## Heegner points on $$X_ 0(N)$$.(English)Zbl 0559.14011

Modular forms, Symp. Durham/Engl. 1983, 87-105 (1984).
[For the entire collection see Zbl 0546.00010.]
Let $$Y_ 0(N)$$ be the open modular curve that parametrizes pairs $$(E,E')$$ of elliptic curves together with a cyclic isogeny $$g: E\to E'$$ of degree $$N$$. The Heegner points of $$Y_ 0(N)(\mathbb C)$$ are those pairs $$(E,E')$$ which have the same ring $$\mathfrak O$$ of complex multiplications. Here $$\mathfrak O$$ is an order in a complex quadratic field. The author discusses in this paper a general conjecture relating the height of a Heegner divisor and the first derivative at $$s=1$$ of an $$L$$-series associated to an automorphic form on $$\text{PGL}(2)\times \text{GL}(2)$$. This identity has been established by the author and D. Zagier in many cases. The author finishes by presenting a general program for work on other modular curves.

### MSC:

 11G05 Elliptic curves over global fields 11G16 Elliptic and modular units 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Zbl 0546.00010