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CATEGORIES:Number Theory Seminar
SUMMARY: Transcendental Brauer-Manin obstructions on Kumme
r surfaces - Rachel Newton (MPIM)
DTSTART;TZID=Europe/London:20150519T161500
DTEND;TZID=Europe/London:20150519T171500
UID:TALK58796AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/58796
DESCRIPTION: In 1970\, Manin observed that the Brauer group Br
(X) of a variety X over a number field K can obstr
uct the Hasse principle on X. In other words\, the
lack of a K-rational point on X despite the exist
ence of points everywhere locally is sometimes exp
lained by non-trivial elements in Br(X). Since Man
in's observation\, the Brauer group and the relate
d obstructions have been the subject of a great de
al of research. \n\nThe 'algebraic' part of the Br
auer group is the part which becomes trivial upon
base change to an algebraic closure of K. It is ge
nerally easier to handle than the remaining 'trans
cendental' part and a substantial portion of the l
iterature is devoted to its study. The transcenden
tal part of the Brauer group is generally more mys
terious\, but it is known to have arithmetic impor
tance – it can obstruct the Hasse principle and we
ak approximation. \n\nI will use class field theor
y together with results of Skorobogatov and Zarhin
to compute the transcendental part of the Brauer
group for certain Kummer surfaces related to produ
cts of elliptic curves with complex multiplication
. I will give examples where there is no Brauer-Ma
nin obstruction coming from the algebraic part of
the Brauer group but a transcendental Brauer class
causes a failure of weak approximation.
LOCATION:MR13
CONTACT:Jack Thorne
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