# Weil's Conjecture for Function Fields

Volume I (AMS-199)

**Dennis Gaitsgory and Jacob Lurie**

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 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 *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.

**Dennis Gaitsgory** is professor of mathematics at Harvard University. He is the coauthor of *A Study in Derived Algebraic Geometry*. **Jacob Lurie** is professor of mathematics at Harvard University. He is the author of *Higher Topos Theory* (Princeton).