Height of Heegner Cycles and Derivatives of L-series

Qingchun Tian^{∗} Department of Mathematics Peking University Beijing 100871

1 Introduction

In [13], Gross and Zagier proved an identity on modular curves between the heights pairings of certain Heegner points and coefficients of certain cusp form of weight 2. As a consequence, they obtained a remarkable formula which relates the first derivative of the L-function of a modular form f of weight 2 on Γ_{0}(N)(over suitable imaginary quadratic field Q(\sqrt{D})) and the Néron-Tate height of a Heegner divisor y\sqrt{D}(over a suitable imaginary quadratic field) on the f-part of the Jacobian of the modular curve X_{0}(N). If w_{N} is the Fricke involution, X_{0}(N)^{∗} = X_{0}(N)/w_{N}, y_{D}^{∗} the image of y_{D} on X_{0}(N)^{∗}, f comes from a modular form with weight \frac{3}{2} under the Shimura lifting, by combining with Waldspurger [23]’s result, then Gross-Zagier formula say that the height of the f-component of Heegner point y_{D}^{∗} is equal, up to a simple factor, to L′(f, 1)c(D)^{2}, where c(D) is the D-th coefficient of the half-integer weight modular form associate to f. In [12], Gross Kohnen and Zagier generalized this relation to the height pairings of the f-component of Heegner divisors with different discrimint.

At the end of [12] [13], Gross and Zagier discussed the possible extension of their work to higher weight modular forms. They mentioned that Deligne has suggested some definitions for CM-cycles and their local pairings. The details of these definitions have been worked out by Brylinski[5]. Since the lack of the global theory of pairings, as well as the theory of self-pairing(ie, adjunction formula), the results of [12] and [13] are difficult to extend to higher weight modular forms by using the theory of Deligne and Brylinski. Recently Zhang[24] used Arakelov theory to extend Gross and Zagier[13]’s calculation to higher weight modular forms. More precisely, Zhang defined the (global) height pairings between CM-cycles in certain Kuga-Sato varieties, and established an identity between the height pairings of Heegner cycles and the coefficients of certain cusp forms of higher weights. Zhang [24] [25] also proved an analogue of Gross-Zagier’s formula for higher weight modular forms and conjecture of Belinson[2] and Bloch[4] by combining with the work of Nekovár[18] or Besser[3]. Nekovár and Besser generalized the Kolyvagin’s work to higher weight modular forms. With the assumption that the height pairing on CM-cycles is positively definite(which is a consequence of the index conjectures of Beilinson conjecture[2] [4] and Gillet-Soule[10]), Zhang also proved the positivity of derivatives of L-series for the modular forms with higher weight.

∗Supported in part by the NSC

In this paper, we will try to generalize Gross, Kohnen and Zagier’s computation to higher weights modular forms by using Zhang’s method. More precisely, we will compute the height pairings between CM cycles in certain Kuga-Sato variety and prove an identity (theorem(1.1)) between the height pairings and coefficients of certain cusp forms of higher weight which comes from Jacobian forms; our theorem(1.2) says that the height pairings of f-component of CMcycles with different discriminants is equal, up to a simple factor, to the product of derivatives of L-series with certain coefficients of Jacobian form associate to modular forms f, and the theorem(1.4) says that the Gross, Kohnen and Zagier’s algebraicity conjecture is right. With the assumption that the height pairing on on CM-cycles is positively definite, the theorem(1.3) says that the f-eigencomponents of Heegner cycle s_{k}(P_{D,r}) all lie a single line in (S_{k}(X^{∗})_{f} ⊗ R)_{f} and the subspace they generate is non-trivial precisely when L′(f, k) ≠ 0.

In [20], Perrin-Riou proved a p-adic version of Gross-Zagier’s formula; and Nékovar[19] has extended the Perrin-Riou’s work to higher weight modular forms.

In the following, we will describe our identity and its application.

Let N ≥ 1, k ≥ 1, D_{0} ≤ 0, D_{1} ≤ 0 be integers and r_{0}r_{1} = ρ(mod2N) with r_{i}^{2} = D_{i}(mod4N).Let Δ = \sqrt{D_{0}D_{1}}. Let S_{2k}(N)^{−} be the space of cusp forms of weight 2k on Γ_{0}(N) with eigenvalues −1 under the action of Fricke involution w_{N}. Let f = ∑_{n=1}^{∞} a(n)e^{2nπz} ∈ S_{2k}(N)^{−} be normalized newform. According to Skorruppa and Zagier[22], there is a non-zero Jacobian form ϕ = ∑ c(n, r)q^{n}ζ^{r} ∈ J_{2k,N}^{cusp} with the same eigenvalues as f under all the Hecke operators T_{m}, (m, N) = 1 and c(n, r_{0}) are real. Define the L-series associate to f by

L(f, s) = ∑_{n=1}^{∞} a(n)n^{−s}

Gross, Kohnen and Zagier[13] proved that L(f, k) = 0 and the linear functional

f ⟶ \frac{(2k − 3)!!N^{k−1}∥f∥^{2}}{2^{k+1}π^{k}Δ^{\frac{k−1}{2}} ∥ϕ∥^{2}} L′(f, k)c(n_{0}, r_{0})c(n_{1}, r_{1})

on the hermitian space S_{2k}(N)^{−} with Peterson product can be represented by an element F_{k,N,Δ,ρ,D_{0}} = ∑_{m≥1} a_{m}q^{m} ∈ S_{2k}(Γ_{0}(N)^{−}. When k = 1, in [12], Gross, Kohnen, Zagier constructed heegner divisors P_{D_{0},r_{o}} and P_{D_{1},r_{1}} on X(N) and proved that

< P_{D_{0},r_{0}}^{∗} − H(D_{0})∞^{∗}, T_{m}(P_{D_{1},r_{1}}^{∗} − H(D_{1})∞^{∗}) >= a_{m}

for (m, N) = 1, where <, > denotes Neron-Tate height pairing on Jacobian of X_{0}(N)/ω_{N}, H(D_{i}) is the class number of the field Q(\sqrt{D_{i}}).

The generalization of Gross, Kohnen and Zagier’s identity in this paper is as follows For each CM- point x^{∗} on X_{Q}^{∗} := X_{0}(N)_{Q[ζ_{N}]}/ω_{N}, we will construct an CM-cycle s_{k}(x^{∗}) on

certain Kuga-Sato variety Y^{∗} with codimension k, and these CM-cycle generated a subgroup of CH^{k}(Y^{∗})_{0}, we denote the subgroup by S_{k}(X^{∗}). We can also define the height pairing <, > on the group S_{k}(X^{∗}) of CM-cycles. We will prove that

Theorem 1.1 Let s_{k}(P_{D,r}^{∗}) = ∑_{x^{∗}∈P_{D,r}} s_{k}(x^{∗}), a_{m} the Fourier coefficient of F_{k,N,Δ,ρ,D_{0}}. Then we have

(D_{0}D_{1})^{\frac{k−1}{2}} < s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >= a_{m}  (1.1)

for (m, N) = 1.

Let V be the subspace of Heegner cycles generated by {T_{m}s_{k}(P_{D_{0},r_{0}}), T_{m}s_{k}(P_{D_{1},r_{1}})} for all (m, N) = 1, V′ be the quotient of V modulo the null subspace with respect to the height pairing, s′_{k}(P_{D_{0},r_{0}}) be the image of s_{k}(P_{D_{0},r_{0}}) in V′. The identity (1.1) will leads to a formula

(D_{0}D_{1})^{\frac{k−1}{2}} < s′_{k}(P_{D_{0},r_{0}}^{∗})_{f}, s′_{k}(P_{D_{1},r_{1}}^{∗})_{f} >= \frac{(f, F_{k,N,Δ,ρ,D_{0}})}{(f, f)}  (1.2)

where s′_{k}(P_{D,r})_{f} is the f-eigencomponents of s′_{k}(P_{D,r}^{∗}), f ∈ S_{2k}(N)^{−} = S_{2k}(Γ_{0}^{∗}(N)) a normalized newforms and <, > has been extended to S_{k}(X^{∗})×R by linearity. From (1.2) and [12], we obtain

Theorem 1.2 Let D_{0}, D_{1} ≤ 0 be the coprime fundamental discriminants, D_{i} = r_{i}^{2} − 4n_{i}N for i = 0, 1, f ∈ S_{2k}(Γ_{0}^{∗}(N)) a normalized newform and ϕ_{f} the Jacobi form associated to f. Then

< s′_{k}(P_{D_{0},r_{0}}^{∗})_{f}, s′_{k}(P_{D_{1},r_{1}}^{∗})_{f} >= \frac{(2k − 3)!!N^{k−1}L′(f, k)}{2^{k+1}π^{k}∥ϕ∥^{2}Δ^{\frac{k−1}{2}}}c(n_{0}, r_{0})c(n_{1}, r_{1}).  (1.3)

Where c(n_{i}, r_{i}) are the coefficients of ϕ_{f}.

On the other hand the main result of [24] implies that

< s′_{k}(P_{D_{0},r_{0}}^{∗})_{f}, s′_{k}(P_{D_{0},r_{0}}^{∗})_{f} >= \frac{(2k − 2)!N^{k−1}∥D_{0}∥^{1/2}}{2^{4k−1}π^{2k}∥f∥^{2}} L(f, D, k)L′(f, k)

, and by [12]’s Corollary 1 in Charp.2 ,

< s′_{k}(P_{D_{0},r_{0}}^{∗})_{f}, s′_{k}(P_{D_{0},r_{0}}^{∗})_{f} >= \frac{(2k − 3)!!N^{k−1}L′(f, k)}{2^{k+1}π^{k}∥ϕ∥^{2}∥D_{0}∥^{k−1}}C(n, r)^{2}(D^{2} = r^{2} − 4Nn)  (1.4)

If we assume further more that the height pairing on S_{k}(X^{∗}) is positively definite, then V = V′. eqs.(1.3) and (1.4) imply that s_{k}(P_{D_{1},r_{1}}^{∗})_{f} and s_{k}(P_{P_{1},r_{1}}^{∗})_{f} are collinear, and this gives the following theorem:

Theorem 1.3 With the assumption that the height pairing on S_{k}(X^{∗}) is positively definite. Let f ∈ S_{2k}(Γ_{0}^{∗}(N)) be a normalized newform. Then the Subspace of S_{k}(X(N)_{Q}^{∗}) ⊗ R generated by the f-eigencomponents of all Heegner cycles (s_{k}(P_{D,r}^{∗})) with (D, 2N) = 1 has dimension 1 if L′(f, k) ≠ 0 and 0 if L′(f, k) = 0. More precisely, s_{k}(D_{D,r}^{∗})_{f} = \frac{c(r^{2}−D/4N,r)}{|D|^{k−1}}s_{k}(P)_{f}, where c(n, r) is the coefficient of q^{n}ζ^{r} in Jacobi form ϕ_{f} ∈ J_{2k,N} and s_{k}(P)_{f} ∈ S_{2k}(Γ_{0}^{∗}(N)_{Q})⊗R is independent of D and r with L′(f, k) = \frac{2^{k+1}π^{k}∥ϕ_{f} ∥^{2}}{(2k−3)!!N^{k−1}} < s_{k}(P)_{f}, s_{k}(P)_{f} >

It would be interesting to discuss the relation between the formula L′(f, k) = \frac{2^{k+1}π^{k}∥ϕ_{f} ∥^{2}}{(2k−3)!!N^{k−1}} < s_{k}(P)_{f}, s_{k}(P)_{f} > and Bloch-Kato’s Tamagawa number conjecture by combining Jannsen[15] with Scholl[21] and Besser[3].

Algebraic conjecture. In [13] Gross-Zagier defined certain Green’s function G_{N,s,m}(x, y) on the complex points of X_{0}(N) for Res ≥ 1. Let

G_{N,s,m}(P_{D_{0},r_{0}}, P_{D_{1},r_{1}}) = ∑_{τ_{0}∈P_{D_{0},r_{0}},τ_{1}∈P_{D_{1},r_{1}}} G_{N,s,m}(τ_{0}, τ_{1})

G_{N,s,m}^{±}(P_{D_{0},r_{0}}, P_{D_{1},r_{1}}) = G_{N,s,m}(P_{D_{0},r_{0}}, P_{D_{1},r_{1}}) + G_{N,s,m}(P_{D_{0},r_{0}}, P_{D_{1},−r_{1}}).

If {λ_{m}}_{m≥1} is a finite collection of integers such that the Hecke correspondence T = ∑_{m≥1} λ_{m}T_{m} vanishes on the space S_{2k}^{new}(Γ(N))^{−} . For Heegner divisors P_{D_{0},r_{0}}, P_{D_{1},r_{1}}, in [12], Gross-Kohnen-Zagier conjectured that there is a rational number a and an algebraic number b_{τ_{0},τ_{1}} for τ_{0} ∈ P_{D_{0},r_{0}}, τ_{1} ∈ P_{D_{1},r_{1}} such that

ln a = (D_{0}D_{1})^{\frac{k−1}{2}} ∑_{m} λ_{m}G_{N,k,m}^{±}(P_{D_{0},r_{0}}, P_{D_{1},r_{1}})  (1.5)
ln b_{τ_{0},τ_{1}} = (D_{0}D_{1})^{\frac{k−1}{2}} ∑_{m} λ_{m}G_{N,k,m}^{±}(τ_{0}, τ_{1})  (1.6)

and they proved that the conjecture is true for k = 1. We will show Theorem 1.4 Assume that k ≥ 2 and T vanishes on S_{k}(X_{Q}^{∗}). Then there is a rational number a such that the (1.5) is true and there exists an algebraic number b_{τ_{0},τ_{1}} for τ_{0} ∈ P_{D_{0},r_{0}}, τ_{1} ∈ P_{D_{1},r_{1}} such that (1.6) is right.

The analogue of the question for D_{0} = D_{1}, r_{0} = r_{1} was discussed by Zhang [24].

Acknowledgments I would like to thank Prof. Shouwu Zhang, he introduced me the subject and shared his some ideas with me. I would like to thank Prof. Faltings, Prof U.Jannsen and Prof. Zagier for their useful discussions. The work was prepared and completed at Max-Planck-Institut für Mathematik in Bonn. I would like to express my gratitude to the institute for her hospitality.

2 Kuga-Sato Varieties and CM-Cycles

In this section, we will review some basic facts about Kuga-Sato varieties and CM-cycles.

2.1 Universal semi-stable elliptic curves

Let N ≥ 3 be an integer and ζ_{N} a primitive N-th root of the unity. Let X(N)_{Q(ζ_{N})} be the compactification of the moduli of elliptic curves E over Q[ζ_{N}]-scheme S, with a canonical full level N-structure, i.e. , an isomorphism of groups

ϕ : (Z/N)^{2} → E(N)

over S, such that the Weil pairing of ϕ(1, 0) and ϕ(0, 1) is ζ_{N}. Let E_{Q[ζ_{N}]}^{0} be the universal elliptic curve over the noncuspidal part X(N)_{Q[ζ_{N}]}^{0} of X(N)_{Q[ζ_{N}]}, and E(N)_{Q[ζ_{N}]} be the Kodaira-Neron minimal compactification of E^{0}(N)_{Q[ζ_{N}]}, which is a semistable elliptic curve over X(N)_{Q[ζ_{N}]} with N-polygons at cusps. Let π_{Q[ζ_{N}]} denote the structure morphism E(N)_{Q[ζ_{N}]} → X(N)_{Q[ζ_{N}]}. Following [7] and [16], S. Zhang[24] showed that if N is a product of two relatively prime integers N_{1} ≥ 3 and N_{2} ≥ 3, then the morphism π_{Q[ζ_{N}]} can be uniquely extended to a morphism π = π_{Z[ζ_{N}]} : E(N) → X(N) of regular, flat and projective Z[ζ_{N}]-schemes, such that π makes E(N) a semistable elliptic curves over X(N). In [24], Zhang also described the fibers of X(N) and E(N) over the closed point spec(k) → Spec(Z[ζ_{N}]). Let p be the character of k. If p |̸N, then X_{k} is smooth and the cuspidal divisor is reduced, so E_{k} is smooth. If p/N, then X(N) is the disjoint union of some smooth curves over k, with crossing at the supersingular points.

Let w_{N} be the Fricke involution of X(N)_{Q[ζ_{N}]}. By [16], w_{N} can be extended to X(N). Let X(N)^{∗∗} be the minimal regulization of X(N)/w_{N}, E(N)′ the pullback of X(N)^{∗} and E(N)/w_{N} over X(N)/w_{N}, at a cusp of X(N)^{∗}, the fiber of E(N)′ has several components. By blowing up E(N)′ along intersections of the components several times, we will obtain a scheme E(N)^{∗∗} which is regular, flat and projective Z[1/N]-scheme, and the morphism π^{∗∗} : E(N)^{∗∗} → X(N)^{∗∗} makes E(N)^{∗∗} a semistable elliptic curves over X(N)^{∗∗}. If N is a product of two relatively prime integers N_{1} andN_{2} which are great than three, by the same argument in the proof of theorem (2.1.1) in [24], we can glue π^{∗∗} to obtain a morphism π^{∗} : E(N)^{∗} → X(N)^{∗} of regular, flat and projective Z-schemes, the morphism makes E(N)^{∗} a semistable elliptic curves over X(N)^{∗} and extends the structure morphism π_{Q}^{∗} : E(N)/w_{N} → X_{Q}^{∗}. The fibers of X(N)^{∗} and E(N)^{∗} over the closed point Spec(k) → Spec(Z) can be also described as follows, let p be the characteristic of k . If p |̸N, then X_{k}^{∗} is smooth, and the cuspidal divisor is reduced, so E(N)_{k}^{∗} is smooth. If p/N, then X(N)_{k}^{∗} is the disjoint union of some smooth curves over k, with crossing at the supersingular points and the fibers of E(N)^{∗} over these smooth curves are semistable elliptic curves.

2.2 Kuga-Sato varieties

Let k be a positive integer ≥ 2, N the product of two relatively prime integers N_{1} ≥ 3 and N_{2} ≥ 3. Let E_{Q[ζ_{N}]}^{2k−2} (E^{∗}_{Q}^{2k−2}) be 2k − 2 tuple fiber product over X(N)(X(N)^{∗}) of E(N)(E(N)_{Q}^{∗}) with itself, and let Y (N)_{Q[ζ_{N}]}(Y (N)_{Q}^{∗}) be the canonical desingularization described in [6]. Let E_{Z[ζ_{N}]}^{2k−2} (E^{∗}_{Z}^{2k−2}) be 2k −2 tuple fiber product over X(N)(X(N)^{∗}) of E(N)(E(N)_{Z}^{∗}) with itself, and let Y (N)_{Z[ζ_{N}]}(Y (N)_{Z}^{∗}) be the regular resolution described in [24] 2.2. It is obvious that Y (Y^{∗}) is the regular model of Y_{Q[ζ_{N}]}(Y_{Q}^{∗}) over Z[ζ_{N}](Z).

We recall the geometric realization of Hecke operators. For an integer m ≥ 1 prime to N, let X(N, m) be the (reduced) curve over Q parameterizing elliptic curves E with a full level N structure C and an isaigeny λ : E → E′ of degree m. There are two finite maps s, t : X(N, m) → X(N) given by

s(E, C, λ) = (E, C), t(E, C, λ) = (E′, λ_{∗}(C))

Then we have the following commutative diagram

E^{2k−2} ⟵^{ϕ_{1}} E^{2k−2} ⊗_{X(N)^{s}} X(N, m) →^{ψ} E^{2r−2} ⊗_{X(N)^{t}} X(N, m) →^{ϕ_{2}} E^{2k−2}
↓  ↓  ↓  ↓
X(N) ⟵  X(N, m)  =  X(N, m)  → X(N)

in which ψ is induced by λ.

On Y,the m-th Hecke correspondence T_{m} is defined as the Zariski closure of the graph of

(ϕ_{1}, ϕ_{2}ψ) : E^{2k−2} ⊗_{X(N)^{s}} X(N, m) → E^{2k−2} × E^{2k−2}

which commutes with the Scholl projector ϵ in [21] and the Fricke involution w_{N}. So the T_{m} also defines a correspondence on Y^{∗}, we also denote this correspondence by T_{m}, and also call it by Hecke correspondence.

It is obvious that the construction of Scholl is also suited to Y^{∗} → X^{∗}, and the results are also right for this case. We still use symbol ϵ to denote the associated Scholl projector.

2.3 CM-cycles

2.3.1 Cycle s_{k}(E)

Let E be a CM elliptic curve over an integral ring R whose generic point has characteristic zero. The ring End_{R}(E) ⊗ Q which depends only on the isogeny class of E, is isomorphic to the imaginary quadratic extension. Fix an embedding τ_{E} : End_{S}(E) × Q → C, If \sqrt{D}(D < 0) ∈ End_{S}(E), let Γ\sqrt{D} be the graph of the multiplication by \sqrt{D}. One defines the divisor

Z(E, \sqrt{D}) = Γ_{E} × \{0\} − |D| \{0\} × E − E × \{0\} ∈ CH^{1}(E × E)

Now Z(E)^{k−1} defines a cycle of codimension k − 1 in E^{2k−2}. Let S_{2k−2} be the symmetric group which acts on E^{2k−2} by permuting the factors. We define

w_{k}(E, \sqrt{D}) = ∑_{g∈S_{2k−2}} sgn(g)g^{∗}(Z(E)^{k−1})

and s_{k}(E, \sqrt{D}) = cw_{k}(E, \sqrt{D})

where c is a positive constant such that the self intersection of s_{k}(E, \sqrt{D})_{η} in E_{η}^{2k−2} is (−1)^{k−1}. Zhang[24] also proved that the class [s_{k}(E, \sqrt{D})] of s_{k}(E, \sqrt{D}) in CH^{k−1}(E^{2k−2}) does not depend on the choice of \sqrt{D} when τ is fixed, if τ is changed to its complex conjugate, then the class [s_{k}(E)] is changed to (−1)^{k−1}[s_{k}(E)] in the Chow group. Thus we can simply denote s_{k}(E, \sqrt{D}) by s_{k}(E).

2.3.2 CM-cycles

Let N be a positive integer. Let R be an algebraic integer ring which is ramified over all the primes dividing N. Let x(x^{∗}) be an irreducible CM divisor in X(X^{∗}), by which we mean that x(x^{∗}) is flat over R and the corresponding elliptic curves pair has complex multiplication. We will recall the construction of s_{k}(x)(s_{k}(x^{∗})) as follows.

We first assume that N is the product of two relatively prime integers ≥ 3. Write E := E(N) ⊗ R, E^{∗} := E(N)^{∗} ⊗ R and Y := Y (N) ⊗ R, Y^{∗} := Y (N)^{∗} ⊗ R. So one has the following morphisms of schemes

\begin{matrix}X ← E^{2k−2} ← Y \\ X^{∗} ← E^{∗}^{2k−2} ← Y^{∗} \end{matrix}

For irreducible CM divisor x(x^{∗}) of X(X^{∗}), the corresponding elliptic curve E = E_{x}(E^{∗} = E^{∗}_{x^{∗}}) has a complex multiplication. Write s_{k}(x)(s_{k}(x^{∗})) for the class of s_{k}(E)(s_{k}(E^{∗})) in CH^{k}(Y) (CH^{k}(Y^{∗})), then we call s_{k}(x)(s_{k}(x^{∗}) the CM-Chow-cycle over x(X^{∗}).

We define the space of CM-cycles S_{k}(X)(S_{k}(X^{∗})) to be the subspace of CH^{k}(Y)(CH^{k}(Y^{∗})) generated by the classes of w_{k}(x)(w_{k}(x^{∗}) over all CM-divisors x(x^{∗}) Of X(X^{∗}). So s_{k}(x) ∈ s_{k}(X) ⊗ R, s_{k}(x^{∗}) ∈ s_{k}(X^{∗}) ⊗ R, and ±s_{k}(x)(±s_{k}(x^{∗})) depends only on x(x^{∗}). S.Zhang also proved the functoriality of CM-cycles. By the functoriality, Zhang also gave the definitions of s_{k}(x) and S_{k}(X) for N is not the product of two relative prime positive integers great than 3. Similarly we can define s_{k}(x^{∗}) and S_{k}(X^{∗}) for the general case. For the details, see Zhang[24]’s 2.4.4. In the following sections, we will use the same symbol s_{k}(x) to denote the cycle and its class in Chow group.

2.3.3 Action of Hecke operator.

Let N, m be a positive integers, (N, m) = 1 and x^{∗} an irreducible CM-divisor of X(N)^{∗}. Then one has

T_{m}(s_{k}(x^{∗})) = ∑_{i} m^{k−1}s_{k}(x_{i}^{∗})  (2.1)

if T_{m}x^{∗} = ∑ x_{i}^{∗}.

Following Zhang[24], it is obvious that the action of Δ_{2k−2} on S_{k}(X^{∗}) has character ϵ.

2.3.4 An Intersection Formula

In the above construction of s_{k}(E, \sqrt{D}), if R = k is of characteristic p ≠ 0 and k is algebraic closed field. E is a supsingularial elliptic curve over k, then End(E) ⊗ Q is a quaternion algebra over Q, not an imaginary quadratic field. For different α, β ∈ End(E), we can construct two Chow-cycles s_{k}(E, α) ∈ CH^{k−1}(E^{2k−2}), s_{k}(E, β) ∈ CH^{k−1}(E^{2k−2}) and they are usually different. By Nekovár[19], or Brylinsky[5], or Zhang[24], we can calculate their intersection number by the following formula

s_{k}(E, α)s_{k}(E, β) = P_{k−1}(\frac{tr(αβ)}{\sqrt{N(α)N(β)}})  (2.2)

where P_{k−1}(x) is the first kind Legrande Polynomial, N(α) (tr(α)) is the norm number (trace) of α in the quaternion algebra End_{k}(E) ⊗ Q. This can be obtained directly from Nekovár[19]’s proposition (3.4) also.

3 Heights for Heegner Cycles

In this chapter, we first review the idea of height pairing for CM-cycles defined by Zhang[24], then define the Heegner cycles s_{k}(P_{D,r}^{∗}) over X(N)^{∗} over Q, these are indexed by negative discriminants D of conductor prime to N and classes r (mod 2N) with r^{2} ≡ D (mod 4N). Their precise definition will be given in sect 2. Our main aim in this chapter is to compute the global height pairing < s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) > on S_{k}(X^{∗}) over Q. In the Height computation, we shall assume that D_{0} and D_{1} are relatively prime. With arithmetic intersection theory, we will express the pairing as a sum of local symbols < s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) >_{v} indexed by the places v of Q. We will compute the non-archimedean symbols using intersection theory on a regular model Y^{∗} over Z and the archimedean symbols using Green’s functions for the Riemann surface X(N)^{∗}.

3.1 Height of CM-cycles

In this section, we will recall some back facts on heights of CM-cycles from [24]. Zhang[24] defined the Néron-Tate type height pairing for S_{k}(X) by using arithmetic intersection theory[9]. This theory was developed by Arakelov[1] and Faltings[8].

Let N be a positive integer, F a number field in which all prime factors of N are unramified. Write X = X(N) ⊗ O_{F}, Zhang defined a height pairings on S_{k}(X_{F}). Let X^{∗} be the minimal desingularization of the quotient of X by the Fricke involution w_{N}. Similar to Zhang[]’s method, we can define a height pairings on S_{k}(X_{F}^{∗}). More preciously, If N is a product of two relatively prime integers ≥ 3, let x^{∗}, y^{∗} be two CM-divisor on X_{F}^{∗} . Let x̄^{∗}, ȳ^{∗} be their closure on X^{∗}. We obtain integer CM-cycles s_{k}(x̄^{∗}), if we let g_{k}(x^{∗}) be the green current for cycle s_{k}(x^{∗}) such that

(a) \frac{∂ ¯∂}{πi}g_{k}(x^{∗}) = δ_{s_{k}(x^{∗})} and that

(b) The integration ∫ g_{k}(x^{∗})η is 0 and any \frac{∂ ¯∂}{πi} closed form η on Y (C)^{∗}, such g_{k}(x^{∗}) is determined up to images of ∂ and ¯∂. We define the arithmetic CM-cycles over x as

ŝ_{k}(x^{∗}) = (s_{k}(x̄^{∗}), g_{k}(x^{∗}))

Now the height pairing of CM-cycles of s_{k}(x^{∗}) and s_{k}(y^{∗}) is

< s_{k}(x^{∗}), s_{k}(y^{∗}) >= (−1)^{k}ŝ_{k}(x̄^{∗}) · ŝ_{k}(ȳ^{∗}) >

If N is not the product of two relatively prime integers ≥ 3. We can also define the height pairing by using projection formula for arithmetic intersection. For the details, see [24].

Positivity. Similar to [24]’s 3.1.2, we can prove the positive definiteness of height pairing on the groups S_{k}(X^{∗}) of CM-cycles under the Arithmetic Hodge Index Conjecture of Gillet-Soule[10] or under the Height Pairing Conjecture of Beilinson[2] and Bloch[4].

Local decomposition. Let the notations and assumption be given as above, we can decompose the global height into local heights

< s_{k}(x^{∗}), s(y^{∗}) >= ∑_{p} < s_{k}(x^{∗}), s_{k}(y^{∗}) >_{p} E_{p}

where

E_{p} = \{ \begin{matrix}log p & if p is finite \\ 1 & if p is real \\ 2 & if p is complex \end{matrix} \}

where we can define

< s_{k}(x^{∗}), s_{k}(y^{∗}) >_{p}= (−1)^{k}(ŝ_{k}(x^{∗}), ŝ_{k}(y^{∗}))_{p}

If x^{∗} ≠ y^{∗}. If x^{∗} = y^{∗}, the cycle ŝ_{k}(x̄^{∗}) is supported in the fiber E^{2k−2} of Y^{∗} over x̄^{∗},where E is an elliptic curve corresponding to x̄^{∗}, since E^{2k−2} is an abelian variety and w_{k}(x̄^{∗}) is a sum of abelian varieties, By the []’s Chapter 1, Theorem 1.3.2, we can define local intersection number (ŝ_{k}(x^{∗}), ŝ_{k}(x^{∗}))_{p} by choosing the same local coordinate as Gross and Zagier[13], for the details see [24].

Since the local decomposition here is compatible with the morphism of modular curves, by the projective formula, one can even define local decomposition of heights even when N is not the product of two relatively prime integers ≥ 3.

3.2 Heegner Cycle

Let D be a negative discriminant which is a square (mod 4N) and has a conductor prime to N and let r be a class (mod 2N) with r^{2} ≡ D (mod 4N), Gross-Kohnen-Zagier[12] defined a rational divisor P_{D,r} on X_{0}(N) over K = Q(\sqrt{D}). Recall that the affine points of X_{0}(N) are given by x = (π : E → E′) where E and E′ are elliptic curves and π a cyclic N-isogeny. Let O_{D} = Z + Z \frac{r+\sqrt{D}}{2} be the ring of discriminant D in K and N the primitive ideal ZN + Z \frac{r+\sqrt{D}}{2} of index N. The point x in P_{D,r} are those which admit endmorphisms by O_{D}[i.e., O_{D} maps into End(E) and End(E′) with the obvious diagram commuting] and such that the kernel of π is annihilated by N. The P_{D,r} = ∑_{x∈P_{D,r}} \frac{1}{e}x, where e is the order of Aut(x)/\{±1\}. Then we can define

s_{k}(P_{D,r}) = ∑_{x∈P_{D,r}} \frac{1}{e}s_{k}(x)

which is a rational K-cycle of Y_{K}.

If we denote by c the non-trivial automorphism of K over Q, then (P_{D,r})^{c} = P_{D,−r} = w_{N}(P_{D,r}), and c induces a map on Y_{K}, we also denote it by c, then s_{k}(P_{D,r})^{c} = s_{k}(P_{D,−r}) = w_{N}(s_{k}(P_{D,r})). If we let P_{D,r}^{∗} be the image of P_{D,r} on the quotient curve X_{0}^{∗}(N) = X_{0}(N)/w_{N}, then P_{D,r}^{∗} is rational over Q and depends only on ±r (mod 2N), It is obvious that s_{k}(P_{P,r}^{∗}) is the image of s_{k}(P_{D,r}) on the quotient variety Y^{∗} = Y_{k}/w_{N}. and it is also rational Q-cycle. Finally, by [24]’s Proposition 2.4.2, and Gross-Kohnen-Zagier[]’s IV 1, we see that

T_{m}(s_{k}(P_{D,r})) = (m)^{k−1} ∑_{m=dd′}(\frac{D}{d′})s_{k}(P_{Dd^{2},rd})

for D fundamental and (m, N) = 1. Since T_{m} commutes with w_{N}, the same formula holds for s_{k}(P_{D,r}^{∗}). In the following, we will compute the globe height pairing of Heegner cycles on Y_{Q}^{∗} as a sum over places of Q, i.e.,

< s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) = ∑_{p} < s_{k}(P_{D_{0},r_{0}}^{∗}, s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}

for (D_{0}, D_{1}) = 1, r_{0} ≠ r_{1}.

3.3 The Contribution from the Finite Places

We fix a finite prime p of Q. To calculate the local height < s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p} in terms of intersection theory, we need regular model X^{∗} for X_{0}(N)^{∗}, and regular model Y^{∗} for Y (N)^{∗} over the valuation ring Z_{p} of Q_{p}. It is obvious that E_{Z_{p}}^{∗} = E^{∗} ⊗ Spec(Z_{p}) is the regular model of E_{Q} over the valuation ring Z_{p} of Q_{p}, X_{Z_{p}}^{∗} = X^{∗} ⊗ Spec(Z_{p}) be the regular model of X_{0}^{∗}(N)_{Q} over the valuation ring Z_{p} of Q_{p}, Y_{Z_{p}}^{∗} = Y^{∗} ⊗ Spec(Z_{p}) is the regular model of Y_{Q}^{∗} over the valuation ring Z_{p} of Q_{p}. When p |̸N, X_{Z_{p}}^{∗} is smooth over Z_{p} , E_{Z_{p}}^{∗} is smooth over X_{Z_{p}}^{∗}, and Y_{Z_{p}}^{∗} is smooth over Z_{p} and X_{Z_{p}}^{∗}. When p/N, the curve X^{∗} ⊗ Z/p may have several components, and over every component C of X^{∗} ⊗ Z/p, the fiber of E^{∗} over C is regular and semistable curve over C. Let P̄_{D,r}^{∗} be the closure of P_{D,r}^{∗} in X_{Z_{p}}^{∗}. By the definition of local height, one has

< s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}= (−1)^{k}(s_{k}(P̄_{D_{0},r_{0}}^{∗}) · s_{k}(P̄_{D_{1},r_{1}}^{∗}))_{p} log p.  (3.1)

Where (s_{k}(P̄_{D_{0},r_{0}}^{∗})·s_{k}(P̄_{D_{1},r_{1}}^{∗}))_{p} is the intersection number of two cycles (s_{k}(P̄_{D_{0},r_{0}}^{∗}) and (s_{k}(P̄_{D_{1},r_{1}}^{∗}) over Y_{Z_{p}}^{∗}. Now we turn to the calculation of the relevant intersection multiplicities. First we have

Proposition 3.1 If p/N, then (s_{k}(P̄_{D_{0},r_{0}}^{∗}) · s_{k}(P̄_{D_{1},r_{1}}^{∗}))_{p} = 0

Proof. We may assume that p |̸D_{0}, the points in P_{D_{0},r_{0}} have ordinary reduction on X. If p is split in Q(\sqrt{D_{1}}), the same is true for points in P_{D_{1},r_{1}}. By the Deuring’s theory[3], singular points with ordinary reduction and distinct quadratic fields of multiplication reduction reduce to distinct points (mod p) and they define different s_{k}(P̄_{D_{0},r_{0}}^{∗})_{p} and s_{k}(P̄_{D_{1},r_{1}})_{p} which have no intersection . If p is not split in Q(\sqrt{D_{1}}), the points in P_{D_{1},r_{1}} have supersingular reduction on X, they are disjoint from the points in the reduction of P_{D_{0},r_{0}} which is ordinary, so s_{k}(P̄_{D_{0},r_{0}}^{∗}) and s_{k}(P̄_{D_{1},r_{1}}^{∗}) have no intersection. Hence the intersection number is zero in all cases.

When p |̸N, X_{Z_{p}} is smooth over Z_{p}, E_{Z_{p}} is smooth over Z_{p} and smooth over X_{Z_{p}}, and X_{Z_{p}}^{∗} ≅ X_{Z_{p}}/w_{N}, E_{Z_{p}}^{∗} ≅ E_{Z_{p}}/w_{N}, and one has the following commutative diagram

(E_{Z_{p}})^{k} ⟶^{F_{E}} (E_{Z_{p}}^{∗})^{k}
↓  ↓
X_{Z_{p}}  ⟶^{F_{X}} X_{Z_{p}}^{∗}.

and

F_{E}_{∗}(s_{k}(P̄_{D,r})) = s_{k}(P̄_{D,r}^{∗})
F_{E}^{∗}(s_{k}(P̄_{D,r}^{∗})) = s_{k}(P̄_{D,r}) + s_{k}(P̄_{D,−r}).

Since D_{0} and D_{1} are relatively prime, we may re-order them such that p |̸D_{0}. Let W be the completion of the maximal unramified extension of Z_{p}, fix a square root \sqrt{D_{0}} of D_{0} in W. A Heegner point x = (ϕ : E → E′) lies in P_{D_{0},r_{0}} if O_{0} = Z + Z \frac{D_{0}+\sqrt{D_{0}}}{2} embeds into End(x) and α_{0} = \frac{r_{0}+\sqrt{D_{0}}}{2} ∈ End(x) annihilates ker(ϕ). Here and the follows, we will write \sqrt{D_{0}} for the endmorphism which gives multiplication by our fixed square root of D_{0} on the tangent space. The points in the P_{D_{0},r_{0}} are all rational over W, for x ∈ P_{D_{0},r_{0}}, the cycle s_{k}(x) is defined over W. A point y lies in the divisor P_{D_{1},r_{1}} + P_{D_{1},−r_{1}} if the order O_{1} of discriminant D_{1} act on y and the cyclic subgroup of order N in its diagram is annihilated by an element α_{1} ∈ O_{1} of trace r_{1}(hence also by −α_{1} of trace −r_{1}. The divisor P_{D_{0},r_{0}} + P_{D_{1},r_{1}} is always rational over W, so s_{k}(P_{D_{0},r_{0}}) + s_{k}(P_{D_{1},r_{1}}) is a rational W -cycle; s_{k}(y) are all rational W -cycle iff p |̸D_{1}. So one has

(s_{k}(P̄_{D_{0},r_{0}}^{∗}) · s_{k}(P̄_{D_{1},r_{1}}^{∗}))_{p} = (s_{k}(P̄_{D_{0},r_{0}})(s(P_{D_{1},r_{1}}) + s_{k}(P̄_{D_{1},−r_{1}}))_{W}  (3.2)

and we now turn to the intersection product on (E)^{k} over W.

First we assume that that p |̸D_{1}. Then we can fix a square root \sqrt{D_{1}} in W. A point y = (ϕ : F → F′) lies in the divisor P_{D_{1},r_{1}} over W if the endmorphism α_{1} = \frac{r_{1}+\sqrt{D_{1}}}{2} annihilates ker(ϕ). If (xy)_{W} > 0 for x ∈ P_{D_{0},r_{0}}, y ∈ P_{D_{1},r_{1}}, then our diagrams reduce to the same isogeny z ≡ (ϕ : E → E′) ≡ (ψ : F → F′) on X ⊗ W/pW. Write R = End_{W/pW} (z) the reduction of endmorphisms gives injections End(x) → R, End(y) → R. It follows from Deuring’s theory that z is a supersingular point on X ⊗ W/pW, and that R ⊗ Q is a quaternion algebra over Q ramified only at p and at infinity. By Gross-Kohnen-Zagier[12], we know that the embedings of End_{W} (x) and End_{W} (y) give elements \sqrt{D_{i}} and α_{i} = (r_{i} + \sqrt{D_{i}})/2(i = 0, 1) in R satisfy

\sqrt{D_{0}}\sqrt{D_{1}} + \sqrt{D_{1}}\sqrt{D_{0}} = 2n, for some n ∈ Z, n ≡ r_{0}r_{1} (mod 2N), n ≤ r_{0}r_{1}.

Lemma 3.1 Let x,y, n, N, D_{0}, D_{1}, r_{0}, r_{1}, R be as above and R^{∗} = {±1}, Δ = D_{0}D_{1}. Then

(s_{k}(x̄) · s_{k}(ȳ))_{W} = \frac{1}{2}ord_{p}(\frac{D_{0}D_{1} − 4n^{2}}{4N})P_{k−1}(\frac{n}{\sqrt{Δ}})

where P_{k}(t) is the first kind Legendre function.

Proof By [24]’s Proposition 3.3.1(a), one have,

(s_{k}(x̄) · s_{k}(ȳ))_{W} = \frac{(−1)^{k}}{2} ∑_{{n≥1 \atop ϕ∈Isom_{n}(x̄,ȳ)}}s_{k}(ȳ)_{0}ϕ^{∗}s_{k}(x̄)_{0}

where s_{k}(x̄)_{0}, s_{k}(ȳ)_{0} denote the reduction of s_{k}(x) and s_{k}(y) over Y ⊗W/pW. By the assumption and Gross-Kohnen-Zagier[12]’s calculation, we have

(s_{k}(x̄) · s_{k}(ȳ))_{W} = \frac{(−1)^{k}}{2} (ord_{p}(M) + 1)s_{k}(x̄)_{0}s_{k}(ȳ)_{0}

where M = \frac{D_{0}D_{1}−n^{2}}{4N}, s_{k}(x̄)_{0} = s_{k}(E_{z}, \sqrt{D_{0}}), s_{k}(ȳ) = s_{k}(E_{z}, \sqrt{D_{1}}), by the (2.2), one has

(s_{k}(x̄) · s_{k}(ȳ))_{W} = \frac{(−1)^{k−1}}{2} (ord_{p}(M) + 1)P_{k−1}(\frac{n}{\sqrt{D_{0}D_{1}}} ).

Conversely, by [12] , for every supsingular point z on X ⊗ W/pW and let R = End_{W/pW} (z), I ⊂ R the ideal annihilating the kernel of cyclic N-isogeny defined by z. Fix a binary quadratic form [D_{0}, 2n, D_{1}] = D_{0}x^{2} + 2nxy + D_{1}y^{2} with n^{2} ≤ D_{0}D_{1} with n^{2} ≤ D_{0}D_{1} and n ≡ r_{0}r_{1} (mod 2N). Let S = S_{[D_{0},n,D_{1}]} be the associated Cliford order and M = \frac{D_{0}D_{1}−n^{2}}{4N}. For each embedding ϕ : S_{[D_{0},n,D_{1}]} → R normalized by insisting that ϕ(e_{0}) = \sqrt{D_{0}} on the tangent space, write ϕ(e_{1}) = ϵ\sqrt{D_{1}} with ϵ = ±1, where e_{0}, e_{1} ∈ S_{[D_{0},n,D_{1}]} with e_{i}^{2} = D_{i} for i = 0, 1 and e_{0}e_{1} + e_{1}e_{0} = n. The idea I contains the elements \frac{s_{0}+\sqrt{D_{0}}}{2} and \frac{s_{1}+E\sqrt{D_{1}}}{2} for some integers s_{0},s_{1} (well defined modulo N), and s_{0}s_{1} ≡ n (mod 2N), and the induce embedding of O_{0} = Q(\sqrt{D_{0}}) and O_{1} = Q(\sqrt{D_{1}}) into R correspond to points x ∈ P_{D_{0},r_{0}}, y ∈ P_{D_{1},r_{1}} which reduce to z (mod p) and are congruent modulo p^{l}, where l = (ord_{p}(M) + 1)/2 . By the calculation of Gross-Kohnen-Zagier[12], one has,

(s_{k}(P̄_{D_{0},r_{0}})(s_{k}(P̄_{D_{1},r_{1}}) + s_{k}(P̄_{D_{1},−r_{1}}))_{W}
= \frac{1}{2^{t+1}} ∑_{{n≡r_{0}r_{1}(2N) \atop n^{2}≤D_{0}D_{1}}} ∑_{R_{i}} # \{S_{[D_{0},n,D_{1}]} → R_{i} (mod R^{∗}/±1)\} × \frac{ord_{p}(M)+1}{2}P_{k−1}(\frac{n}{\sqrt{Δ}})

where t is the number of prime divisor of N, # {S_{[D_{0},n,D_{1}]} → R_{i} (mod R)_{i}^{∗}/±1} is the order of set of equivalent class for the embedding S_{[D_{0},n,N_{1}]} → R_{i} under the act of R_{i}^{∗}/±1, R_{i} is right order of left ideal classes for a fixed Eichler order of index N in the quaternion algebra B(p) ramified at p and ∞. Similar to Gross-Kohnen-Zagier[12]’s discussion in page pp -pp , the formula is still true without the assumption that R_{i}^{∗} = {±1} and p |̸D_{1}. We now combine (3.1) with formula (3.2) to obtain the following formula for the local height pairing:

< s_{k}(P_{D_{0},r_{0}}), s_{k}(P_{D_{1},r_{1}}) >_{p} = ∑_{{n≡r_{0}r_{1} (mod 2N) \atop n^{2}≥D_{0}D_{1}}}2^{−t−1} ∑_{R_{i}} # \{ϕ : S → R_{i} (mod R_{i}^{∗}/±1)\}(3.3)
×\frac{ord_{p}(M) + 1}{2}P_{k−1}(\frac{n}{\sqrt{Δ}})(− log p)  (3.4)

For M = \frac{n^{2}−D_{0}D_{1}}{4N}, any p |̸NM, we define ϵ(p) as follows, let D be any integer prime to p which can be represented by q = D_{0}x^{2} +2nxy +D_{1}y^{2} and define ϵ(p) = (\frac{D}{p}). This is independent of the choice of D and equals to 1 if p/N. For any positive divisor d = ∏_{i} p_{i}^{a_{i}} of M, we let ϵ(d) = ∏_{i} ϵ(p_{i})^{a_{i}} and introduce the Dirichlet series

l(s) = ∑_{d/M} ϵ(d)d^{s}

Since the algebra B = S ⊗ Q is definite and ramifies at at least finite place. Hence l(0) = 0 and the derivative l′(0) = ∑_{d} ϵ(d) log d is nonzero precisely. In this case, Gross-Kohnen-Zagier showed in [12]’s Proposition 3 of Chap. 1, Sect.3, that

l′(0) = 2^{−t−1} ∑_{R_{i}} # \{S → R_{i}^{∗} (mod R)_{i}/±\} \frac{1}{2}(ord_{p}(M) + 1)(− log p).

Comparing this with our formula (3.3) for the local height, we obtain our final result: Theorem 3.1 The total non-archimedean contribution to the height pairing of Heegner cycles s_{k}(P_{D_{0},r_{0}}^{∗}) and s_{k}(P_{D_{1},r_{1}}^{∗}) is given by

∑ ; p prime < s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}= ∑_{{n≡r_{0}r_{1}(2N) \atop n^{2}≤D_{0}D_{1}}} ( ∑_{d/M} ϵ(d) log d)P_{k−1}(\frac{n}{\sqrt{D_{0}D_{1}}} )

3.4 Archimedean Contribution

In this section, we will, following Gross-Zagier[13] , Gross-Kohnen-Zagier[12] and Zhang[24], use Green function constructed in [13] to represent the archimedean height. We recall the definition as follows, for s ∈ C with Re(s) > 1 , define the point-pair invariant

g_{s}(z_{0}, z_{1}) = −Q_{s−1}(1 + \frac{∥z_{0} − z_{1}∥^{2}}{2Im(z_{0})Im(z_{1})})z_{0}, (z_{1} ∈ H, z_{0} ≠ z_{1}.)

Where Q_{s−1}(t) is the Legendre function of second kind. The Green function is defined by the convergent sum

G_{N,s}(z_{0}, z_{1}) = ∑_{γ∈Γ_{0}(N)} g_{s}(z_{0}, γz_{1})

Let G_{N,s}^{∗}(z_{0}, z_{1}) be the correspond function when Γ_{0}(N) is replaced by Γ_{0}^{∗}(N), we have

G_{N,s}^{∗} = G_{N,s}(z_{0}, z_{1}) + G_{N,s}(z_{0}, w_{N}z_{1})

Proposition 3.2 For any two CM-points x and y on X^{∗}, one has

< s_{k}(x), s_{k}(y) >= \frac{1}{2}G_{k}^{∗}(x, y)

In particular,

< s_{k}(P_{D_{0},r_{0}}), s_{k}(P_{D_{1},r_{1}} >= \frac{1}{2}G_{k}^{∗}(P_{D_{0},r_{0}}^{∗}, P_{D_{1},r_{1}}^{∗})

= \frac{1}{2}[G_{k}(P_{D_{0},r_{0}}, P_{D_{1},r_{1}}) + G_{k}(P_{D_{0},r_{0}}, P_{D_{1},−r_{1}})]

Proof Step 1. By the same argument as [24]’s 3.4.2, we can reduce to the case N ≥ 3.

Step 2. By the same argument as in [24]’s (3.4.3), one can construct a function H_{k}(z, z′) on X^{∗} × X^{∗} − diagram such that < s_{k}(x^{∗}), s_{k}(y^{∗}) >= H_{k}(x, y) for different CM-points x, y ∈ X^{∗}. By using the same method in Zhang[24] we can prove that H_{k}(x, y) satisfy

(1) H_{k}(z, z′) is killed by Δ − k(k − 1) operate on the first variable where Δ = t^{2}(\frac{∂^{2}}{∂s^{2}} + \frac{∂^{2}}{∂t^{2}} ) where z = s + ti

(2)H_{k}(z, z′) = log ∥z − z′∥ + O(1) as z → z′

(3) In a neighborhood of a cusp γ^{−1}∞, the function im(γz)^{k−1}H_{k}(z, z′) extends to a continuous function.

step 3. By the same argument as in [13] or [22]’s 3.4.4, one can prove that H_{k}(x, y) = \frac{1}{2}G_{k}(x, y).

4 Heights and L-Series

In this section we will prove our main theorems and their consequence. In the first paragraph, we will give the main identity. After assuming the existence of f- isotropic components of heegner cycles, we give several consequences about the derivatives of L-Series and the heights of Heegner cycles in paragraph 4.2.

4.1 Identity.

Let N ≥ 1 and Δ > 0 be a discriminant which is a product of two negative discriminant D_{0},D_{1}. D_{i} = r_{i}^{2} − 4Nn_{i}, for i = 0, 1. ρ is an integer (mod 2N) with ρ^{2} ≡ Δ (mod 4N). Let P_{D_{0},r_{0}}, P_{D_{1},r_{1}} be the Heegner divisor defined in 3.2, s_{k}(P_{D_{0},r_{0}}^{∗}), s_{k}(P_{D_{1},r_{1}}^{∗}) the associated Heegner cycles. Let S_{2k}(N)^{−} be the (−1)^{k−1}-eigenspace of w_{N} in the space of cusp forms of weight 2k, Jcusp_{k+1,N} the space of Jacobian cusp forms of weight k +1 and index N. By [22], for any normalized new forms f = ∑_{m} a_{m}q^{m} ∈ S_{2k}(N)^{−}, there is a non-zero Jacobian form

Φ_{f} = ∑_{{n,r∈Z \atop r^{2}≤4Nn}}c(n, r)q^{n}ζ^{r} ∈ Jcusp_{k+1,N}

which has the same eigenvalues as f under all Hecke operators T_{m}, (m, N) = 1 and c(n, r) are all real. Define the L-series associate to f by L(f, s) = ∑_{n≥1} a_{m}m^{−s}. Gross-Kohnen-Zagier[12] proved the following theorem:

Theorem 4.1 (1) L(f, k) = 0 (2) There is a holomorphic cusp form F_{k,N,Δ,ρ,D_{0}} = ∑_{m≥1} a(m)q^{m} ∈ S_{2k}(Γ(N))^{−} satisfies

(f, F_{k,N,Δ,ρ,D_{0}}) = \frac{(2k − 3)!!N^{k−1}c(n_{0}, r_{0})c(n_{1}, r_{1})}{2^{k+1}π^{k}∥ϕ∥}L′(f, k)

for all new form f ∈ Snew_{2k}(Γ_{0}(N))^{−} and with

a(m) = (m\sqrt{Δ})^{k−1}( ∑_{{|n|<m\sqrt{Δ} \atop n≡mρ(2N)}}σ′_{χ}((\frac{n+m\sqrt{Δ}}{2})N^{−1})P_{k−1}(\frac{n}{m\sqrt{Δ}}) −

∑_{{n>m\sqrt{Δ} \atop n≡mρ(2N)}}σ_{0,χ}((\frac{n+m\sqrt{Δ}}{2})N^{−1})Q_{k−1}(\frac{n}{m\sqrt{Δ}}) − ∑_{{n>m\sqrt{Δ} \atop n≡−mρ(2N)}}σ_{0,χ}((\frac{n+m\sqrt{Δ}}{2})N′^{−1})Q_{k−1}(\frac{n}{m\sqrt{Δ}}))

where c(n_{i}, r_{i}) are the coefficients of Jacobi forms Φ_{f}, σ′_{0,χ}(a) and σ_{χ}(a) for a an integral ideal of Q(\sqrt{Δ}) denote the value and derivative respectively of

σ_{s,χ}(a) = ∑_{b|a} χ(b)N(b)^{s}  (4.1)

at s = 0, and P_{k−1}(t) and Q_{k−1}(t) are Legendre functions of first and second kinds, N = ZN + Z(\frac{ρ+\sqrt{Δ}}{2}) ⊂ O_{K}.

In the following, we will give an arithmetic interpretation of Fourier coefficients of F_{k,N,Δ,ρ,D_{0}} for k > 1

Theorem 4.2 < s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >= (\sqrt{Δ})^{−k+1}a(m)

proof. Step 1, In fact one has the following fact: For any decomposition m = m_{0}m_{1} with m_{0} and m_{1} coprime, (m_{1}, D_{0}) = 1, (m_{0}, D_{1}) = 1 and any place p of Q, one has

< s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}= ∑_{{m_{0}=d_{0}d′_{0} \atop m_{1}=d_{1}d′_{1}}}(\frac{D_{0}}{d′_{0}} )(\frac{D_{1}}{d′_{1}} ) < s_{k}(P_{D_{0}d_{0}^{2},r_{0}d_{0}}^{∗}), s_{k}(P_{D_{1}d_{1}^{2},r_{1}d_{1}}^{∗}) >_{p}

In fact, for any decomposition m = m_{0}m_{1} as above, one have

< s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}=< T_{m_{0}}s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m_{1}}s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}

This follows that T_{m} = T_{m_{0}}T_{m_{1}} and T_{m_{0}} is adjoint with respect to the local height pairing.

Step 2. Non-archimedean parts of the height pairing. First we recall a notation in [12]. For integers M, and all of whose prime factors p satisfy (\frac{Δm^{2}}{p}) ≠ −1, we define l(M, s) = ∑_{d/M} ϵ(d)d^{s} , where ϵ is the multiplicative function on such integers given on primes p with (\frac{Δm^{2}}{p}) ≠ −1 by ϵ(p) = (\frac{D_{i}}{p}) if p |̸D_{i}m, By the theorem 3.1 and Step 1, one has

∑_{p} < s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >_{p}

= m^{k−1} ∑_{{m_{0}=d_{0}d′_{0}, \atop m_{1}=d_{1}d′_{1}}}(\frac{D_{0}}{d′_{0}} )(\frac{D_{1}}{d′_{1}} ) ∑_{{x<m\sqrt{Δ} \atop n≡ρΔ (mod 2N)}}l′(\frac{Δd_{0}^{2}d_{1}^{2} − n^{2}}{4N} , 0)P_{k−1}(\frac{n}{m\sqrt{Δ}})

= m^{k−1} ∑_{{n^{2}<m^{2}Δ \atop n≡mρ(2N)}} ∑_{{d′_{0}/(n,m_{0}) \atop d′_{1}/(n,m_{1})}}(\frac{D_{0}}{d′_{0}} )(\frac{D_{1}}{d′_{1}} )l′(\frac{Δd_{0}^{2}d_{1}^{2} − n^{2}}{4N} , 0)P_{k−1}(\frac{n}{m\sqrt{Δ}})

= m^{k−1} ∑_{{n^{2}<m^{2}Δ \atop n≡mρ(2N)}}σ′_{χ}((\frac{n + m\sqrt{Δ}}{2})N^{−1})P_{k−1}(\frac{n}{m\sqrt{Δ}})

The last step is followed from [12], Chap V’s Lemma.

Step 3. The archimedean part of the height pairing.By the Step 1 and theorem(3.2) one has

< s_{k}(P_{D_{0},r_{0}}^{∗}),  T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >_{∞}= ∑_{{m=d_{0}d′_{0}, \atop m_{1}=d_{1}d′_{1}}}(\frac{D_{0}}{d′_{0}} )(\frac{D_{1}}{d′_{1}} ) < s_{k}(P_{D_{0}d_{0}^{2},r_{0}d_{0}}^{∗}), s_{k}(P_{D_{1}d_{1}^{2},r_{1}d_{1}}^{∗}) >_{∞}

= m^{k−1} ∑_{{m=d_{0}d′_{0}, \atop m_{1}=d_{1}d′_{1}}}(\frac{D_{0}}{d′_{0}} )(\frac{D_{1}}{d′_{1}} )(G_{N,k}(P_{D_{0}d_{0}^{2},r_{0}d_{0}}, P_{D_{1}d_{1}^{2},r_{1}d_{1}}) + G_{N,k}(P_{D_{0}d_{0}^{2},r_{0}d_{0}}, P_{D_{1}d_{1}^{2},−r_{1}d_{1}}))

By Gross-Kohnen-Zagier[12] chapter IV ’s proposition 2 and chap V ’s formula (6), one has

< s_{k}(P_{D_{0},r_{0}}^{∗}), T_{m}s_{k}(P_{D_{1},r_{1}}^{∗}) >_{∞}=

− ∑_{{n>m\sqrt{Δ} \atop n≡mρ(2N)}}σ_{0,χ}((\frac{n+m\sqrt{Δ}}{2})N^{−1})Q_{k−1}(\frac{n}{m\sqrt{Δ}}) − ∑_{{n>m\sqrt{Δ} \atop n≡−mρ(2N)}}σ_{0,χ}((\frac{n+m\sqrt{Δ}}{2})N^{−1})P_{k−1}(\frac{n}{m\sqrt{Δ}})

Combining step 2 and step 3, we complete the proof of the theorem.

4.2 Consequence

Before we turn to the consequences, we first give a property about the actions of Hecke correspondences on cusp forms and CM-cycles which is analogous to theorem 1 in [25].

Proposition 4.1 Let v_{1}, v_{2}, ·, v_{h} ∈ S_{k}(X^{∗}). Assume for each i, j there is an element g_{i,j} ∈ S_{2k}(Γ(N))^{−} such that for any positive integer m prime to N, the number , v_{i}, T_{m}v_{j} > is the m-th coefficient of g_{i,j}. Let V be the subspace of Heegner cycles generated by {T_{m}v_{j}, j = 1, ·, h} for all (m, N) = 1. Let V′ be the quotient of V modulo the null subspace with respect to the height pairing. Then the Hecke module V′ is isomorphic to a sub-quotient module of (S_{2k}(Γ_{0}(N))^{−})^{+h}.

Let D_{i} = r_{i}^{2} − 4n_{i}N < 0(i = 0, 1) be coprime fundamental discriminants and write s(p_{i}^{∗}) for s_{k}(P_{D_{i},r_{i}}) ∈ S_{k}(X^{∗}). With the assumption that the Hecke algebra on S_{k}(X^{∗}) should be the the quotient of the Hecke algebra over S_{2k}(N)^{−}, the power series

G(z) = ∑_{m=1}^{∞} < s_{k}(P_{0}^{∗}), T_{m}s_{k}(P_{1}^{∗}) > q^{m}(q^{m} = e^{2πi})

defines a modular form in S_{2k}(Γ_{0}^{∗}(N)) = S_{2k}(N)^{−}. By the formal argument given in section 5 in [24] or on P306 of [13], the theorem (4.2) says that the coefficients of q^{m} in F and G agree for all m prime to N . In particular F −G is an oldform , so (G, f) = (F, f), for any normalized newform f in S_{2k}(N)^{−}. Applying theorem (4.2) and proposition (4.1) to cycles {s_{k}(P_{0}), s_{k}(P_{1})}, we have that V′ is a finite dimensional T_{k} algebra. Let s′_{k}(P_{0}) be the image of s_{k}(P_{0}) in V′. The arguments on P308 of [9] or [25] show that (G, f) = (f, f) times the height pairing < s′_{k}(P_{0}^{∗})_{f}, s′_{k}(P_{1}^{∗})_{f} >. Where s_{k}(P_{i}^{∗})_{f} is the f-eigencomponent of s_{k}(P_{i}^{∗}) on S_{k}(X^{∗}) × R. Hence

< s′_{k}(P_{0}^{∗})_{f}, s′_{k}(P_{1}^{∗})_{f} >= \frac{1}{∥f∥^{2}} (F, f)

Let Φ_{f} = ∑ c(n, r)q^{n}ζ^{r} be the Jacobi form associated to f. By the theorems in the section 3 and section 4 of ChapterII in [12], one has

< s′_{k}(P_{0}^{∗})_{f}, s′_{k}(P_{1}^{∗})_{f} >= \frac{(2k − 3)!!N^{k−1}}{2^{k+1}π^{k}∥Φ_{f}∥Δ^{\frac{k−1}{2}}} L′(f, k)c(n_{0}, r_{0})c(n_{1}, r_{1})

so we obtain the following theorem.

Theorem 4.3 Let D_{0}, D_{1} < 0 be coprime fundamental discriminants, D_{i} ≡ r_{i}^{2} (mod 4N), and f ∈ S_{2k}(Γ_{0}^{∗}(N)) a normalized newform. Then

< s′_{k}(P_{0}^{∗})_{f}, s′_{k}(P_{1}^{∗})_{f} >= \frac{(2k − 3)!!N^{k−1}}{2^{k+1}π^{k}∥Φ_{f}∥Δ^{\frac{k−1}{2}}} L′(f, k)c(n_{0}, r_{0})c(n_{1}, r_{1})

By S.Zhang[24], we have

< s′_{k}(P_{0}^{∗}), s′_{k}(P_{0}^{∗}) >= \frac{(2k − 2)!|D_{0}|^{\frac{1}{2}}}{2^{4k−1}π^{2k}∥f∥^{2}} L(f, D_{0}, k)L′(f, k).

[where L(f, D_{0}, s) denotes the twist of L(f, s) by (\frac{D_{0}}{·})] By ChapterII, section 3 Corollary 1 of [12]

\frac{c(n_{0}, r_{0})^{2}}{∥Φ∥^{2}} = \frac{(k − 1)!|D_{0}|^{k−1/2}}{2^{2k−1}π^{k}N^{k−1}} · \frac{L(f, D_{0}, k)}{∥f∥^{2}},

one gets

< s′_{k}(P_{0}^{∗})_{f}, s′_{k}(P_{0}^{∗})_{f} >= \frac{(2k − 3)!!N^{k−1}}{2^{k+1}π^{k}∥Φ_{f}∥^{2}} L′(f, k)c(n_{0}, r_{0})^{2}

If we let w_{k}(P^{∗}) = (D)^{\frac{k−1}{2}} s_{k}(P^{∗}), L = \frac{(2k−3)!!N^{k−1}}{2^{k+1}π^{k}∥Φ_{f} ∥^{2}} L′(f, k), then,

< w′_{k}(P_{0}^{∗})_{f}, w′_{k}(P_{0}^{∗})_{f} >= c_{0}^{2}L, < w′_{k}(P_{1}^{∗}), w′_{k}(P_{1}^{∗}) >= c_{1}^{2}L
< w_{k}(P_{0}^{∗})_{f}, w′_{k}(P_{1}^{∗})_{f} >= c_{0}c_{1}L

With the assumption that the height pairing on S_{k}(X^{∗}) ⊗ R is positive definite, then V′ = V, the above formulas imply that c_{1}w_{k}(P_{0}^{∗}) − c_{0}w_{k}(P_{1}^{∗}) has height 0, hence s_{k}(P_{0}^{∗})_{f} and s_{k}(P_{1}^{∗})_{f} is colinear. In order to obtain theorem (1.3), we need the same statement when D_{0} and D_{1} are not necessary coprime. But this follows, since by lemma 3.2 of [22] we can choose a fundamental discriminants D_{2} = r_{2}^{2} − 4n_{2}N prime to D_{0}, D_{1} and 2N with c_{2} = c(n_{2}, r_{2}) ≠ 0, and then (w_{k}(P_{0}^{∗}))_{f} = \frac{c_{0}}{c_{2}} (w_{k}(P_{2}^{∗}))_{f}, (w_{k}(P_{1}^{∗}))_{f} = \frac{c_{1}}{c_{2}} (w_{k}(P_{2}^{∗}))_{f}. By the special case of coprime discriminants, we deduce that c_{i}^{−1}(w_{k}(P_{i}^{∗}))_{f} is independent of i(for c_{i} ≠ 0), i.e., there exists a vector y_{f} ∈ S_{k}(X^{∗}) ⊗ R)_{f} with height < y_{f}, y_{f} >= L such that

S_{k}(P_{D,r}^{∗}) = \frac{c(n, r)}{∥D∥^{k−1}} y_{f}, (r^{2} − 4nN = D)  (4.2)

for all fundamental discrimint D prime to 2N, this is Theorem (1.3).

Similar to [13]’s argument, (4.1) is still true for D not fundamental, and D not prime to 2N. If one knows (4.1) for all D and r, one could restate it in the form

∑_{{n,r∈Z \atop r^{2}<4Nn}}(4Nn − r^{2})^{\frac{k−1}{2}} (s_{k}(P_{r^{2}−4Nn,r})_{f}q^{n}ζ^{r} = ϕ_{f}(τ, z)y_{f}  (4.3)

Summing this over all new forms would say that the power series ∑_{n,r} D^{\frac{k−1}{2}} (s_{k}(P_{D,r})_{new}q^{n}ζ^{r} belongs to S_{k}(X^{∗}) ⊗ J_{2k,N}^{cusp,new}, where D = r^{2} − 4Nn. Presumably the same relation would hold for the old parts by induction on N, this would give the following theorem which can be considered as an analogy of the theorem in [14]. Theorem 4.4 With the assumption that the the height pairing on S_{k}(X^{∗})⊗R is positive definite,

then the formal power series ∑_{{r,n∈Z, \atop r^{2}−4Nn<0}}(4nN − r^{2})^{k−1/2}s_{k}(P_{D,r}^{∗})q^{n}ζ^{r} ∈ S_{k}(X^{∗}) ⊗ J_{2k,N}^{cusp} with non

vanishing f-component if and only if L′(f, k) ≠ 0

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