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Weil's Conjecture for Function Fields: Volume I by Dennis Gaitsgory

Weil's Conjecture for Function Fields: Volume I

by Dennis Gaitsgory
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Book cover type: Paperback
  • ISBN13: 9780691182148
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Princeton University Press
  • Publisher Imprint: Princeton University Press
  • Publication Date:
  • Pages: 320
  • Original Price: USD 88.0
  • Language: English
  • Edition: N/A
  • Item Weight: 522 grams
  • BISAC Subject(s): Geometry / Algebraic and Number Theory

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.

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).

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