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CATEGORIES:Number Theory Seminar
SUMMARY:Effective proof of the theorem of André on the com
plex multiplication points on curves - Yuri Bilu (
Bordeaux)
DTSTART;TZID=Europe/London:20140311T161500
DTEND;TZID=Europe/London:20140311T171500
UID:TALK49035AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/49035
DESCRIPTION: A complex multiplication point (hereinafter CM-po
int) on the complex affine plane C^2 is a point of
the form (j(a)\, j(b))\, where a and b are imagin
ary quadratic irrationalities and j denotes the mo
dular 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 cur
ve 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 Shi
mura varieties.\n\nLater several other proofs of t
he the Theorem of Andre were discovered\; mention
especially a remarkable proof by Plia\, which read
ily extends to the multidimensional case. But\, un
til recently\, all known proof of the Theorem of A
ndre 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-B
rauer inequality on the class number of an imagina
ry quadratic field\, which is known to be ineffect
ive.\n\nRecently Lars Kühne and others suggested t
wo new approaches to the Theorem of André\, which
are both effective. One approach uses the method o
f Baker and completely avoids the inequality Siege
l-Brauer. In the other approach\, the Siegel-Braue
r inequality is replaced by the "semi-effective" t
heorem of Siegel-Tatuzawa.\n\nIn my talk I will di
scuss these new approaches to the Theorem of André
.
LOCATION:MR13
CONTACT:James Newton
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