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SUMMARY: Transcendental Brauer-Manin obstructions on Kummer surfaces - Rac
 hel Newton (MPIM)
DTSTART:20150519T151500Z
DTEND:20150519T161500Z
UID:TALK58796@talks.cam.ac.uk
CONTACT:Jack Thorne
DESCRIPTION: In 1970\, Manin observed that the Brauer group Br(X) of a var
 iety X over a number field K can obstruct the Hasse principle on X. In oth
 er words\, the lack of a K-rational point on X despite the existence of po
 ints everywhere locally is sometimes explained by non-trivial elements in 
 Br(X). Since Manin's observation\, the Brauer group and the related obstru
 ctions have been the subject of a great deal of research. \n\nThe 'algebra
 ic' part of the Brauer group is the part which becomes trivial upon base c
 hange to an algebraic closure of K. It is generally easier to handle than 
 the remaining 'transcendental' part and a substantial portion of the liter
 ature is devoted to its study. The transcendental part of the Brauer group
  is generally more mysterious\, but it is known to have arithmetic importa
 nce – it can obstruct the Hasse principle and weak approximation. \n\nI 
 will use class field theory together with results of Skorobogatov and Zarh
 in to compute the transcendental part of the Brauer group for certain Kumm
 er surfaces related to products of elliptic curves with complex multiplica
 tion. I will give examples where there is no Brauer-Manin obstruction comi
 ng from the algebraic part of the Brauer group but a transcendental Brauer
  class causes a failure of weak approximation.
LOCATION:MR13
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