  We prove the Andre-Oort conjecture on special points of Shimura varieties for
arbitrary products of modular curves, assuming the Generalized Riemann
Hypothesis. More explicitly, this means the following. Let n be a positive
integer, and let S be a subset of C^n (with C the complex numbers) consisting
of points all of whose coordinates are j-invariants of elliptic curves with
complex multiplications. Then we prove (under GRH) that the irreducible
components of the Zariski closure of S are ``special subvarieties'', i.e.,
determined by isogeny conditions on coordinates and pairs of coordinates. A
weaker variant is proved unconditionally.
