A central concern of number theory is the study of local-to-global
principles, which describe the behavior of a global field K in
terms of the behavior of various completions of K. This book looks
at a specific example of a local-to-global principle: Weil's
conjecture on the Tamagawa number of a semisimple algebraic group G
over K. In the case where K is the function field of an algebraic
curve X, this conjecture counts the number of G-bundles on X
(global information) in terms of the reduction of G at the points
of X (local information). The goal of this book is to give a
conceptual proof of Weil's conjecture, based on the geometry of the
moduli stack of G-bundles. Inspired by ideas from algebraic
topology, it introduces a theory of factorization homology in the
setting -adic sheaves. Using this theory, Dennis Gaitsgory and
Jacob Lurie articulate a different local-to-global principle: a
product formula that expresses the cohomology of the moduli stack
of G-bundles (a global object) as a tensor product of local
factors. Using a version of the Grothendieck-Lefschetz trace
formula, Gaitsgory and Lurie show that this product formula implies
Weil's conjecture. The proof of the product formula will appear in
a sequel volume.
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