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SUMMARY:Understanding the conjecture of Birch and Swinnerton-Dyer - Ivan F
 esenko (Nottingham)
DTSTART:20110208T143000Z
DTEND:20110208T153000Z
UID:TALK28536@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:To every elliptic curve over a global field one can associate 
 a regular proper\nmodel\nwhich is geometrically a two-dimensional object a
 nd which reveals more\nunderlying\nstructures and dualities than its gener
 ic fibre.\nUnlike the classical adelic structure on one-dimensional arithm
 etic schemes\, \nthere are two adelic structures on arithmetic surfaces: \
 none is more suitable for geometry and another is more suitable for analys
 is and\narithmetic. \nThe two-dimensional adelic analysis studies the zeta
  function of the surface\nlifting it to a zeta integral \nusing the second
  adelic structure.\nIts main theorem reduces the study of analytic propert
 ies of the zeta integral\nto those of a boundary term which is an integral
  over the weak boundary of\nadelic spaces of the second type.\nTo study th
 e latter one uses the symbol map from K_1 of the first adelic\nstructure a
 nd K_1 of the second adelic structure \nto K_2 of the first adelic structu
 re. \nThe (known in some partial cases but not really understood) equality
  of the\nanalytic and arithmetic ranks\nbecomes much more transparent and 
 natural in the language of the two adelic\nstructures on the surface and t
 heir interplay. \nMoreover\, the adelic approach includes a potential to e
 xplain the finiteness of\nthe Brauer-Grothendieck group of the surface \na
 nd hence of Shah. \n
LOCATION:MR13
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