Help Privacy Policy Disclaimer
  Advanced SearchBrowse




Journal Article

Heegner points and derivatives of L-series.


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

External Resource
No external resources are shared
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)
There are no public fulltexts stored in PuRe
Supplementary Material (public)
There is no public supplementary material available

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
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.