Sobre o livro
A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global fieldK in terms of the behavior of various completions ofK. This book looks at a specific example of a local-to-global principle: Weils conjecture on the Tamagawa number of a semisimple algebraic groupG overK. In the case whereK is the function field of an algebraic curveX, this conjecture counts the number ofG-bundles onX (global information) in terms of the reduction ofG at the points ofX (local information). The goal of this book is to give a conceptual proof of Weils conjecture, based on the geometry of the moduli stack ofG-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 ofG-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 Weils conjecture. The proof of the product formula will appear in a sequel volume.