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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.

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 |