Weil's Conjecture for Function Fields : Volume I (AMS-199) / Dennis Gaitsgory, Jacob Lurie.

Format:

Book

Author/Creator:

Gaitsgory, Dennis, Author.

Lurie, Jacob, Author.

Series:

Annals of mathematics studies ; number 199.

Princeton scholarship online.

Annals of Mathematics Studies ; 199

Language:

English

Subjects (All):

Weil conjectures.

Physical Description:

1 online resource (321 pages).

Place of Publication:

Princeton, NJ : Princeton University Press, [2019]

Language Note:

In English.

Summary:

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.

Contents:

Frontmatter

Contents

Chapter One. Introduction

Chapter Two. The Formalism of ℓ-adic Sheaves

Chapter Three. E∞-Structures on ℓ-Adic Cohomology

Chapter Four. Computing the Trace of Frobenius

Chapter Five The Trace Formula for BunG(X)

Bibliography

Notes:

Also issued in print: 2019.

Includes bibliographical references.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 06. Apr 2020)

ISBN:

9780691184432

0691184437

OCLC:

1079759075