An effective “Theorem of André” for <i>CM</i>-points on a plane curve

Abstract

Abstract It is a well known result of Y. André (a basic special case of the André-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM -invariants is either a horizontal or vertical line, or a modular curve Y 0 ( n ). André's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X 1 + X 2 =1 or the hyperbola X 1 X 2 =1 it may be shown quite quickly that there are no imaginary quadratic τ 1 ,τ 2 with j (τ 1 )+ j (τ 2 )=1 or j (τ 1 ) j (τ 2 )=1, where j is the classical modular function.