(x, uq(y)/q(x)) 6= 0

only if −C < v(q(x)) < C. Extend mφv to Bv × F× by taking zero outside B×v × F×. We

only need to show that it is locally constant in B×v × F×.

64

We have φv(EvUv, F×v ) = 0 by the assumption that φv(Ev, F

×v ) = 0 and φv is invariant

under Uv. Thus

mφv(y, u) =

∫B×vm(y−1, x)(1− 1E×v Uv

(x))φv(x, uq(y)/q(x))dx.

It is locally constant in B×v × F×

v , since m(y−1, x)(1 − 1E×v Uv(x)) is locally constant as a

function on B×v × B×

v . This completes the proof.

5.3 Superspecial case

Let v be a finite prime of F non-split in both B and E. In this section we consider the localheight

iv(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)aiv(t1x, t2).

Since most of the computations are similar to the supersingular case, sometimes we will onlylist the results.

Denote by B = B(v) the nearby quaternion algebra. We fix identifications Bv 'M2(Fv)and B(Av

f ) ' Bvf . The intersection pairing is given by a multiplicity function m on

Hv := B×v ×E×v

B×v /Uv.

More precisely, the intersection of two points (b1, β1), (b2, β2) ∈ Hv is given by

gv((b1, β1), (b2, β2)) = m(b−11 b2, β

−11 β2).

The multiplicity function m is defined everywhere on Hv except at the image of (1, 1). Itsatisfies the property

m(b, β) = m(b−1, β−1).

For any two distinct CM-points β1 ∈ CMU and t2 ∈ CU , their local height is given by

iv(β1, t2) =∑

γ∈µU\B×m(γt2v, β

−11v )1Uv((βv1)

−1γtv2).

If β1 = t2, then the summation is not defined at γ ∈ E× such that β−11 γt2 ∈ U . So we

extend the definition as

iv(β1, t2)

=∑

γ∈µU\(B×−E×∩β1Ut−12 )

m(γt2v, β−11v )1Uv((βv1)

−1γtv2)

=∑

γ∈µU\(B×−E×)

m(γt2v, β−11v )1Uv((βv1)

−1γtv2) +∑

γ∈µU\(E×−β1Uvt−12 )

m(γt2v, β−11v )1Uv((βv1)

−1γtv2).

65

Analogous to Proposition 5.2.2, we have the following result.

Proposition 5.3.1.

iv(Z∗φ(g)t1, t2) = M (v)

φ (g, (t1, t2)) + N (v)φ (g, (t1, t2))

where

M (v)φ (g, (t1, t2)) =

∑u∈µ2

U\F×

∑y∈B−E

r(g, (t1, t2))φv(y, u) mr(g,(t1,t2))φv(y, u),

N (v)φ (g, (t1, t2)) =

∑u∈µ2

U\F×

∑y∈E×

r(g, (t1, t2))φv(y, u) r(t1, t2)nr(g)φv(y, u).

Here we use the same notations:

mφv(y, u) =

∫B×vm(y−1, x)φv(x, uq(y)/q(x))dx, (y, u) ∈ (Bv − Ev)× F×

v

nφv(y, u) =

∫B×v −yUv

m(y−1, x)φv(x, uq(y)/q(x))dx, (y, u) ∈ E×v × F×

v

Lemma 5.2.3 is still true. It says that the action of P (Fv)×(E×v ×E×

v ) onmr(g,(t1,t2))φv(y, u)and the action of P (Fv) on nr(g)φv(y, u) behave like Weil representation. The main resultbelow is parallel to Proposition 5.2.7.

Proposition 5.3.2. Assume φv ∈ S 0(Bv × F×v ) is invariant under the right action of Uv.

Then:

(1) The function mφv(y, u) can be extended to a Schwartz function for (y, u) ∈ Bv × F×v .

(2) The function nφv(y, u) can be extended to a Schwartz function for (y, u) ∈ Ev × F×v .

Proof. The proof is very similar to Proposition 5.2.7. We only sketch one for mφv . The prooffor nφv is similar.

By the argument of Proposition 5.2.7, there is a constant C > 0 such that mφv(y, u) 6= 0only if −C < v(q(y)) < C and −C < v(u) < C. Extend mφv to Bv × F× by taking zerooutside B×

v ×F×. The same method shows that it is locally constant on B×v ×F×. But this

is not enough to conclude that it is compactly supported in y, since Bv is the matrix algebrathis time. However, there is a compact subgroup Dv of B×

v such that m(y, x−1) 6= 0 only ify ∈ E×

v Dv. By this we can have our conclusion.

66

5.4 Ordinary case

Assume that v is a finite prime of F such that Ev is split. Then Bv is split because ofthe embedding Ev → Bv. Let v1 and v2 be the two primes of E lying over v. Fix an

identification Bv∼= M2(Fv) under which Ev =

(Fv

Fv

). Assume that v1 corresponds to

the ideal

(Fv

0

)and v2 corresponds to

(0

Fv

)of Ev.

We will make use of results of [Zh2]. The reduction map of CM-points to ordinary pointsat v1 is given by

E×\B×f /U −→ E×\(N(Fv)\GL2(Fv))× Bv×

f /U.

The intersection multiplicity is a function on N(Fv)Uv/Uv. By [Zh2, Lemma 6.3.2], thelocal pairing of β1, β2 ∈ GL2(Fv)/Uv is given by

gv1(β1, β2) =

∫GL2(Fv)

∫GL2(Fv)

1β1Uv(hy)1β2Uv(y)dydh =

∫GL2(Fv)

1Uv(β−11 hβ2)dh.

Here the measure dy is the usual Haar measure on GL2(Fv), and the measure dh is definedas follows: ∫

GL2(Fv)

f(h)dh :=

∫F×v

f(n(b))d×b =

∫F×v

f

(1 b

1

)d×b.

A nice property is that dh is invariant under the substitution h 7→ tht−1 for any t ∈ E×v .

And thus ∫GL2(Fv)

f(ht)dh =

∫GL2(Fv)

f(th)dh.

By this pairing, the local height pairing of two distinct CM points β1, β2 ∈ E×\B×f /U is

given by

iv1(β1, β2) =∑

γ∈µU\E×gv1(β1, γβ2)1Uv(β−1

1 γβ2).

Just like the other cases, the above summation is only well-defined for β1 6= β2. But wecan extend the definition as

iv1(β1, β2) =∑

γ∈µU\(E×−β1Uβ−12 )

gv1(β1, γβ2)1Uv(β−11 γβ2) =

∑γ∈µU\(E×−β1Uvβ

−12 )

gv1(β1, γβ2)1Uv(β−11 γβ2).

We want to compute

iv1(Z∗φ(g)t1, t2) =

∑a∈F×

∑x∈B×f /U

r(g)φ(x)a iv1(t1x, t2)

=∑a∈F×

∑x∈B×f /U

r(g)φ(x)a∑

γ∈µU\(E×−t1xUvt−12 )

gv1(t1x, γt2)1Uv(x−1t−11 γt2).

67

By 1Uv(x−1t−11 γt2) = 1, we have xv ∈ t−1

1 γt2Uv; by γ /∈ t1xUvt

−12 , we have xv /∈ t−1

1 γt2Uv.Thus

iv1(Z∗φ(g)t1, t2) =

∑a∈F×

∑γ∈µU\E×

r(g)φv(t−11 γt2)a

∑xv∈(B×v −t−1

1 γt2Uv)/Uv

r(g)φv(xv)agv1(t1xv, γt2).

Recall that

gv1(b1, b2) =

∫GL2(Fv)

1Uv(b−11 hb2)dh.

The last summation is equal to∑xv∈(B×v −t−1

1 γt2Uv)/Uv

r(g)φv(xv)a

∫GL2(Fv)

1Uv(x−1v t−1

1 hγt2)dh

=

∫GL2(Fv)

∑xv∈(B×v −t−1

1 γt2Uv)/Uv

r(g)φv(xv)a1Uv(x−1v t−1

1 γt2h)dh

=

∫GL2(Fv)−Uv

r(g)φv(t−11 γt2h)adh.

Here we explain the last equality above. We have xv ∈ t−11 γt2hUv by 1Uv(x

−1v ht−1

1 γt2) = 1.Then h /∈ Uv by xv /∈ t−1

1 γt2Uv.In summary, we have obtained:

Proposition 5.4.1.

iv1(Z∗φ(g)t1, t2) =

∑u∈µ2

U\F×

∑y∈E×

r(g, (t1, t2))φv(y, u) r(t1, t2)nr(g)φv ,v1(y, u).

where

nφv ,v1(y, u) =

∫GL2(Fv)−Uv

φv(yh, u)dh

=φ1,v(y, u)

∫F×v

φ2,v

(y

(0 b

0

), u

)(1− 1Uv

((1 b

1

)))d×b.

for any (y, u) ∈ E×v × F×

v .

The above result is only for the place v1. By symmetry, we have a formula for v2 afterreplacing the upper triangular matrices by lower triangular matrices. The average of thesetwo results yields:

Proposition 5.4.2.

iv(Z∗φ(g)t1, t2) = N (v)

φ (g, (t1, t2)) =∑

u∈µ2U\F×

∑y∈E×

r(g, (t1, t2))φv(y, u) r(t1, t2)nr(g)φv(y, u)

68

where

nφv(y, u) =1

2φ1,v(y, u)

∫F×v

φ2,v

(y

(0 b

0

), u

)(1− 1Uv

((1 b

1

)))d×b

+1

2φ1,v(y, u)

∫F×v

φ2,v

(y

(0b 0

), u

)(1− 1Uv

((1b 1

)))d×b

for any (y, u) ∈ E×v × F×

v .

In the unramified case, we have

Proposition 5.4.3. Assume that Uv is maximal and φv is the standard characteristic func-tion. Then nφv(y, u) 6= 0 only if (y, u) ∈ OEv ×O×

Fv. In this case,

nφv(y, u) =1

2ordvq(y).

Proof. Assume that (y, u) ∈ OEv × O×Fv

. Otherwise, both sides are zero. Assume that

y =

(a1

a2

)under the identification.

Since Uv is maximal,

(1 b

1

)∈ Uv if and only if b ∈ OFv . Then

nφv ,v1(y, u) =

∫Fv−OFv

φ2,v

((0 ba1

0

), u

)d×b = ordv(a1).

Similarly, the second integral is equal to ordv(a2).

The above result looks exactly the same as the result of nφv(y, u) in Proposition 5.2.5in the supersingular case. This is the reason that we use the same notation for them. Forexample, we have the counterpart of Lemma 5.2.3 (2) as follows:

Lemma 5.4.4. The function nr(g)φv(y, u) behaves like Weil representation under the actionof P (Fv) on (y, u), i.e.,

nr(m(a))φv(y, u) = |a|2nφv(ay, u), a ∈ F×v

nr(n(b))φv(y, u) = ψ(buq(y))nφv(y, u), b ∈ Fvnr(d(c))φv(y, u) = |c|−1nφv(y, c

−1u), c ∈ F×v .

Analogous to Proposition 5.2.6, we have:

Proposition 5.4.5. For the maximal Uv and the standard φv as in Proposition 5.4.3, wehave

nr(g)φv(y, u) = −(log δ(gv) +1

2log |uq(y)|v) r(g)φv(y, u).

69

Proof. We have the identity for g = 1 by Proposition 5.4.5. To extend the result to generalg ∈ GL2(Fv), we only need to apply Lemma 5.4.4 to the Iwasawa decomposition of g.

Parallel to Proposition 5.2.7, we have

Proposition 5.4.6. For general Uv, the function nφv(y, u) can be extended to a Schwartzfunction for (y, u) ∈ Ev × F×

v if φv ∈ S 0(Bv × F×v ).

Proof. It suffices to show the result for

nφv ,v1(y, u) =

∫F×v

φv

((a1 a1b

a2

), u

)(1− 1Uv

((1 b

1

)))d×b.

Here we still write y =

(a1

a2

).

It is easy to see that nφv ,v1 is locally constant at any (y, u) with a1 6= 0. The problem isto show that, for fixed (y0, u0) ∈ Fv × F×

v , the function nφv ,v1 extends to a locally constant

function at (y0, u0). Here y0 denotes the element

(0

y0

)∈ Ev.

Consider the behavior when (y, u) is closed to (y0, u0). Since v(a1) is large, we see that

a1b is closed to zero when

(1 b

1

)∈ Uv, and thus φv

((a1 a1b

a2

), u

)= φv(y0, u0) = 0.

It follows that

nφv ,v1(y, u) =

∫F×v

φv

((a1 a1b

a2

), u

)d×b =

∫F×v

φv

((a1 b

a2

), u

)d×b.

It is independent of (y, u) in a neighborhood of (y0, u0), and we can extend the function to(y0, u0).

5.5 The j -part

Now we consider the pairing jv(Z∗φ(g)t1, t2). It is nonzero only if v is a finite prime such that

Uv is not maximal or Bv is nonsplit. The goal is to write it as a pseudo-theta series andcontrol the singularities. It turns out that it has no singularity if φv ∈ S 0(Bv × F×

v ). Inthat case, an integration of the associated theta series becomes an Eisenstein series.

For any β1, β2 ∈ CMU , the pairing j(β1, β2) 6= 0 only if β1, β2 are in the same geometricallyconnected components of XU . Once this is true, the pairing is given by a symmetric locallyconstant function l(·, ·) on (B×

v /Uv)2. In summary, we have

jv(β1, β2) = l(β1v, β2v)1F×+ q(β2)q(U)(q(β1)) = l(β1v, β2v)1(Bf )adU(β−12 β1).

70

Here we denote

(Bf )ad =x ∈ Bf : q(x) ∈ F×+

(Bf )a =x ∈ Bf : q(x) = a, a ∈ A×f .

Now we go back to

jv(Z∗φ(g)t1, t2)

=∑a∈F×

∑x∈B×f /U

r(g)φ(x)a jv(t1x, t2)

=∑a∈F×

∑x∈B×f /U

r(g)φ(x)a l(t1xv, t2)1(Bf )adU(t−12 t1x)

=∑a∈F×

∑x∈(Bf )adU/U

r(g)φ(t−11 t2x)a l(t2xv, t2)

=∑a∈F×

∑x∈(Bf )ad/Uad

r(g, (t1t−12 , 1))φ(x)a l(t2xv, t2).

Apparently µUU1 ⊂ Uad. The quotient Uad/µUU

1 = (q(U) ∩ F×+ )/µ2

U is finite since it is aquotient of two finitely generated abelian groups of the same rank. Hence we have

[q(U) ∩ F×+ : µ2

U ] jv(Z∗φ(g)t1, t2)

=∑a∈F×

∑x∈(Bf )ad/µUU1

r(g, (t1t−12 , 1))φ(x)a l(t2xv, t2)

=∑

u∈µ2U\F×

∑x∈(Bf )ad/U1

r(g, (t1t−12 , 1))φ(x, u) l(t2xv, t2)

=∑

u∈µ2U\F×

∑a∈F×+

∑x∈(Bf )a/U1

r(g, (t1t−12 , 1))φ(x, u) l(t2xv, t2)

=1

vol(U1)

∑u∈µ2

U\F×

∑a∈F×+

∫(Bf )a

r(g, (t1t−12 , 1))φ(x, u) l(t2xv, t2)dx.

The integral above is a local product, which is nice at every place except v.We first consider the case that Bv is a matrix algebra. We introduce a function

lφv(t2, y, u) =1

vol(B1v)

∫(Bv)q(y)

φv(x, u) l(t2x, t2)dx, (y, u) ∈ B×v × F×

v .

Note that Bv is a division algebra and thus B1v is compact and has finite volume.

Fix t2 ∈ T (Af ). In general, lφv(t2, y, u) is not defined at y = 0. But if φv ∈ S 0(Bv×F×v ),

then φv(x, F×v ) = 0 when v(q(x)) is large. It follows that lφv(t2, y, u) = 0 for y closed to 0.

It is automatically a Schwartz function for (y, u) ∈ Bv × F×v .

71

Now we go back to the computation. We have:

[q(U) ∩ F×+ : µ2

U ]vol(U1) jv(Z∗φ(g)t1, t2)

=∑

u∈µ2U\F×

∑a∈F×+

∫(Bf )a

r(g, (t1t−12 , 1))φv(y, u)lr(g,(t1t−1

2 ,1))φv(t2, y, u)dy

=∑

u∈µ2U\F×

∑y∈B×/B1

∫B1(Af )

r(g, (t1t−12 , 1))φv(yh, u)lr(g,(t1t−1

2 ,1))φv(t2, yh, u)dh

=∑

u∈µ2U\F×

∫B1\B1(Af )

∑y∈B×

r(g, (t1t−12 , 1))φv(yh, u)lr(g,(t1t−1

2 ,1))φv(t2, yh, u)dh.

The integral on B1\B1(Af ) essentially reduces to a finite sum. When lφv is a Schwartzfunction, we may replace the pseudo-theta series above by the associated theta series. Wewill get an integral of this theta series, which gives an Eisenstein series.

Next we consider the case that Bv is a division algebra. In this case, Bv is a matrixalgebra and the function lφv above is never Schwartz in y. To resolve this problem, we takean open compact subset D of Bv such that Da = D ∩ B(Fv)a is nonempty for all a in therange

q(x) : x ∈ Bv, φv(x, F×v ) 6= 0.

This is possible for φv ∈ S 0(Bv × F×v ) and D depends on φv. Then we define

lφv(t2, y, u) =1

vol(Dq(y))1D(y)

∫(Bv)q(y)

φv(x, u) l(t2x, t2)dx, (y, u) ∈ B×v × F×

v .

It is easy to check that it is a Schwartz function for (y, u) ∈ Bv × F×v . And we also have∫

B(Fv)a

lφv(t2, y, u)dy =

∫(Bv)a

φv(x, u) l(t2x, t2)dx.

Then the argument is the same as before. We still get an integral of a non-singular pseudo-theta series. The integration of the associated theta series becomes an Eisenstein series.

Remark. The matching between lφv(t2, y, u) and φv(x, u) l(t2x, t2) so that they have the sameintegral looks very similar to the situation of Fundamental Lemma.

72

6 Pseudo-theta series and Gross–Zagier formula

In this section we prove the main result, Theorem 1.3.1. We argue that the computationand estimation of singularities we have made are enough to make the conclusion. Takingadvantage of the modularity of both sides, we use some approximation to take care the termswe don’t know how to compute.

Besides the linear independence of quasi-multiplicative Whittaker functions and theirquasi-derivatives used in our previous paper [Zh1], the new ingredient involved in the preciseproof is a theory of pseudo-theta series described in §6.1. Such series arise naturally in thecomputation of the derivative and the local height pairing. Roughly speaking, there aremany series we don’t know how to compute explicitly, and their local components are reallymessed. Then Lemma 6.1.1 asserts that, if a finite sum of pseudo-theta series is automorphic,then it can actually be written as a finite sum of usual theta series. The lemma simplifies theproof significantly. We expect that it also helps with the computation of derivative formulafor triple product L-functions in our another joint work.

6.1 Pseudo-theta series

We use the name “pseudo-theta series” because it looks like theta series and can be comparedwith some theta series associated to it. Note that it is usually not automorphic.

Pseudo-theta series

Now we introduce pseudo-theta series. Let V be a positive definite quadratic space over F ,and V0 ⊂ V1 ⊂ V be two subspaces over F with induced quadratic forms. We allow V0 tobe empty. Let S be a finite set of finite places of F , and φS ∈ S (V (AS)×AS×)GO(V∞) be aSchwartz function with the standard infinite components.

A pseudo-theta series is a series of the form

A(S)φ′ (g) =

∑u∈µ2\F×

∑x∈V1−V0

φ′S(g, x, u)rV(g)φS(x, u), g ∈ GL2(A).

We explain the notations as follows:

• The Weil representation rV

is not attached to the space V1 but to the space V ;

• φ′S(g, x, u) =∏

v∈S φ′v(gv, xv, uv) as a product of local terms;

• For each v ∈ S, the function

φ′v : GL2(Fv)× (V1 − V0)(Fv)× F×v → C

is locally constant. And it is smooth in the sense that there is an open compactsubgroup Kv of GL2(Fv) such that

φ′v(gκ, x, u) = φ′v(g, x, u), ∀(g, x, u) ∈ GL2(Fv)× (V1 − V0)(Fv)× F×v , κ ∈ Kv.

73

• µ is a subgroup of O×F with finite index such that φS(x, u) and φ′S(g, x, u) are invariant

under the action α : (x, u) 7→ (αx, α−2u) for any α ∈ µ. This condition makes thesummation well-defined.

• For any v ∈ S and g ∈ GL2(Fv), the support of φ′S(g, ·, ·) in (V1 − V0)(Fv) × F×v is

bounded. This condition makes the sum convergent.

The pseudo-theta series A(S) sitting on the triple V0 ⊂ V1 ⊂ V is called non-degenerateif V1 = V , and is called non-truncated if V0 is empty. It is called to be of Whittaker type ifφ′S(g, ·, ·) transfer according to Weil representation under the unipotent group, i.e.,

φ′S(n(b)g, x, u) = φ′S(g, x, u) ψ(buq(x)), ∀b ∈ A.

We use this name because, if the above is true, then A(S) is invariant under the left actionof N(F ), and its first Fourier coefficient is exactly∑

(x,u)∈µ\((V1−V0)×F×)uq(x)=1

φ′S(g, x, u)rV(g)φS(x, u).

We call it the Whittaker function of A(S).The pseudo-theta series A(S) is called non-singular if for each v ∈ S, the local component

φ′v(1, x, u) can be extended to a Schwartz function on V1(Fv)× F×v .

Assume that A(S)φ′ is non-singular. Then there are two usual theta series associated to

A(S). Extending by zero we may view φ′v(1, ·, ·) as a Schwartz function on V1(Fv) × F×v for

each v ∈ S, and by restriction we may view φw as a Schwartz function on V1(Fw)× F×w for

each w /∈ S. Then the theta series

θA,1(g) =∑

u∈µ2\F×

∑x∈V1

rV1

(g)φ′S(1, x, u)rV1(g)φS(x, u)

is called the first theta series associated to A(S)φ′ . Replacing the space V1 by V0, we get the

theta seriesθA,0(g) =

∑u∈µ2\F×

∑x∈V0

rV0

(g)φ′S(1, x, u)rV0(g)φS(x, u).

We call it the second theta series associated to A(S)φ′ . We set θA,0 = 0 if V0 is empty.

We introduce these theta series because the difference between θA,1 and θA,0 somehowapproximates A(S).

Basic properties

Now we consider the relation between the non-singular pseudo-theta series A(S)φ′ and its

associated theta series θA,1 and θA,0.

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We first consider the non-truncated case. Then V0 is empty, and

A(S)φ′ (g) =

∑u∈µ2\F×

∑x∈V1

φ′S(g, x, u)rV(g)φS(x, u).

Apparently we have A(S)φ′ (1) = θA,1(1). But of course we can get more.

A simple computation using Iwasawa decomposition asserts that, if φw is the standardSchwartz function on V (Fw)× F×

w , then for any (x, u) ∈ V1(Fw)× F×w ,

rV(g)φw(x, u) =

δw(g)

d−d12 r

V1(g)φw(x, u) if w - ∞;

λw(g)d−d1

2 δw(g)d−d1

2 rV1

(g)φw(x, u) if w | ∞.

Here we write d = dimV and d1 = dimV1. Here δw and λw coming from Iwasawa decompo-sition are explained in Section 1.6.

This result implies that,

A(S)φ′ (g) = λ∞(g)

d−d12 δ(g)

d−d12 θA,1(g), ∀ g ∈ 1S′GL2(AS′).

Here S ′ is a finite set consisting of finite places v such that v ∈ S or φv is not standard.Now we consider the general

A(S)φ′ (g) =

∑u∈µ2\F×

∑x∈V1−V0

φ′S(g, x, u)rV(g)φS(x, u).

We have to compare it with the difference between the theta series

θA,1(g) =∑

u∈µ2\F×

∑x∈V1

rV1

(g)φ′S(1, x, u)rV1(g)φS(x, u)

and the non-truncated pseudo-theta series

B(S)φ′ (g) =

∑u∈µ2\F×

∑x∈V0

rV1

(g)φ′S(1, x, u)rV1(g)φS(x, u).

Note that B(S) is just part of θA,1, where the summation is taken over V0 but the representa-tion is taken over V1. By the discussion above, we should compare B(S) with the associatedtheta series

θB,1(g) =∑

u∈µ2\F×

∑x∈V0

rV0

(g)φ′S(1, x, u)rV0(g)φS(x, u).

But this is exactly the same as θA,0. By the same argument, there exists a finite set S ′ offinite places such that

A(S)φ′ (g) = λ∞(g)

d−d12 δ(g)

d−d12 (θA,1(g)−B

(S)φ′ (g)), ∀ g ∈ 1S′GL2(AS′);

B(S)φ′ (g) = λ∞(g)

d1−d02 δ(g)

d1−d02 θA,0(g), ∀ g ∈ 1S′GL2(AS′).

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Our conclusion is that for any g ∈ 1S′GL2(AS′),

A(S)φ′ (g) = λ∞(g)

d−d12 δ(g)

d−d12 θA,1(g)− λ∞(g)

d−d02 δ(g)

d−d02 θA,0(g). (6.1.1)

By the smoothness condition of pseudo-theta series, there exists an open compact subgroupKS′ of GL2(FS′) such that the above identity is actually true for any g ∈ KS′GL2(AS′).

If furthermore A(S) is of Whittaker type, the above remains true for any g ∈ KS′GL2(AS′)if we replace A(S), θA,1, θA,0 by their Whittaker functions. It holds by the same argumentsince the Whittaker functions are just a part of their corresponding pseudo-theta series ortheta series.

Now we can state our main result for pseudo-theta series.

Lemma 6.1.1. Let A(S`)` ` be a finite set of non-singular pseudo-theta series of Whittaker

type sitting on vector spaces V`,0 ⊂ V`,1 ⊂ V`. Let (Ei, Di) be a finite set of pairs ofEisenstein series for some non-equivalent pairs of characters (µi, νi) and their derivations.Assume that the sum

f :=∑`

A(S`)` (g) +

∑i

(Ei +Di)(g)

is cuspidal in g ∈ GL2(A). Then

(1)∑`

A(S`)` =

∑`∈L0,1

θA`,1,

(2)∑`∈Lk,1

θA`,1 −∑`∈Lk,0

θA`,0 = 0, ∀k ∈ Z>0.

(3) Di = 0.

Here Lk,1 is the set of ` such that dimV` − dimV`,1 = k, and Lk,0 is the set of ` such thatdimV` − dimV`,0 = k. In particular, L0,1 is the set of ` such that V`,1 = V`.

Proof. In the equation of f , replace each A(S`)` by its corresponding combinations of theta se-

ries on the right-hand side of equation (6.1.1). After recollecting these theta series accordingto the powers of λ∞(g)δ(g), we end up with an equation of the following form:

n∑k=0

λ∞(g)kδ(g)kfk(g) = 0, ∀g ∈ KSGL2(AS). (6.1.2)

Here S is some finite set of finite places, KS is an open compact subgroup of GL2(FS),and f0, f1, · · · , fn are some automorphic forms on GL2(A) coming from combinations of fand theta series. In particular,

f0 = f −∑`∈L0,1

θA`,1 −∑

(Ei +Di).

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We will show that fk = 0 = Di identically, which is exactly the result of (1), (2), and (3).As all theta series fk are defined by positive definite quadratic space, fk has a decompo-

sitionfk =

∑m

fk,m, k > 0

f0 =∑m

f0,m +∑

Dm.

into a finite sum of forms and their derivations in the irreducible automorphic representationsπm of GL2(A). The equation of the Whittaker functions corresponding to 6.1.2 is given by∑

k,m

λ∞(g)kδ(g)kWfk,m(g) +

∑m

WDm(g) = 0, ∀g ∈ KSGL2(AS). (6.1.3)

This is a sum of quasi-multiplicative functions on GL2(AS) as δ(g) is already multiplicative.Now we can apply Lemma 4.5.1 in our previous paper [Zh1] to functions on Kirillov models

αk,m(a) := δkWfk,m

(a 00 1

), hm(a) := WDm

(a 00 1

)to conclude the following:

• Dm = 0 for all m;

• for any pair (k,m), there is another pair (k′,m′) such that

δ(g)kWfk,m(g) + δ(g)k

′Wfk′,m′

(g) = 0

for g ∈ GL2(AS).

The second equation implies an equality between spherical Whittaker functions W 0m(g) for

almost all unramified places v:

W 0m,v(g) = cδ(g)k

′−kW 0m′,v(g)

where c is a nonzero constant. It follows that πm,v and πm′,v have the same central character.

Evaluate the above identity for g =

(πm

1

)to conclude that πm = πm′ ⊗ | · |(k′−k)/2. Thus

we must have k = k′, πm,v = πm′,v. By strong multiplicity one, we have π = π′ and thenfk,m = 0.

6.2 Proof of Gross–Zagier formula

Recall that

P (φ, ϕ, χ, s) = vol(UT )

(∫CU

I(s, g, r(t, 1)φ)χ(t)dt, ϕ(g)

).

We are going to show:

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Theorem 6.2.1.

P ′(φ, ϕ, χ, 0) =− 2vol(UT )

(∫CU

〈Zφ(g)t, 1〉χ(t)dt, ϕ(g)

)=− 2

vol(UT )

|CU |

(∫CU

∫CU

〈Zφ(g)t1, t2〉χ(t1t−12 )dt1dt2, ϕ(g)

).

Here we write t = t− deg(t)ξ.

Note that for any c ∈ A×f , we have

Z∗φ(cg)[t] =

∑a∈F×

∑x∈B×f /U

r(cg)φ(x)a[tx] =∑a∈F×

∑x∈B×f /U

r(g)φ(cx)a[tx]

=∑a∈F×

∑x∈B×f /U

r(g)φ(x)a[tc−1x] = Z∗

φ(g)[c−1t].

It already implies that Zφ(cg)[t] = Zφ(g)[c−1t] by modularity. It follows that∫

CU

〈Zφ(g)t, 1〉χ(t)dt

has the same central character as ϕ, and thus the Petersson inner product in the theoremmakes sense.

The theorem is equivalent to that∫CU

(I ′(0, g, r(t, 1)φ) + 2〈Zφ(g)t, 1〉)χ(t)dt

is perpendicular to ϕ. Now we are going to compare those two kernels.By Proposition 3.7.1, replacing φ by r(t, 1)φ, we get

Pr(I ′(0, g, r(t, 1)φ))

=−∑v|∞

I ′(0, g, r(t, 1)φ)(v)−∑

v-∞ non−split

I ′(0, g, r(t, 1)φ)(v)

+ 2∑

(y,u)∈µU\E××F×(log δf (gf ) +

1

2log |uq(y)|f )r(g, (t, 1))φ(y, u)

−∑v

∑u∈µ2

U\F×,y∈E×cφv(g, y, u)r(g, (t, 1))φv(y, u)− c1

∑u∈µ2

U\F×,y∈E×r(g, (t, 1))φ(y, u)

+ Eisenstein series and their derivations.

Here

I ′(0, g, r(t, 1)φ)(v) = 2

∫Z(A)T (F )\T (A)

K (v)φ (g, (tt′, t′))dt′, v|∞,

I ′(0, g, r(t, 1)φ)(v) = 2

∫Z(A)T (F )\T (A)

K (v)φ (g, (tt′, t′))dt′, v - ∞.

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Now we list our computational results for the geometric part. Our place-to-place com-putation and comparison show that we also have

〈Zφ(g)t, 1〉 =∑v

iv(Z∗φ(g)t, 1) logNv +

∑v

jv(Z∗φ(g)t, 1) logNv

+i0(1, 1)∑a∈F×

∑y∈E×/µU

r(g, (t, 1))φ(y)a + 〈Zφ,0(g)t, 1〉

+ Eisenstein series and their derivations

=∑

v non−split

logNv

∫CU

M (v)φ (g, (tt′, t′))dt′ +

∑v/∈Σ

logNv

∫CU

N (v)φ (g, (tt′, t′))dt′

+∑v

logNv

∫CU

jv(Z∗φ(g)tt

′, t′)dt′

+i0(1, 1)∑a∈F×

∑y∈E×/µU

r(g, (t, 1))φ(y)a + 〈Zφ,0(g)t, 1〉

+ Eisenstein series and their derivations.

Since the archimedean parts are standard, we see that the average integrals∫Z(A)T (F )\T (A)

=

∫Z(A)T (F )\T (A)/UT

=

∫T (F )\T (A)/UT

=

∫CU

.

Furthermore, CU is finite and the integral reduces to a finite sum.In summary, we have proved that

Pr′∫CU

(I ′(0, g, r(t, 1)φ) + 2〈Zφ(g)t, 1〉)χ(t)dt =

∫CU

Dφ(g, (t, 1))χ(t)dt,

whereDφ(g, (t, 1)) = Pr(I ′(0, g, r(t, 1)φ)) + 2〈Zφ(g)t, 1〉

is a sum of non-singular pseudo-theta series, Eisenstein series and their derivations.Here is a checklist of the comparison:

(1) For v|∞, Proposition 5.1.1 shows that

K (v)φ (g, (t1, t2))−M (v)

φ (g, (t1, t2)) = 0.

(2) For v finite and non-split in E, we have showed that

K (v)φ (g, (t1, t2))−M (v)

φ (g, (t1, t2)) logNv

is a nonsingular pseudo-theta series and vanishes for almost all v. This pseudo-thetaseries is non-degenerate. See Proposition 5.2.6 and Proposition 5.2.7(1) for the super-singular case and Proposition 5.3.2 for the superspecial case.

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(3) For v /∈ Σ, we have showed that the difference

N (v)φ (g, (t1, t2)) +

∑(y,u)∈µU\E××F×

(log δf (gf ) +1

2log |uq(y)|f )r(g, (t1, t2))φ(y, u)

is a nonsingular pseudo-theta series and vanishes for almost all v. This pseudo-theta se-ries is degenerate. See Proposition 5.2.6 and Proposition 5.2.7(2) for the supersingularcase and Proposition 5.4.5 and Proposition 5.4.6 for the ordinary case.

(4) For all finite v, we have showed that jv(Z∗φ(g)tt

′, t′) is an integral of a nonsingularpseudo-theta series on B(v) if φv. This pseudo-theta series is non-degenerate. Theintegration of the associated theta series gives a Siegel–Einseisten series. See Section5.5.

(5) The rest of terms are explicitly expressed as non-singular pseudo-theta series. Someare non-singular automatically, and some are non-singular if φv ∈ S 0(Bv × F×

v ).

(6) The Einsenstein series and their derivation comes from Proposition 3.6.3 for holomor-phic projection and Proposition 4.6.1 for the height pairing related to the Hodge classξ. See §4.4.4 in our previous paper [Zh1] for a general explanation of derivations.

Only finitely many terms are left after the comparison. Note that the integral on CU isjust a finite sum. Then we are in the situation to apply Lemma 6.1.1 to∫

CU

Dφ(g, (t, 1))χ(t)dt =∑

A` +∑i

(Ei +Di).

Since their sum is modular and cuspidal, it is equal to a finite sum of usual theta seriesand Eisenstein series by Lemma 6.1.1. These theta series are coming from K (v)

φ (g, (t1, t2))−M (v)

φ (g, (t1, t2)) logNv and jv(Z∗φ(g)tt

′, t′), the only non-degenerate pseudo-theta series in thelist. Note that the quadratic spaces for both of them are the nearby quaternion algebra B(v).Therefore, we end up with

Dφ(g, (t, 1)) =

∫CU

∑v

θφ(v)(g, (tt′, t′))dt′ + Eisenstein Series.

We understand that the equality is true after integrating against χ(t) on CU , and that thetheta series is given by

θφ(v)(g, (t1, t2)) =∑

u∈µ2U\F×

∑y∈B(v)

r(g, (t1, t2))φ′v(yv, uv)r(g, (t1, t2))φ

v(y, u)

for some Schwartz function φ(v) = φ(v)v ⊗ φv ∈ S (B(v)A ×A×) with φ

(v)v ∈ S (B(v)v × F×

v ).

80

Therefore, we get that(∫CU

χ(t)Dφ(g, (t, 1))dt, ϕ(g)

)=∑v

(∫CU

∫CU

χ(t)θφ(v)(g, (tt′, t′))dt′dt, ϕ(g)

).

Now the vanishing of the right-hand side following from the criterion of the existence of localχ-linear vectors by the result of Tunnell [Tu] and Saito [Sa], Proposition 1.1.1.

Note that to control the singularities for some ramified non-archimedean place v, wehave assumed that φv ∈ S 0(Bv × F×

v ). But we can easily extend the result to all Schwartzfunction φv ∈ S (Bv×F×

v ). We have showed that the main theorem is true for all pairs (ϕ, φ)with ϕ ∈ π and φ = ⊗vφv such that φv is standard for good v, and φv ∈ S 0(Bv × F×

v ) forbad v. By Proposition 3.3.1, all theta liftings θϕφ given by such pairs span the space π′ ⊗ π′.On the other hand, the result of the main theorem depends only on the theta lifting θϕφ . Weconclude that the main theorem is true for all ϕ, φ.

81

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