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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

##### MSC:
 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