×

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

Citations:

Zbl 0546.00010