Heegner points and the modular curve of prime level (Q1091439)

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 \textit{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.