* 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.