Weil's Conjecture for Function Fields: Volume I (AMS-199)
Dennis Gaitsgory and Jacob Lurie
Abstract
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 ... More
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, the authors 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, the book shows that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.
Keywords:
number theory,
Weil's conjecture,
G-bundles,
algebraic topology,
factorization homology,
local-to-global principle,
cohomology
Bibliographic Information
| Print publication date: 2019 |
Print ISBN-13: 9780691182148 |
| Published to Princeton Scholarship Online: September 2019 |
DOI:10.23943/princeton/9780691182148.001.0001 |
Authors
Affiliations are at time of print publication.
Dennis Gaitsgory, author
Harvard University
Jacob Lurie, author
Harvard University
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