Heights and the special values of \(L\)-series. (English) Zbl 0623.10019

Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 115-187 (1987).
[For the entire collection see Zbl 0611.00005.]
The author uses ideas similar to those that he successfully exploited in his well known joint work with D. Zagier [Heegner points and derivatives of \(L\)-series, Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)] to give another approach to Waldspurger’s results on central critical values of \(L\)-series. More precisely, let \(N\) be a prime number and let \(f\) be a weight two cusp form on \(\Gamma_ 0(N)\). Let \(K\) be an imaginary quadratic field of discriminant \(-D\) in which \(N\) is inert, and let \(A\) be a fixed ideal class of the integers of \(K\). The main result is an identity relating the special value at \(s=1\) of the \(L\)-series \(L(f,A,s)\) to the heights of so-called special points of discriminant \(-D\). The proof of the identity closely follows the proof of the Gross-Zagier formula. The author uses Rankin’s method and explicit computation to obtain a formula for \(L(f,A,1)\). He then computes the height pairing on the special points, compares the result with his previous formula, and thus obtains the desired identity.
Reviewer: S.Kamienny


11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
11R52 Quaternion and other division algebras: arithmetic, zeta functions
11F11 Holomorphic modular forms of integral weight
14H52 Elliptic curves