Special points on curves in products of modular curves

In the talk we report on a recent work on the André–Oort conjecture in the special case that the Shimura variety is a product of modular curves. André has shown that there are only finitely many special points. In a recent Annals paper, Kühne has given an alternative proof which is effective. We shall explain how the result can be obtained in a more systematic and therefore more elegant way. Linear forms in logarithms and the analytic subgroup theorem are the main ingredients for the proof. Our approach opens the view for trying to deal also with more general Shimura varieties.