Weil's Conjecture for Function Fields

Volume I

Dennis Gaitsgory, Jacob Lurie

• Publisher: Princeton University Press
• Year: 2019
• Print ISBN: 9780691182148, 9780691182131
• eBook ISBN: 9780691184432
• Topic: Matematica
• Subtopic: Geometria algebrica

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.

Table of contents

• Cover
• Title
• Copyright
• Contents
• 1 Introduction
• 2 The Formalism of ℓ-adic Sheaves
• 3 E∞-Structures on ℓ-Adic Cohomology
• 4 Computing the Trace of Frobenius
• 5 The Trace Formula for BunG(X)
• Bibliography