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Heegner points and derivatives of L-series.

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Zagier,  Don B.
Max Planck Institute for Mathematics, Max Planck Society;

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Gross, B. H., & Zagier, D. B. (1986). Heegner points and derivatives of L-series. Inventiones Mathematicae, 84, 225-320.


Cite as: https://hdl.handle.net/21.11116/0000-0004-3962-E
Abstract
The authors give a very beautiful and important relation between the heights of Heegner points on the Jacobian J of the modular curve X\sb 0(N) and the first derivative at s=1 of Rankin L-series of certain modular forms. \par 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\sb 0(N), i.e. x is the image in Γ\sb 0(N)\setminus \frak H\subset X\sb 0(N)(\bbfC) of a number τ in the upper half-plane \frak H which satisfies a quadratic equation aτ\sp 2+bτ +c=0, (a,b,c\in \bbfZ, agt;0, N \vert a, b\sp 2-4ac=D). Then x is defined over the Hilbert class field H of K. Let c be the class of the divisor (x)-(∞) in J. Let S\sb 2(N) be the space of cusp forms of weight\sp 2 on Γ\sb 0(N), and let f(z)= \sum\sbn≥ 1a(n) e\sp2π inz be an element in the subspace of newforms S\sb 2\spnew(N). If σ\in G, the Galois group of H over K, then under the Artin map of class field theory σ corresponds to an ideal class A of K, and we define L\sbσ(f,s)=\sum\sbn≥ 1,\ (n,DN)=1ε (n)n\sp1-2s\ ⋅ \sum\sbn≥ 1a(n)r\sb A(n)n\sp-s\ (Re(s)gt;\frac32), where ε is the quadratic character of K/\bbfQ and r\sb A(n) is the number of integral ideals in A with norm n. If χ is a complex character of G and f is a normalized Hecke eigenform, we put L(f,χ,s)=\sum\sbσ \in Gχ (σ)L\sbσ(f,s). One can show ((5.5) proposition) that the L-series L\sbσ(f,s) and L(f,χ,s) have holomorphic continuations to \bbfC, satisfy functional equations under s\mapsto 2-s and vanish at s=1. Let c\sbχ=\sum\sbσ \in Gχ\sp-1(σ) c\spσ\in (J(H)\otimes \bbfC)\sbχ and denote by c\sbχ,f the f- isotypical component of c\sbχ with respect to the action of the Hecke algebra on J(H)\otimes \bbfC. Let lt;\quad,\quad gt; be the global height pairing of J over H and write (.,.) for the Petersson product on S\sb 2(N). \par Main result: (i) The function g\sbσ(z)= \sum\sbm≥ 1 lt;c,T\sb mc\spσgt; e\sp2π imz (T\sb m= Hecke operator), which is in S\sb 2(N), satisfies (f,g\sbσ)=u\sp 2 \sqrt\vert D\vert L\sbσ'(f,1)/8π\sp 2 for all f\in S\sb 2\spnew(N). Here u is half the number of roots of unity in K. (ii)\quad The formula L'(f,χ,1)=8π\sp 2(f,f)lt;c\sbχ,f,c\sbχ,fgt;/hu\sp 2 \sqrt\vert D\vert\quad holds, where h is the class number of K. - For the proof, the height pairing lt;c,T\sb mc\spσgt; 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 φ\sbσ\in S\sb 2(N) with the property (f,φ\sbσ)= \sqrt\vert D\vert L\sbσ'(f,1)/8π\sp 2 (for all f\in S\sb 2\spnew(N)), computes the Fourier coefficients a\sbm,σ of φ\sbσ and finds (!) that u\sp 2a\sbm,σ for m≥ 1, (m,N)=1 is equal to the expression giving lt;c,T\sb mc\spσgt;. The assertions (i) and (ii) then follow easily. \par Among the important corollaries to (ii) we mention only two: \par (1.) Application to elliptic curves: Let E/\bbfQ be an elliptic curve and assume that E is modular of level N (i.e. E occurs as a \bbfQ- isogeny factor of J) so that the Hasse-Weil zeta function of E/\bbfQ has a holomorphic continuation to s=1. (According to the conjecture of Shimura-Taniyama-Weil, every elliptic curve E/\bbfQ should be modular of level N for some N.) Suppose ord\sbs=1L(E/\bbfQ,s)=1. Then (in combination with a non-vanishing result for L-series at the central point due to \it J.-L. Waldspurger ["Correspondances de Shimura et quaternions" (preprint))] it follows from (ii) that the Mordell-Weil group E(\bbfQ) 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 rank\sb\bbfZE(\bbfQ)=1. \par (2.) Application to the class number problem of Gauß: From (ii) the authors deduce the existence of a modular elliptic curve E/\bbfQ with ord\sbs=1L(E/\bbfQ,s)=3. Combining this with the earlier work of \it 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 ε gt;0 there is an \it effectively computable constant κ (ε)gt;0 such that h(d)gt;κ (ε)(\log \vert d\vert)\sp1- ε. Using the refinement of Goldfeld's method due to \it J. Oesterlé [Sémin. Bourbaki, 36e année, Vol. 1983/84, Exposé 631, Astérisque 121/122, 309-323 (1985; Zbl 0551.12003)] one obtains, in particular, that h(d)gt;(\log \vert d\vert)/55 for d prime.