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CATEGORIES:Number Theory Seminar
SUMMARY:Sansuc’s formula and Tate global duality (d’après
Rosengarten). - Brian Conrad (Stanford University)
DTSTART;TZID=Europe/London:20161025T160000
DTEND;TZID=Europe/London:20161025T170000
UID:TALK67454AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/67454
DESCRIPTION:Tamagawa numbers are canonical (finite) volumes at
tached to smooth connected affine groups G over gl
obal fields k\; they arise in mass formulas and lo
cal-global formulas for adelic integrals. A conjec
ture of Weil (proved long ago for number fields\,
and recently by Lurie and Gaitsgory for function f
ields) asserts that the Tamagawa number of a simpl
y connected semisimple group is equal to 1\; for s
pecial orthogonal groups this expresses the Siegel
Mass Formula. Sansuc pushed this further (using
a lot of class field theory) to give a formula for
the Tamagawa number of any connected reductive G
in terms of two finite arithmetic invariants: its
Picard group and degree-1 Tate-Shafarevich group.
\n\nOver number fields it is elementary to remove
the reductivity hypothesis from Sansuc’s formula\
, but over function fields that is a much harder p
roblem\; e.g.\, the Picard group can be infinite.
Work in progress by my PhD student Zev Rosengarten
is likely to completely solve this problem. He h
as formulated an alternative version\, proved it i
s always finite\, and established the formula in
many new cases. We will discuss some aspects of t
his result\, including one of its key ingredients:
a generalization of Tate local and global duality
to the case of coefficients in any positive-dimen
sional (possibly non-smooth) commutative affine al
gebraic k-group scheme and its (typically non-repr
esentable) GL_1-dual.
LOCATION:MR13
CONTACT:Jack Thorne
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