An effective result of André-Oort type

Abstract

Using transcendence theory we prove the André-Oort conjecture in case of the Shimura variety $\mathbb{A}_{\mathbb{C}}^{2}$. It is well known that this result implies the André-Oort conjecture for a product of two arbitrary modular curves. In contrast to all previous proofs we obtain a result that is at once effective and unconditional.