Weils Conjecture for Function Fields: Volume I (Annals of Mathematics Studies)

A central concern of number theory is the study of localtoglobal 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 localtoglobal principle: Weils 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 Gbundles 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 Weils conjecture, based on the geometry of the moduli stack of Gbundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ladic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different localtoglobal principle: a product formula that expresses the cohomology of the moduli stack of Gbundles (a global object) as a tensor product of local factors.Using a version of the GrothendieckLefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weils conjecture. The proof of the product formula will appear in a sequel volume.