Special points on products of modular curves

Abstract

We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the generalized Riemann hypothesis (GRH). More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of ℂⁿ 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 Σ are special subvarieties, that is, are determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant (Th. 1.3) is proved unconditionally.