Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

The Tamagawa number formula over function fields

by Prof. Dennis GAITSGORY (Harvard & IHÉS)

Centre de conférences Marilyn et James Simons (IHES)

Centre de conférences Marilyn et James Simons


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

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.