The Tamagawa number formula over function fields
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Metadata
Formal Metadata
Title  The Tamagawa number formula over function fields 
Title of Series  Conférences Paris Pékin Tokyo 
Part Number  13 
Number of Parts  15 
Author 
Gaitsgory, Dennis

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20482 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Séminaire Paris Pékin Tokyo / Mercredi 17 novembre 2015 Let G be a semisimple and simply connected group and X an algebraic curve. We consider Bun G(X), the moduli space of Gbundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of Bun G, namely H^*(Bun G)=Sym(H *(X)\otimes V), where V is the space of generators for H^* G(pt). When we take our ground field to be a finite field, the AtiyahBott formula implies the Tamagawa number conjecture for the function field of X. The caveat here is that the AB 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 extend to any field of characteristic zero). In the talk we will outline an algebrogeometric proof that works over any ground field. As its main geometric 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 statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2categories leads to an equality of numbers. 