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portada Weil's Conjecture for Function Fields: Volume i (Ams-199) (Annals of Mathematics Studies)
Type
Physical Book
Publisher
Author
Year
2019
Language
English
Pages
320
Format
Hardcover
ISBN13
9780691182131
Edition No.
1
Categories

Weil's Conjecture for Function Fields: Volume i (Ams-199) (Annals of Mathematics Studies)

Dennis Gaitsgory; Jacob Lurie (Author) · Princeton University Press · Hardcover

Weil's Conjecture for Function Fields: Volume i (Ams-199) (Annals of Mathematics Studies) - Dennis Gaitsgory; Jacob Lurie

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Synopsis "Weil's Conjecture for Function Fields: Volume i (Ams-199) (Annals of Mathematics Studies) "

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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