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Heegner points and derivatives of \(L\)-series. (English) Zbl 0608.14019
The authors give a very beautiful and important relation between the heights of Heegner points on the Jacobian \(J\) of the modular curve \(X_ 0(N)\) and the first derivative at \(s=1\) of Rankin \(L\)-series of certain modular forms.
Let \(K\) be an imaginary quadratic field with discriminant \(D\) prime to \(2N\), and assume that \(D\) is congruent to a square modulo \(4N\). Let \(x\) be a Heegner point of discriminant \(D\) on \(X_ 0(N)\), i.e. \(x\) is the image in \(\Gamma_ 0(N)\setminus \mathfrak H\subset X_ 0(N)(\mathbb C)\) of a number \(\tau\) in the upper half-plane \(\mathfrak H\) which satisfies a quadratic equation \(a\tau^ 2+b\tau +c=0\), (\(a,b,c\in\mathbb Z\), \(a>0\), \(N\mid a\), \(b^ 2-4ac=D\)). Then \(x\) is defined over the Hilbert class field \(H\) of \(K\). Let \(c\) be the class of the divisor \((x)-(\infty)\) in \(J\). Let \(S_ 2(N)\) be the space of cusp forms of weight \(2\) on \(\Gamma_ 0(N)\), and let \(f(z)= \sum_{n\geq 1}a(n) e^{2\pi inz}\) be an element in the subspace of newforms \(S_ 2^{\text{new}}(N)\). If \(\sigma\in G\), the Galois group of \(H\) over \(K\), then under the Artin map of class field theory \(\sigma\) corresponds to an ideal class \(A\) of \(K\), and we define \[ L_{\sigma}(f,s)=\sum_{n\geq 1,\quad (n,DN)=1}\varepsilon (n)n^{1-2s}\;\cdot \sum_{n\geq 1}a(n)r_ A(n)n^{-s}\;(Re(s)>\frac{3}{2}), \] where \(\varepsilon\) is the quadratic character of \(K/\mathbb Q\) and \(r_ A(n)\) is the number of integral ideals in \(A\) with norm \(n\). If \(\chi\) is a complex character of \(G\) and \(f\) is a normalized Hecke eigenform, we put \(L(f,\chi,s)=\sum_{\sigma \in G}\chi (\sigma)L_{\sigma}(f,s)\). One can show ((5.5) proposition) that the \(L\)-series \(L_{\sigma}(f,s)\) and \(L(f,\chi,s)\) have holomorphic continuations to \(\mathbb C\), satisfy functional equations under \(s\mapsto 2-s\) and vanish at \(s=1\). Let \[ c_{\chi}=\sum_{\sigma \in G}\chi^{-1}(\sigma) c^{\sigma}\in (J(H)\otimes\mathbb C)_{\chi} \] and denote by \(c_{\chi,f}\) the \(f\)-isotypical component of \(c_{\chi}\) with respect to the action of the Hecke algebra on \(J(H)\otimes\mathbb C\). Let \(\langle\, ,\,\rangle\) be the global height pairing of \(J\) over \(H\) and write \((\cdot,\cdot)\) for the Petersson product on \(S_ 2(N)\).
Main result: (i) The function \(g_{\sigma}(z)= \sum_{m\geq 1} \langle c, T_ mc^{\sigma}\rangle e^{2\pi imz}\) (\(T_ m=\) Hecke operator), which is in \(S_ 2(N)\), satisfies \[ (f,g_{\sigma})=u^ 2 \sqrt{| D|} L_{\sigma}'(f,1)/8\pi^ 2 \] for all \(f\in S_ 2^{\text{new}}(N)\). Here \(u\) is half the number of roots of unity in \(K\).
(ii) The formula \(L'(f,\chi,1)=8\pi^ 2(f,f) \langle c_{\chi,f}, c_{\chi,f}\rangle/hu^ 2 \sqrt{| D|}\) holds, where \(h\) is the class number of \(K\).
For the proof, the height pairing \(\langle c,T_ mc^{\sigma}\rangle\) is computed by means of the theory of local symbols due to Néron, and one ends up with a complicated expression involving many transcendental terms. On the other hand, by means of Rankin’s method and the theory of holomorphic projection one constructs a cusp form \(\phi_{\sigma}\in S_ 2(N)\) with the property \[ (f,\phi_{\sigma})= \sqrt{| D|} L_{\sigma}'(f,1)/8\pi^ 2 \] (for all \(f\in S_ 2^{\text{new}}(N))\), computes the Fourier coefficients \(a_{m,\sigma}\) of \(\phi_{\sigma}\) and finds (!) that \(u^ 2a_{m,\sigma}\) for \(m\geq 1\), \((m,N)=1\) is equal to the expression giving \(\langle c,T_ mc^{\sigma}\rangle\). The assertions (i) and (ii) then follow easily.
Among the important corollaries to (ii) we mention only two:
1. Application to elliptic curves: Let \(E/\mathbb Q\) be an elliptic curve and assume that \(E\) is modular of level \(N\) (i.e. \(E\) occurs as a \(\mathbb Q\)-isogeny factor of \(J\)) so that the Hasse-Weil zeta function of \(E/\mathbb Q\) has a holomorphic continuation to \(s=1\). (According to the conjecture of Shimura-Taniyama-Weil, every elliptic curve \(E/\mathbb Q\) should be modular of level \(N\) for some \(N\).) Suppose \(\text{ord}_{s=1}L(E/\mathbb Q,s)=1\). Then (in combination with a non-vanishing result for \(L\)-series at the central point due to J.-L. Waldspurger [”Correspondances de Shimura et quaternions” (preprint))] it follows from (ii) that the Mordell-Weil group \(E(\mathbb Q)\) has a point of infinite order. This result is in accordance with the conjecture of Birch and Swinnerton-Dyer, which under the above hypothesis predicts \(\text{rank}_{\mathbb Z}E(\mathbb Q)=1\).
2. Application to the class number problem of Gauss: From (ii) the authors deduce the existence of a modular elliptic curve \(E/\mathbb Q\) with \(\text{ord}_{s=1}L(E/\mathbb Q,s)=3\). Combining this with the earlier work of D. M. Goldfeld [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 3, 623–663 (1976; Zbl 0345.12007)] one obtains: Let \(d\) be the discriminant of an imaginary quadratic field and \(h(d)\) be the class number. Then for every \(\varepsilon >0\) there is an effectively computable constant \(\kappa (\varepsilon)>0\) such that \(h(d)>\kappa (\varepsilon)(\log | d|)^{1- \varepsilon}\). Using the refinement of Goldfeld’s method due to J. Oesterlé [Sémin. Bourbaki, 36e année, Vol. 1983/84, Exp. No. 631, Astérisque 121/122, 309–323 (1985; Zbl 0551.12003)] one obtains, in particular, that \(h(d)>(\log | d|)/55\) for \(d\) prime.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F11 Holomorphic modular forms of integral weight
11G05 Elliptic curves over global fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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