Weil's Conjecture for Function Fields: Volume I (Ams-199) Contributor(s): Gaitsgory, Dennis (Author), Lurie, Jacob (Author) |
|
![]() |
ISBN: 0691182132 ISBN-13: 9780691182131 Publisher: Princeton University Press OUR PRICE: $174.80 Product Type: Hardcover - Other Formats Published: February 2019 |
Additional Information |
BISAC Categories: - Mathematics | Geometry - Algebraic - Mathematics | Number Theory |
Physical Information: 1" H x 6.3" W x 9.3" (1.45 lbs) 320 pages |
Descriptions, Reviews, Etc. |
Publisher Description: 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. |