Gross, B.; Kohnen, W.; Zagier, Don Heegner points and derivatives of \(L\)-series. II. (English) Zbl 0641.14013 Math. Ann. 278, 497-562 (1987). [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 Cited in 16 ReviewsCited in 184 Documents 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 Keywords:derivatives of \(L\)-series; modular curve; Jacobian; Heegner divisor; conjectures of Birch and Swinnerton-Dyer Citations:Zbl 0608.14019; Zbl 0651.10020 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Abramowitz, N., Stegun, I.: Handbook of mathematical functions. New York: Dover 1965 · Zbl 0171.38503 [2] Cohen, H.: A lifting of modular forms in one variable to Hilbert modular forms in two variables. In: Modular functions of one variable. VI. Serre, J.-P., Zagier, D. (eds.), 175-196. Lect. Notes. Math. 627. 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