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SUMMARY:The Tamagawa number formula over function fields
DTSTART;VALUE=DATE-TIME:20151117T090000Z
DTEND;VALUE=DATE-TIME:20151117T100000Z
DTSTAMP;VALUE=DATE-TIME:20190717T225511Z
UID:indico-event-858@indico.math.cnrs.fr
DESCRIPTION:Let G be a semi-simple and simply connected group and X an alg
ebraic curve. We consider Bun_G(X)\, the moduli space of G-bundles on X. I
n their celebrated paper\, Atiyah and Bott gave a formula for the cohomolo
gy of Bun_G\, namely H^*(Bun_G)=Sym(H_*(X)\\otimes V)\, where V is the spa
ce of generators for H^*_G(pt). When we take our ground field to be a fini
te field\, the Atiyah-Bott formula implies the Tamagawa number conjecture
for the function field of X.\n\nThe caveat here is that the A-B proof uses
the interpretation of Bun_G as the space of connection forms modulo gauge
transformations\, and thus only works over complex numbers (but can be ex
tend to any field of characteristic zero). In the talk we will outline an
algebro-geometric proof that works over any ground field. As its main geom
etric ingredient\, it uses the fact that the space of rational maps from X
to G is homologically contractible. Because of the nature of the latter s
tatement\, the proof necessarily uses tools from higher category theory. S
o\, it can be regarded as an example how the latter can be used to prove s
omething concrete: a construction at the level of 2-categories leads to an
equality of numbers.\n\nhttps://indico.math.cnrs.fr/event/858/
LOCATION:IHES Centre de confĂ©rences Marilyn et James Simons
URL:https://indico.math.cnrs.fr/event/858/
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