Special Points on the Product of Two Modular Curves

Abstract

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C², containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m ≥ 1such that X is the image, under the usual map, of the modular curve Y₂0(m). The proof uses some number theory and some topological arguments.