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Heegner points and derivatives of \(L\)-series. II. (English) Zbl 0641.14013
[For part I see B. H. Gross and D. B. Zagier, Invent. Math. 84, 225–320 (1986; Zbl 0608.14019).]
Let \(X_ 0(N)\) be the modular curve of level \(N\) and \(J_ N\) its Jacobian, and denote by \(J^*_ N\) the Jacobian of \(X_ 0(N)/w_ N\) where \(w_ N\) is the Fricke involution. For a fundamental discriminant \(D<0\) and \(r\in {\mathbb Z}\) with \(D\equiv r^ 2 (4N)\) we let \(Y_{D,r}\) be the corresponding Heegner divisor in \(J_ N\) and \(Y^*_{D,r}\) be its image in \(J^*_ N\). Let \(f\) be a normalized newform in \(S_ 2(\Gamma_ 0(N))\) and \(L(f,s)\) be its \(L\)-series. Suppose that the root number of \(L(f,s)\) is \(-1\).
The main result of the paper then states that the subspace of \(J^*_ N({\mathbb Q})\otimes {\mathbb R}\) generated by the \(f\)-eigencomponents of all Heegner divisors \((y^*_{D,r})_ f\) with \((D,2N)=1\) has dimension \(1\) if \(L'(f,1)\neq 0\).
More precisely, \((y^*_{D,r})_ f=c((r^ 2- D)/4N,r)y_ f\), where \(c(n,r)\) is the coefficient of \(e^{2\pi i(n\tau +rz)}\) in a Jacobi form \(\phi_ f\) of weight 2 and index \(N\) corresponding to \(f\) in the sense of N.-P. Skoruppa and D. Zagier [Invent. Math. 94, 113–146 (1988; Zbl 0651.10020)] and \(y_ f\in (J^*({\mathbb Q})\otimes {\mathbb R})_ f\) is independent of \(D\) and \(r\) with \(\langle y_ f,y_ f\rangle =L'(f,1)/4\pi \| \phi_ f\|^ 2\) \((\langle\cdot, \cdot\rangle\)=canonical height pairing).
This result is in accordance with the conjectures of Birch and Swinnerton-Dyer which in the above situation (i.e. under the assumption \(\text{ord}_{s=1}L(f,s)=1)\) would predict that \(\dim (J^*_ N({\mathbb Q})\otimes {\mathbb R})_ f=1\).
Reviewer: W.Kohnen

MSC:
14H25 Arithmetic ground fields for curves
11F50 Jacobi forms
11G18 Arithmetic aspects of modular and Shimura varieties
14H40 Jacobians, Prym varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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