# The Tamagawa Number Formula Via Chiral Homology (joint with J. Lurie)

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Dennis Gaitsgory, Harvard University
IAS Room S-101

Let $X$ a curve over $F_q$ and $G$ a semi-simple simply-connected group. The initial observation is that the conjecture of Weil's which says that the volume of the adelic quotient of $G$ with respect to the Tamagawa measure equals 1, is equivalent to the Atiyah-Bott formula for the cohomology of the moduli space $Bun_G(X)$ of principal G-bundles on $X$. The latter formula makes sense over an arbitrary ground field and says that $H^*(BunG(X))$ is given by the chiral homology of the commutative chiral algebra corresponding to $H^*(BG)$, where BG is the classifying space of $G$. When the ground field is $C$, the Atiyah-Bott formula can be easily proved by considerations from differential geometry, when we think of G-bundles as connections on the trivial bundle modulo gauge transformations. In algebraic geometry, we will give an alternative proof by approximating $Bun_G(X)$ by means of the multi-point version of the affine Grassmannian of $G$ using a recent result on the contractibility of the space of rational maps from $X$ to $G$.