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SUMMARY:Sansuc’s formula and Tate global duality (d’après Rosengarten
 ). - Brian Conrad (Stanford University)
DTSTART:20161025T150000Z
DTEND:20161025T160000Z
UID:TALK67454@talks.cam.ac.uk
CONTACT:Jack Thorne
DESCRIPTION:Tamagawa numbers are canonical (finite) volumes attached to sm
 ooth connected affine groups G over global fields k\; they arise in mass f
 ormulas and local-global formulas for adelic integrals. A conjecture of We
 il (proved long ago for number fields\, and recently by Lurie and Gaitsgor
 y for function fields) asserts that the Tamagawa number of a simply connec
 ted semisimple group is equal to 1\; for special orthogonal groups this ex
 presses 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 co
 nnected reductive G in terms of two finite arithmetic invariants: its Pica
 rd group and degree-1 Tate-Shafarevich group.  \n\nOver number fields it i
 s elementary to remove the reductivity hypothesis from Sansuc’s formula\
 , but over function fields that is a much harder problem\; e.g.\, the Pica
 rd group can be infinite. Work in progress by my PhD student Zev Rosengart
 en is likely to completely solve this problem.  He has formulated an alter
 native version\, proved it is always finite\, and  established the formula
  in many new cases.  We will discuss some aspects of this result\, includi
 ng one of its key ingredients: a generalization of Tate local and global d
 uality to the case of coefficients in any positive-dimensional (possibly n
 on-smooth) commutative affine algebraic k-group scheme and its (typically 
 non-representable) GL_1-dual.
LOCATION:MR13
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