  We prove, assuming the Generalized Riemann Hypothesis for imaginary quadratic
fields, that irreducible curves in the product of two modular curves that
contain infinitely many complex multiplication points are either a Hecke
correspondence or a fibre for one of the two projections. This gives evidence
for a conjecture of Oort that says that irreducible components of the Zariski
closure of a set of CM points in a Shimura variety are sub Shimura varieties.
