## The Tamagawa number formula over function fields

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 Title The Tamagawa number formula over function fields Title of Series Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo Author Contributors 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. Identifiers 10.5446/20482 (DOI) Publisher Release Date 2015 Language English

 Subject Area Mathematics Abstract Séminaire Paris Pékin Tokyo / Mercredi 17 novembre 2015 Let G be a semi-simple and simply connected group and X an algebraic curve. We consider Bun G(X), the moduli space of G-bundles 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 Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X. The 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 extend 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 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 2-categories leads to an equality of numbers.
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