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CATEGORIES:Number Theory Seminar
SUMMARY:Understanding the conjecture of Birch and Swinnert
on-Dyer - Ivan Fesenko (Nottingham)
DTSTART;TZID=Europe/London:20110208T143000
DTEND;TZID=Europe/London:20110208T153000
UID:TALK28536AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/28536
DESCRIPTION:To every elliptic curve over a global field one ca
n associate a regular proper\nmodel\nwhich is geom
etrically a two-dimensional object and which revea
ls more\nunderlying\nstructures and dualities than
its generic fibre.\nUnlike the classical adelic s
tructure on one-dimensional arithmetic schemes\, \
nthere are two adelic structures on arithmetic sur
faces: \none is more suitable for geometry and ano
ther is more suitable for analysis and\narithmetic
. \nThe two-dimensional adelic analysis studies th
e zeta function of the surface\nlifting it to a ze
ta integral \nusing the second adelic structure.\n
Its main theorem reduces the study of analytic pro
perties of the zeta integral\nto those of a bounda
ry term which is an integral over the weak boundar
y of\nadelic spaces of the second type.\nTo study
the latter one uses the symbol map from K_1 of the
first adelic\nstructure and K_1 of the second ade
lic structure \nto K_2 of the first adelic structu
re. \nThe (known in some partial cases but not rea
lly understood) equality of the\nanalytic and arit
hmetic ranks\nbecomes much more transparent and na
tural in the language of the two adelic\nstructure
s on the surface and their interplay. \nMoreover\,
the adelic approach includes a potential to expla
in the finiteness of\nthe Brauer-Grothendieck grou
p of the surface \nand hence of Shah. \n
LOCATION:MR13
CONTACT:Tom Fisher
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