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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.

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)
Full Text: DOI EuDML
[1] Atkin, A.O.L., Lehner, J.: Hecke operators on? 0(m). Math. Ann.185, 134-160 (1970) · Zbl 0185.15502
[2] Birch, B.J.: Elliptic curves and modular functions. Symp. Math. Ist. Alta Mat.4, 27-32 (1970) · Zbl 0225.14016
[3] Birch, B.J.: Heegner points of elliptic curves. Symp. Math.15, 441-445 (1975) · Zbl 0317.14015
[4] Birch, B.J., Stephens, N.: Heegner’s construction of points on the curvey 2=x3-1728 e2. Séminaire de Théorie des Nombres, Paris 1981-82, Progress in Math.38, 1-19. Boston: Birkhäuser 1983
[5] Birch, B.J., Stephens, N.: Computation of Heegner points. In: Modular forms (ed. R.A. Rankin), pp. 13-41. Chichester: Ellis Horwood 1984 · Zbl 0559.14010
[6] Deligne, P.: Valeurs de fonctionsL et périodes d’intégrales. Symp. Pure Math. A.M.S.33, II, 313-346 (1979)
[7] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable II (ed P. Deligne, W. Kuyk), Lect. Notes Math.349, 143-316. Berlin-Heidelberg-New York: Springer 1973
[8] Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamb.14, 197-272 (1941) · Zbl 0025.02003
[9] Drinfeld, V.G.: Elliptic modules (Russian). Math. Sbornik94, 596-627 (1974). English translation: Math. USSR, Sbornik23, (4), 1973
[10] Eichler, M.: Lectures on modular correspondences. Lect. Notes of the Tata Institute9 (1956), Bombay · Zbl 0073.26501
[11] Eichler, M.: Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math.5, 355-366 (1954) · Zbl 0059.03804
[12] Goldfeld, D.: The class numbers of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. Sc. Norm. Super. Pisa3, 623-663 (1976) · Zbl 0345.12007
[13] Gross, B.: Heegner points onX 0 (N). In: Modular Forms (ed. R.A. Rankin), pp. 87-106. Chichester: Ellis Horwood 1984
[14] Gross, B.: Local heights on curves. (To appear in Proceedings of the Conference on Arithmetic Algebraic Geometry, Storrs. Springer-Verlag) · Zbl 0605.14027
[15] Gross, B.: On canonical and quasi-canonical liftings. Invent Math.84, 321-326 · Zbl 0597.14044
[16] Gross, B.: Heights and the special values ofL-series. (To appear in Conference Proceedings of the CMS Vol. 7 (1986). AMS publication)
[17] Gross, B., Zagier, D.: Points de Heegner et dérivées de fonctionsL. C. R. Acad. Sci. Paris297 85-87 (1983) · Zbl 0538.14023
[18] Gross, B., Zagier, D.: On singular moduli. J. Reine Angew. Math.355, 191-220 (1985) · Zbl 0545.10015
[19] Hecke, E.: Analytische Arithmetik der positiven quadratischen Formen. In: Mathematische Werke, pp. 789-918. Göttingen: Vandenhoeck und Ruprecht 1959
[20] Hejhal, D.: The Selberg trace formula forPSL (2, ?). Lect. Notes Math.1001, Berlin-Heidelberg-New York-Tokyo: Springer 1983 · Zbl 0543.10020
[21] Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. Ann. Math. Stud.108, Princeton: University Press 1985 · Zbl 0576.14026
[22] Kramer, K.: Arithmetic of elliptic curves upon quadratic extension. Trans. Am. Math. Soc.264, 121-135 (1981) · Zbl 0471.14020
[23] Lang, S.: Elliptic functions. Reading: Addison-Wesley 1973 · Zbl 0316.14001
[24] Lang, S.: Les formes bilinéaires de Néron et Tate. Sém. Bourbaki No.274 (1964) · Zbl 0138.42101
[25] Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves. Invent. Math.25, 1-61 (1974) · Zbl 0281.14016
[26] Mestre, J.-F.: Courbes de Weil de conducteur, 5077. C.R. Acad. Sc. Paris300, 509-512 (1985) · Zbl 0589.14026
[27] Néron, A.: Quasi-fonctions et hauteurs sur les variétés abéliennes. Ann. Math.82, 249-331 (1965) · Zbl 0163.15205
[28] Oesterlé, J.: Nombres de classes des corps quadratiques imaginaires. Sém. Bourbaki No.631 (1984)
[29] Serre, J.-P.: Complex multiplication. In: Algebraic number theory (ed. J.W.S. Cassels, A. Fröhlich), pp. 292-296. London-New York: Academic Press 1967
[30] Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math.88, 492-517 (1968) · Zbl 0172.46101
[31] Serre, J.-P., Tate, J. (Mimeographed notes from the AMS Summer Institute at Woods Hole, 1964)
[32] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58-159 (1967) · Zbl 0204.07201
[33] Sturm, J.: Projections ofC ? automorphic forms. Bull. Am Math. Soc.2, 435-439 (1980) · Zbl 0433.10013
[34] Swinnerton-Dyer, H.P.F., Birch, B.J.: Elliptic curves and modular functions. In: Modular functions of one variable IV (ed. B.J. Birch, W. Kuyk) Lect Notes Math.476, pp. 2-32, Berlin-Heidelberg-New York: Springer 1975 · Zbl 1214.11081
[35] Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018
[36] Vignéras, M.-F.: Valeur au centre de symétrie des fonctionsL associées aux formes modulaires. Séminare de Théorie des Nombres. Paris 1979-80, Progress in Math. 12, pp. 331-356, Boston: Birkhäuser 1981
[37] Waldspurger, J.-L.: Correspondances de Shimura et quaternions. (Preprint) · Zbl 0567.10020
[38] Weil, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.168, 149-156 (1967) · Zbl 0158.08601
[39] Zagier, D.:L-series of elliptic curves, the Birch-Swinnerton-Dyer conjecture, and the class number problem of Gauss. Notices Am. Math. Soc.31, 739-743 (1984) · Zbl 0547.14014
[40] Drinfeld, V.G.: Coverings ofp-adic symmetric regions. Funct. Anal. Appl.10, 29-40 (1976)
[41] Manin, Y.: Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR6 (1972), Am. Math. Soc. translations 19-64 · Zbl 0243.14008
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