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
. 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
. In the case where
is the function field of an algebraic curve
X
, this conjecture counts the number of
-bundles on
(global information) in terms of the reduction of
at the points of
(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
-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-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
-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.