49 for some numbers ˜b(σ)∈C. Since the coefficientsc(n) are real, from the additive formula (1.24) for the Naganuma map we get
eb(σ) = X
l|(σ)da−1
lk−1δp
n(σ)p2 N l2
c
n(σ)p2 N l2
. It follows from (3.8) that
eb(σ) =X
l|σ
X
4σσ′/l2−t2≥0 p|4σσ′/l2−t2
lk−1δ σσ′
l2
b
(4σσ′−l2t2)p l2
.
Note that for any sequence {γ(n)}n∈Z we have X
n≡amodp
γ(n2) = 1
2δ(a) X
n≡±amodp
γ(n2).
Thus, we have X
4σσ′/l2−t2≥0 p|4σσ′/l2−t2
lk−1δ σσ′
l2
b
4σσ′−l2t2 p·l2
= 1 2
X
4σσ′−l2t2≥0 p·l|trσ−l·t
l|σ
lk−1b
4σσ′−l2t2 p·l2
.
Hence, we arrive at
Nag(τ1, τ2) = X
σ∈ad−1 σ≫0
eb(σ)e(στ1+σ′τ2),
where
eb(σ) = 1 2
X
4σσ′−l2t2≥0 p·l|trσ−l·t
l|σ
lk−1b
4σσ′−l2t2 p·l2
. (3.9)
Comparing the Fourier expansions (3.7) and (5.3) we finish the proof of the Theorem.
with N prime, and suppose that the power series
gp(τ) :=h(pτ)θ(τ)|U4N
is a modular form of weight k, level p and Nebentypus (p·) for every prime p≡1 (mod 4) with (Np) = 1. Then h belongs to Mk+−1/2(N).
In this section we prove Theorem 3.2 under additional assumptions on h. Namely, we assume that coefficients b(D) have moderate growth and power seriesh converges for
|q|<1. Then h(τ) is a holomorphic function on H.
Although we don’t know that the function h is a modular form, we can define its Gritsenko lift by formula (3.1)
F(Z) =X
T
B(T)e tr(T Z) . For the function F we prove the following
Lemma 3.3. Let h(τ) be a holomorphic function on H. Suppose that the function h satisfies the hypotheses of Theorem 2. Then the function F defined above is a Siegel modular form for the paramodular group ΓN.
Proof. We have to show that
F|kM(Z) =F(Z) (3.10)
for all M ∈ΓN and Z ∈ H(2). Suppose thatZ =ρ(τ1, τ2)∈ H(2) for some map ρ defined in (1.6) and some (τ1, τ2)∈H×H. We will check the identity (3.10) for the generators of ΓN. It is shown in [32] that the paramodular group is generated by
JN =
0 0 1 0
0 0 0 N−1
−1 0 0 0 0 −N 0 0
and the elements of ΓN ∩Γ∞(Q), where
Γ∞(Q) =
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 ∗ ∗
∈Sp4(Q)
.
First we show that for Z =ρ(τ1, τ2) the identity
F|kJN(Z) =F(Z) (3.11)
holds. In this case
F|kJN(Z) =N−kdetZ−kF
1 0 0 N−1
tR−1
−1 τ1 0 0 −τ21
R−1
1 0 0 N−1
.
51 It follows from the definition of the Gritsenko lift (1.34) that
F
τ1 z z τ2
=F
!N τ2 −z
−z N−1τ1
#
= (3.12)
=F
!!
0 −N1/2 N−1/2 0
# τ1 z z τ1
! 0 N−1/2
−N1/2 0
##
. Thus, we can write
F(JN(Z)) = F
1 0 0 N−1
tR−1
−1 τ1 0 0 −τ21
R−1
1 0 0 N−1
=
=F
0 −1 1 0
tR−1
−1 N τ1 0
0 N τ−1
2
R−1
0 1
−1 0
. Note that
tR−1 = det(R−1)
0 −1 1 0
R
0 1
−1 0
. Using this identity we arrive at
F(JN(Z)) =F
R
−p N τ2 0
0 N τ−p1
tR
=
=F ◦ρ −p
N τ2, −p N τ1
.
It follows from Theorem 3.1 that F ◦ρ= 12Na(g). It follows from (1.30) that 1
2Na(g) −p
N τ2
, −p N τ1
= 1
2Nkτ1kτ2kNa(g) (τ1, τ2) =
= Nkτ1kτ2kF ◦ρ(τ1, τ2).
Now it remains to prove (3.10) forM ∈ΓN∩Γ∞(Q). Each element ofM ∈ΓN∩Γ∞(Q) can be written as M =MUMS, where
MU =
U 0 0 U
, with UtU =E, U =
∗ N∗ N∗ ∗
, and
MS =
U 0 0 U
, withS= tS, S =
∗ ∗
∗ N−1∗
.
Here all ∗ are in Z. The matrix P = U R satisfies (1.5) and defines a map π(τ1, τ2) = P
τ1 0 0 τ2
tP. Obviously,
F(MU(Z)) =F ◦π(τ1, τ2).
It follows from Theorem 3.1 that F ◦π(τ1, τ2) = 1
2Na(g)(τ1, τ2) = F ◦ρ(τ1, τ2).
Thus, we obtain
F|kMU(Z) =F(Z). (3.13)
Finally, it follows easily from (3.1) that
F|kMS(Z) = F(Z +S) = F(Z). (3.14) It follows from (3.11), (3.13), and (3.14) that (3.10) holds for allM ∈ΓN andZ =ρ(τ1, τ2).
The set of all matricesZ =ρ(τ1, τ2) for all primesp, mapsρdefined by (1.6) and all points (τ1, τ2) ∈ H×H is dense in H(2). Since the function F is continuous, the identity (3.10) holds for all Z ∈H(2). Lemma 3.3 is proved.
Using Theorem 3.2 we can deduce the modularity of the generating series of the classes of the Heegner divisors in the Jacobian of X0(N) from the modularity of the generating series of the classes of Hirzebruch-Zagier curves in the homology group of a certain Hilbert surface. This idea is explained in [76] and is applied there in the case N = 37. The Hirzebruch-Zagier curve TN on the Hilbert surface Yδ = H×H\SL2(o) is given by the equations
Aτ1τ2+ λ
√pτ1− λ′
√pτ2+B = 0
with AB − λλ′/p = N. The curve TN is isomorphic to X0(N). Denote by [TNc] the (compact) homology class ofTN. It is proved in [41] thatP∞
N=1[TN]qN is a modular form of weight 2, level p and nebentypus (p). Since the surface Yp is simply connected, the generating series P∞
N=1TNqN of classes of TN in its first Chow group is also a modular form. Since
TN ∩TM = [
r2<4M N p|4M N−r2
P4M N−r2 p
,
the series
X
M
! X
4M N−r2>0 p|4M N−r2
P4M N−r2 p
# qM
is also a modular form of weight 2, level pand nebentypus (p).
53 Next, an estimate of the naive height of points Pd in the Jacobian shows that the
power series X
d
Pdqd converges for |q|<1.
Hence, it follows from Theorem 3.2 that P
dPdqd is a modular form of half integral weight. This is one of the main results of [36], where it is proved by a much more difficult computation of the height pairings of Heegner points.
Chapter 4
Heegner points and Siegel Eisenstein series
4.1 Intoduction
In this chapter we study a relation between Fourier coefficients of the degree 3 Siegel Eisenstein series of weight 2 and heights of Heegner points on modular curves. Such a relation was conjectured by B. Gross and S. Kudla many years ago and it became one of the motivating examples of Kudla’s program connecting special cycles on Shimura varieties with Eisenstein series. In this chapter we explain how this idea can give a new approach to Gross-Kohen-Zagier formula that is both easier and more conceptual than the original one. Some of our results overlap with results given in [53].
In the paper [1] Gross and Zagier showed that the height of Heegner point on an elliptic curveEis an explicit(and in general non-zero) multiple of the derivativeL′(E/Q,1). This implies that for any given elliptic curve E with ords=1L(E/Q, s) = 1 there are Heegner points of non-zero height, which therefore are non-torsion, in particular the rank ofE(Q) is then at least one. In a subsequent paper [36] the same authors and W. Kohnen proved a more general formula involving L′(E(Q),1) and height pairings between two different Heegner points. We now explain this in more detail.
Let X0(N) be the modular curve with complex points Γ0(N)\H and J∗ be the Jaco-bian of X0∗(N), the quotient of X0(N) by the Fricke involution wN. For each imaginary quadratic field K whose discriminant D is a square modulo N and to each r ∈ Z/2NZ with r2 ≡ D(mod 2N), we associate a Heegner divisor yD,r∗ ∈ J∗ as follows. If τ ∈ H = {z ∈C| ℑ(z)>0}is the root of a quadratic equation
aτ2+bτ +c= 0, a, b, c∈Z, a >0, a ≡0(mod N), b ≡r(mod 2N), b2−4ac=D
then the image of τ in H/Γ0(N)⊂X0(N)(C) is defined over H, the Hilbert class field of K. There are exactly h = [H : K] such images and their sum is a divisor PD,r of degree h defined over K. We write yD,r for the divisor PD,r −h·(∞) of degree 0 on X0(N) and for its class in the Jacobian, and y∗D,r for the image of yD,r in J∗. The action of the
55
non-trivial element of Gal(K/Q) on TrH/K((y)) is the same as that of wN, therefore the image yD,r∗ of yD,r in J∗ is defined over Q. Its f-component is non-trivial only if f is a modular form on Γ∗(N), and this is the case precisely when L(f, s) has a minus sign in its functional equation and hence a zero (of odd order) at s = 1.
A striking coincidence is that the Heegner divisors yD,r∗ and Fourier coefficients of Jacobi cusp forms of weight 2 and level N are indexed by the same set of pairs (D, r), where D is a square modulo N and r ∈Z/2NZ satisfies r2 ∼=D(mod 2N). Moreover, it is shown in [67] that the new part of the space of Jacobi cusp forms Jk,Ncusp is isomorphic as a Hecke module to the new part of the space S2k−2(N)− space of cusp forms of weight 2k−2 on Γ0(N) with eigenvalue−1 under the involution f(z)→(−N z2)−k+1f(−1/N z).
This led the authors of [36] to guess that the height pairing of the f-components of y∗D0,r0 and yD∗1,r1 for different discriminants D0 and D1 should be related to the product L′(f,1)c(n0, r0)c(n1, r1), where Di =ri2−4N ni and c(ni, ri) are the Fourier coefficients of a unique up to scalar Jacobi form φ∈J2,Ncusp having the same eigenvalues as f under all Hecke operators.
More precisely, letD0,D1 <0 be coprime fundamental discriminants,Di =r2i−4N ni, and f ∈ S2(Γ∗0(N)) a normalized newform, in [36] the authors prove the formula for the height pairings of the f-eigencomponents of yD∗0,r0 and y∗D1,r1
h(y∗D0,r0)f,(y∗D1,r1)fi= L′(f,1)
4πkφk2 c(n0, r0)c(n1, r1). (4.1) where c(ni, ri) denote the coefficient of e(niτ +riz) in φ ∈ J2,Ncusp, the Jacobi form corre-sponding to f (i= 0,1).
In the case whenfis a modular form of weight 2k > 2 P. Deligne has found a definition of Heegner vectors Sx in the stalks above Heegner pointsx of the local coefficient system Sym2k−2(H1)(H1 = first cohomology group of the universal elliptic curve overX0(N)) and suggested an interpretation of the right-hand side of (4.1) as some sort of height pairing between these Heegner vectors. In [16] Brylinsky worked out some definitions of local heights suggested by Deligne. In [80] Zhang extended the result of [36] to higher weights by using the arithmetic intersection theory of Gillet and Soul´e [30]. More precisely, for a CM-divisor on X0(N)Z Zhang defined a CM-cycle Sk(x) on a certain Kuga-Sato variety.
He defined the (global) height pairing between CM-cycles in these Kuga-Sato varieties, and showed an identity between the height pairings of Heegner cycles and coefficients of certain cusp forms of higher weights. We consider the following two generating functions
Hk,N(τ0, z0;τ1, z1) := X
n0,r0,n1,r1
hSk−1(yD0,r0), Sk−1(yD1,r1)ie(n0τ0+r0z0 +n1τ1+r1z1) and
Lk,N(τ0, z0;τ1, z1) :=
dimS2kX−2(N)− i=1
L′(fi, k−1)
4πkφik2 φi(τ0, z0)φi(τ1, z1), (4.2) where the sum is taken over a set of normalized Hecke eigenforms fi ∈ S2k−2(N)− and φi ∈Jk,N are the corresponding Jacobi forms.
57 The Gross-Kohnen-Zagier formula (4.1) says that the Fourier coefficients of H2,N and L2,N ate(n0τ0+r0z0+n1τ1+r1z1) coincide if the discriminantsD0, D1 are fundamental and coprime. This formula is proved in [36] by showing that bothH (the value related to height pairing of two Heegner points) and L (the special value of L-function) are equal to a complicated “seventeen term” expression X. The idea of B. Gross and S. Kudla was show that H and L are both equal to E, the Fourier coefficients of a certain Siegel Eisenstein series. This idea is shown symbolically in the following picture:
HGross−Kohnenformula−ZagierL
X E
(4.3)
Denote by E2(3)(·;s) the non-holomorphic Siegel Eisenstein series of degree 3 and weight 2. The definition of Siegel Eisenstein series is given in Section 4.6. The upper half-space H(3) consists of symmetric 3×3 complex matrices
Z =
τ0 z z0
z τ1 z1
z0 z1 τ
!
with positive definite imaginary part.
The starting point of our research is the observation made many years ago by B. Gross and S. Kudla that
E2(3)(Z; 0)≡0, Z ∈H(3). (4.4) We will show that the single function of 6 variables
E∗(Z) := ∂
∂sE2(3)(Z;s)
s=0
encodes information about the height of Heegner points and Heegner cycles for all weights k and levels N.
The function E∗(Z) is naturally related to both sides of the Gross-Kohnen-Zagier formula. Firstly, using the Rankin-Selberg method one can find a connection between E∗(Z) and a derivatives of L-functions. This has been shown by T. Arakawa and B.
Heim [5]. Secondly, the height pairing of two divisors is defined as a sum of local heights for all primes including infinity (see Section 4.3). At the same time the Fourier coefficients ofE2(3)(·;s) can be written as a product of local densities (see Section 4.6). The miraculous identity (4.4) leads to a natural decomposition of the the Fourier coefficients of E∗ into a sum of local contributions (see Theorems 4.1 and 4.2). Thus, we hope to restore from the function E∗ not only the global height of Heegner points but also the local contribution for each place of Q. In this chapter we show that this is the case when the discrminants of these two Heegner points are fundamental and coprime.