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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)
##### Citations:
Zbl 0345.12007; Zbl 0551.12003
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