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CATEGORIES:Number Theory Seminar
SUMMARY:Evaluating the wild Brauer group - Rachel Newton
(King's College London)
DTSTART;TZID=Europe/London:20211123T143000
DTEND;TZID=Europe/London:20211123T153000
UID:TALK162208AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/162208
DESCRIPTION:The local-global approach to the study of rational
points on varieties over number fields begins by
embedding the set of rational points on a variety
X into the set of its adelic points. The Brauer--M
anin pairing cuts out a subset of the adelic point
s\, called the Brauer--Manin set\, that contains t
he rational points. If the set of adelic points is
non-empty 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 local points. If an el
ement of the Brauer group has order coprime to p\,
then its evaluation at a p-adic point factors via
reduction of the point modulo p. For p-torsion el
ements this is no longer the case: in order to com
pute 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 describe using work of Bloch a
nd Kato. Applications of our work include addressi
ng Swinnerton-Dyer's question about which places c
an play a role in the Brauer-Manin obstruction. Th
is is joint work with Martin Bright.
LOCATION:MR13
CONTACT:Rong Zhou
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