Weil's conjecture on Tamagawa numbers for function fields

Weil's conjecture on Tamagawa numbers for function fields

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Amina Abdurrahman, Princeton University
Fine Hall 110

We sketch some beautiful topological ideas in Gaitsgory's and Lurie's proof of Weil's conjecture for function fields (2014). We first discuss how the Siegel mass formula which counts particular equivalence classes of quadratic forms motivates the conjecture for number fields (entirely proven in 1988). Gaitsgory and Lurie reformulate the conjecture for function fields as a similar counting problem of principal G-bundles on an algebraic curve X and reduce the problem to understanding the topology of the space that these bundles give rise to. We discuss different formulations of Weil's conjecture and a topological local-to-global principle that is used to compute the cohomology of the moduli stack of G-bundles on X.