Weil's Conjecture for Function Fields

Gaitsgory, Dennis; Lurie, Jacob

ISBN 10: 0691182140 / ISBN 13: 9780691182148

Edité par Princeton University Press, 2019

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N° de réf. du vendeur 32759715-n

Détails bibliographiques

• Titre: Weil's Conjecture for Function Fields
• Éditeur: Princeton University Press
• Date de parution: 2019
• Langue: anglais
• ISBN à 10 chiffres: 0691182140
• ISBN à 13 chiffres: 9780691182148
• Reliure: Couverture souple
• État de l'article: New

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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.

À propos de l?auteur

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).