Weil's conjecture for function fields. Volume I / Dennis Gaitsgory, Jacob Lurie.
By: Gaitsgory, D. (Dennis) [author.].
Contributor(s): Lurie, Jacob [author.].
Material type: BookSeries: Annals of mathematics studies: no. 199.Publisher: Princeton : Princeton University Press, 2019Description: 1 online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9780691184432; 0691184437.Subject(s): Weil conjectures | Conjectures de Weil | MATHEMATICS -- Geometry -- General | MATHEMATICS -- Geometry -- Algebraic | Weil conjecturesGenre/Form: Electronic books.DDC classification: 516.3/52 Online resources: Click here to access onlineOnline resource; title from PDF title page (EBSCO, viewed December 21, 2018).
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.
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