Heights and L-series. (English) Zbl 0665.14010

Gross, Benedict H.

Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 425-433 (1987).

[For the entire collection see Zbl 0657.00005.]
This paper is a summary of the author’s address to the 1986 International Congress of Mathematicians. It describes a limit formula - due to the author and D. Zagier - which connects heights of special divisor classes on modular curves and the behavior of certain Rankin L-series in the center of their critical strips. This limit formula has important arithmetic consequences for e.g., the conjectures of Birch and Swinnerton-Dyer for elliptic curves over \({\mathbb{Q}}\).

Reviewer: D.Gross

MSC:

14H25	Arithmetic ground fields for curves
11F03	Modular and automorphic functions
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52	Elliptic curves

Keywords:

heights; divisor classes; modular curves; Rankin L-series; conjectures of Birch and Swinnerton-Dyer

Citations:

Zbl 0657.00005