BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Effective proof of the theorem of André on the complex multiplica
 tion points on curves - Yuri Bilu (Bordeaux)
DTSTART:20140311T161500Z
DTEND:20140311T171500Z
UID:TALK49035@talks.cam.ac.uk
CONTACT:James Newton
DESCRIPTION: 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 an
 d b are imaginary quadratic irrationalities and j denotes the modular inva
 riant. 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.\n\nLater several other proofs of the the Theorem of Andre were
  discovered\; mention especially a remarkable proof by Plia\, which readil
 y extends to the multidimensional case. But\, until recently\, all known p
 roof of the Theorem of Andre were ineffective\; that is\, they did not all
 ow\, 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 imagi
 nary quadratic field\, which is known to be ineffective.\n\nRecently 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 comple
 tely avoids the inequality Siegel-Brauer. In the other approach\, the Sieg
 el-Brauer inequality is replaced by the "semi-effective" theorem of Siegel
 -Tatuzawa.\n\nIn my talk I will discuss these new approaches to the Theore
 m of André. 
LOCATION:MR13
END:VEVENT
END:VCALENDAR