CM-points on straight lines and hyperbolas

Abstract.
A CM-point is a point on the complex affine plane C^2 whose both coordinates are j-invariants of elliptic curves with complex multiplications. Yves Andre proved in 1998 that, with "obvious" exceptions, a plane algebraic curve can have only finitely many CM-points. This was the first non-trivial case of the celebrated conjecture of Andre-Oort.
Recently, in a series of joint articles with B. Allombert, F. Luca, D. Masser and A. Pizarro much more explicit results were obtained for plane curves of small degree like straight lines and hyperbolas. In my talk I will give the exact statement of the theorem of Andre and will address this more recent progress.