Weil's Conjecture for Function Fields

Name: Weil's Conjecture for Function Fields | Volume I (AMS-199)
Brand: Princeton University Press
Price: 391.95 EUR
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Volume I (AMS-199)

Dennis Gaitsgory Jacob Lurie(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 19. Februar 2019

320 Seiten

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Inhalt

Cover
Title
Copyright
Contents
1 Introduction
1.1 The Mass Formula
1.1.1 Quadratic Forms
1.1.2 Classification of Quadratic Forms
1.1.3 The Smith-Minkowski-Siegel Mass Formula
1.1.4 The Unimodular Case
1.2 Adelic Formulation of the Mass Formula
1.2.1 The Adelic Group Oq(A)
1.2.2 Adelic Volumes
1.2.3 Digression: Local Fields
1.2.4 Tamagawa Measure
1.2.5 The Mass Formula and Tamagawa Numbers
1.2.6 Weil's Conjecture
1.3 Weil's Conjecture for Function Fields
1.3.1 Tamagawa Measure in the Function Field Case
1.3.2 Principal Bundles
1.3.3 Weil's Conjecture as a Mass Formula
1.4 Cohomological Formulation of Weil's Conjecture
1.4.1 The Moduli Stack of G-Bundles
1.4.2 Counting Points on Algebraic Varieties
1.4.3 The Trace Formula for BunG(X)
1.4.4 Weil's Conjecture
1.5 Computing the Cohomology of BunG(X) over C
1.5.1 Bundles on a Riemann Surface
1.5.2 The Atiyah-Bott Formula
1.5.3 Digression: Rational Homotopy Theory
1.5.4 The Product Formula
1.5.5 Proof of the Product Formula
1.5.6 Proof of the Atiyah-Bott Formula
1.6 Summary of This Book
2 The Formalism of l-adic Sheaves
2.1 Higher Category Theory
2.1.1 Motivation: Deficiencies of the Derived Category
2.1.2 The Differential Graded Nerve
2.1.3 The Weak Kan Condition
2.1.4 The Language of Higher Category Theory
2.1.5 Example: Limits and Colimits
2.1.6 Stable 8-Categories
2.2 Étale Sheaves
2.2.1 Sheaves of ?-Modules
2.2.2 The t-Structure on Shv(X
?)
2.2.3 Functoriality
2.2.4 Compact Generation of Shv(X
?)
2.2.5 The Exceptional Inverse Image
2.2.6 Constructible Sheaves
2.2.7 Sheaves of Vector Spaces
2.2.8 Extension of Scalars
2.2.9 Stability Properties of Constructible Sheaves
2.3 l-adic Sheaves
2.3.1 l-Completeness
2.3.2 Constructible l-adic Sheaves
2.3.3 Direct and Inverse Images
2.3.4 General l-adic Sheaves
2.3.5 Cohomological Descent
2.3.6 The t-Structure on l-Constructible Sheaves
2.3.7 Exactness of Direct and Inverse Images
2.3.8 The Heart of Shvcl (X)
2.3.9 The t-Structure on Shvl (X)
2.4 Base Change Theorems
2.4.1 Digression: The Beck-Chevalley Property
2.4.2 Smooth and Proper Base Change
2.4.3 Direct Images and Extension by Zero
2.4.4 Base Change for Exceptional Inverse Images
3 E8-Structures on l-Adic Cohomology
3.1 Commutative Algebras
3.1.1 Commutative Monoids
3.1.2 Symmetric Monoidal 8-Categories
3.1.3 Commutative Algebra Objects
3.1.4 Tensor Products of Chain Complexes
3.1.5 E8-Algebras
3.1.6 E8-Structures on Cochain Complexes
3.1.7 The Topological Product Formula with General Coefficients
3.2 Cohomology of Algebraic Stacks
3.2.1 l-Adic Cohomology of Algebraic Varieties
3.2.2 Digression: Tensor Products of l-Adic Sheaves
3.2.3 E8-Structures on l-adic Cochain Complexes
3.2.4 Algebraic Stacks and Fibered Categories
3.2.5 l-Adic Cohomology of Algebraic Stacks
3.2.6 Digression: Fibered 8-Categories
3.3 The !-Tensor Product
3.3.1 The Künneth Formula
3.3.2 Associativity of the !-Tensor Product
3.3.3 Dualizing Sheaves
3.3.4 The 8-Category Shv*l
3.3.5 The 8-Category Shv!l
3.4 The Cohomology Sheaf of a Morphism
3.4.1 The Sheaf [Y]F
3.4.2 Functoriality
3.4.3 Compatibility with Exceptional Inverse Images
3.4.4 Functoriality Revisited
3.4.5 Compatibility with External Tensor Products
3.4.6 Tensor Functoriality
3.4.7 The Algebra Structure on [Y]X
4 Computing the Trace of Frobenius
4.1 The Product Formula
4.1.1 Factorization Homology
4.1.2 Formulation of the Product Formula
4.2 The Cotangent Fiber
4.2.1 Augmented Commutative Algebras
4.2.2 Cotangent Fibers and Square-Zero Extensions
4.2.3 Examples of Cotangent Fibers
4.2.4 The m-adic Filtration
4.2.5 Convergence of the m-adic Filtration
4.2.6 Application: Linearizing the Product Formula
4.3 Convergent Frob-Modules
4.3.1 Definitions
4.3.2 The Case of an Augmented Algebra
4.3.3 The Proof of Proposition 4.3.2.1
4.4 The Trace Formula for BG
4.4.1 The Motive of an Algebraic Group
4.4.2 Digression: The Eilenberg-Moore Spectral Sequence
4.4.3 The Motive as a Cotangent Fiber
4.4.4 Proof of the Trace Formula
4.4.5 The Proof of Lemma 4.4.2.3
4.5 The Cohomology of BunG(X)
4.5.1 The Motive of a Group Scheme
4.5.2 Estimating the Eigenvalues of Frobenius
4.5.3 The Proof of Theorem 4.5.0.1
5 The Trace Formula for BunG(X)
5.1 The Trace Formula for a Quotient Stack
5.1.1 A Convergence Lemma
5.1.2 The Proof of Proposition 5.1.0.1
5.1.3 Application: Change of Group
5.2 The Trace Formula for a Stratified Stack
5.2.1 Stratifications of Algebraic Stacks
5.2.2 Convergent Stratifications
5.2.3 The Proof of Proposition 5.2.2.5
5.3 The Harder-Narasimhan Stratification
5.3.1 Semistable G-Bundles
5.3.2 The Harder-Narasimhan Stratification: Split Case
5.3.3 Rationality Properties of the Harder-Narasimhan Stratification
5.3.4 Digression: Inner Forms
5.3.5 The Harder-Narasimhan Stratification: The Inner Case
5.4 Quasi-Compactness Properties of Moduli Spaces of Bundles
5.4.1 Quasi-Compact Substacks of BunG(X)
5.4.2 Varying G and X
5.4.3 Quasi-Compactness of Harder-Narasimhan Strata
5.5 Comparison of Harder-Narasimhan Strata
5.5.1 Parabolic Reductions
5.5.2 Digression: Levi Decompositions
5.5.3 Twisting Parabolic Reductions
5.5.4 Existence of Twists
5.5.5 Twisting as a Morphism of Moduli Stacks
5.5.6 Classification of Untwists
5.6 Proof of the Trace Formula
5.6.1 The Case of a Split Group Scheme
5.6.2 Reductive Models
5.6.3 The Proof of Theorem 5.0.0.3
5.6.4 The Proof of Proposition 5.6.3.2
Bibliography

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