Weil\ s Conjecture for Function Fields: Volume I (Ams-199)

Dennis Gaitsgory|Jacob Lurie

ISBN 10: 0691182132 / ISBN 13: 9780691182131

Published by PRINCETON UNIV PR, 2019

About this Item

&Uumlber den AutorDennis 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.

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Bibliographic Details

• Title: Weil\ s Conjecture for Function Fields: Volume I (Ams-199)
• Publisher: PRINCETON UNIV PR
• Publication Date: 2019
• Language: English
• ISBN 10: 0691182132
• ISBN 13: 9780691182131
• Binding: Gebunden
• Condition: New

About this title

Synopsis

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.

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).