Weil's conjecture for function fields. Volume I /
Dennis Gaitsgory, Jacob Lurie.
- 1 online resource
- Annals of mathematics studies ; number 199 .
- Annals of mathematics studies ; no. 199. .
Includes bibliographical references.
The formalism of l-adic sheaves -- E∞-structures on l-adic cohomology -- Computing the trace of Frobenius -- The trace formula for BunG(X).
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.
9780691184432 0691184437
22573/ctv4t8349 JSTOR 9452475 IEEE
Weil conjectures. Conjectures de Weil. MATHEMATICS--Geometry--General. MATHEMATICS--Geometry--Algebraic. Weil conjectures.