Abstract

A complex multiplication point (hereinafter CM-point) on the complex affine plane C^2 is a point of the form (j(a), j(b)), where a and b are imaginary quadratic irrationalities and j denotes the modular invariant. In 1998, Yves André proved that the irreducible plane curve f(x,y)=0 can contain only finitely many CM-points, except when the curve is a horizontal or vertical line, or a modular curve. It was the first proven case of the famous André-Oort hypothesis about special points on Shimura varieties.

Later several other proofs of the the Theorem of Andre were discovered; mention especially a remarkable proof by Plia, which readily extends to the multidimensional case. But, until recently, all known proof of the Theorem of Andre were ineffective; that is, they did not allow, in principle, to determine all CM-points on the curve. This was due to the use of the Siegel-Brauer inequality on the class number of an imaginary quadratic field, which is known to be ineffective.

Recently Lars Kühne and others suggested two new approaches to the Theorem of André, which are both effective. One approach uses the method of Baker and completely avoids the inequality Siegel-Brauer. In the other approach, the Siegel-Brauer inequality is replaced by the “semi-effective” theorem of Siegel-Tatuzawa.

In my talk I will discuss these new approaches to the Theorem of André.