Heegner points and the modular curve of prime level. (English) Zbl 0623.14010

Gross, Benedict H.

J. Math. Soc. Japan 39, 345-362 (1987).

Let N be a prime and let \(Y=\Gamma_ 0(N)\setminus {\mathcal H}\) be the classifying space of N-isogenies \(x=(\phi: E\to E')\) of elliptic curves E, E’ over \({\mathbb{C}}\). The compactification X of Y obtained by adjoining two cusps is a projective curve defined over \({\mathbb{Q}}\). Specific points x on this modular curve X, the so-called Heegner points, have played an important role for arithmetic questions in connection with the conjectures of Birch and Swinnerton-Dyer. (See the joint work of D. Zagier and the author.) A Heegner point is simply given by an isogeny \(\phi: E\to E'\) where \(End(E)=End(E')={\mathcal O}\) is an order of conductor prime to N in an imaginary quadratic field. The purpose of the present article is to show that Heegner points also supply a powerful tool to study the geometry of X. So for instance the classical model for X in \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) given by the N-th modular polynomial, is shown to have only ordinary double points as singularities. This result had been obtained by Dwork using p-adic methods. For \(N\equiv 3 (4)\) a certain fibre system of elliptic curves over X is identified with an elliptic modular surface defined previously by Shioda. This answers a question of T. Shioda in J. Math. Soc. Japan 24, 20-59 (1972; Zbl 0226.14013). Furthermore the fibres over certain Heegner points are related to \({\mathbb{Q}}\)-curves.

Reviewer: C.-G.Schmidt

MSC:

14H15	Families, moduli of curves (analytic)
14G05	Rational points
14K22	Complex multiplication and abelian varieties
11F03	Modular and automorphic functions
14H25	Arithmetic ground fields for curves
14H52	Elliptic curves
14H45	Special algebraic curves and curves of low genus
11G15	Complex multiplication and moduli of abelian varieties

Keywords:

modular curve; Heegner points; isogeny; elliptic modular surface

Citations:

Zbl 0226.14013