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SUMMARY:Evaluating the wild Brauer group  - Rachel Newton (King's College 
 London)
DTSTART:20211123T143000Z
DTEND:20211123T153000Z
UID:TALK162208@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION:The local-global approach to the study of rational points on v
 arieties over number fields begins by embedding the set of rational points
  on a variety X into the set of its adelic points. The Brauer--Manin pairi
 ng cuts out a subset of the adelic points\, called the Brauer--Manin set\,
  that contains the rational points. If the set of adelic points is non-emp
 ty but the Brauer--Manin set is empty then we say there's a Brauer--Manin 
 obstruction to the existence of rational points on X. Computing the Brauer
 -Manin pairing involves evaluating elements of the Brauer group of X at lo
 cal points. If an element of the Brauer group has order coprime to p\, the
 n its evaluation at a p-adic point factors via reduction of the point modu
 lo p. For p-torsion elements this is no longer the case: in order to compu
 te the evaluation map one must know the point to a higher p-adic precision
 . Classifying p-torsion Brauer group elements according to the precision r
 equired to evaluate them at p-adic points gives a filtration which we desc
 ribe using work of Bloch and Kato. Applications of our work include addres
 sing Swinnerton-Dyer's question about which places can play a role in the 
 Brauer-Manin obstruction. This is joint work with Martin Bright.
LOCATION:MR13
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