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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.
About the author
Dennis Gaitsgory is professor of mathematics at Harvard University. He is the coauthor of
A Study in Derived Algebraic Geometry.
Jacob Lurie is professor of mathematics at Harvard University. He is the author of
Higher Topos Theory (Princeton).
Summary
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 wher
Additional text
"The book is written in a clear and vivid style, pays attention to foundations and details, and yet elucidates motivations and ideas. It should be highly useful for researchers working with stacks and higher category theory."---Stefan Schröer, Zentralblatt MATH
