Arakelov Geometry and Diophantine Applications (Lecture Notes in Mathematics)

✍ Scribed by Emmanuel Peyre (editor), Gaël Rémond (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 473
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry.  Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research.

 This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.

✦ Table of Contents

Preface
Contents
Introduction
1 Part A: Concepts of Arakelov Geometry
2 Part B: Distribution of Rational Points and Dynamics
3 Part C: Shimura Varieties
References
Part A Concepts of Arakelov Geometry
Chapter I: Arithmetic Intersection
1 Introduction
2 Definition of the Height
2.1 Algebraic Preliminaries
2.2 Analytic Preliminaries
2.3 Heights
3 Existence of the Height
4 Arithmetic Chow Groups
4.1 Definition
4.2 Example
4.3 Products
4.4 Functoriality
4.5 Heights and Intersection Numbers
References
Chapter II: Minima and Slopes of Rigid Adelic Spaces
1 Introduction
2 Rigid Adelic Spaces
2.1 Algebraic Extensions of Q
2.2 Rigid Adelic Spaces
3 Minima and Slopes
3.1 Successive Minima
Examples
3.2 Slopes
4 Comparisons Between Minima and Slopes
4.1 Lower Bounds
4.2 Upper Bounds
4.3 Transference Theorems
5 Heights of Morphisms and Slope-Minima Inequalities
5.2 Tensor Product
References
Chapter III: Introduction aux théorèmes de Hilbert-Samuel arithmétiques
1 Introduction
Remerciement
2 Méthode combinatoire
2.1 Algèbre de polynômes
2.2 Algèbre de semi-groupes
2.3 Semi-groupe d'un corps convexe
2.4 Fonction de Hilbert d'une algèbre de semi-groupe
2.5 Cas d'une algèbre graduée générale
3 Approche géométrique
3.1 Interprétation géométrique des algèbres graduées
3.2 Nombre d'intersection et théorème de Hilbert-Samuel
3.3 Diviseurs de Cartier et systèmes linéaires
3.4 Cas torique
3.5 Corps convexe de Newton-Okounkov
4 Version métrique
4.1 Norme déterminant
4.2 Espace de Berkovich
4.3 Faisceaux inversibles métrisés
4.4 Version métrique du théorème de Hilbert-Samuel
4.5 Méthode combinatoire
5 Cas arithmétique
5.1 Fibré vectoriels adéliques sur un corps de nombres
5.2 Fibrés inversibles adéliques sur une variété arithmétique
5.3 Théorème de Hilbert-Samuel arithmétique
5.4 Cas sans hypothèse d'amplitude
Références
Chapter IV: Euclidean Lattices, Theta Invariants, and Thermodynamic Formalism
1 Introduction
2 Euclidean Lattices
2.1 Un peu d'histoire
2.2 The Classical Invariants of Euclidean Lattices
2.3 Euclidean Lattices as Hermitian Vector Bundles Over Spec Z
3 Reduction Theory for Euclidean Lattices
3.1 A Theorem of Hermite, Korkin and Zolotarev
3.2 Complements
3.3 An Application to Transference Inequalities
4 Theta Series and Banaszczyk's Transference Estimates
4.1 Poisson Formula and Theta Series of Euclidean Lattices
4.2 Banaszczyk's Transference Estimates
4.3 The Key Inequalities
4.4 Proof of the Transference Inequality (4.7)
5 Vector Bundles on Curves and the Analogy with Euclidean Lattices
5.1 Vector Bundles on Smooth Projective Curves and Their Invariants
5.2 Euclidean Lattices as Analogues of Vector Bundles Over Projective Curves
5.3 The Invariants h0Ar(E), h0θ(E) and h1θ(E)
5.4 How to Reconcile the Invariants h0Ar(E) and h0θ(E)
5.5 Some Further Analogies Between h0θ(E) and h0(C, E)
5.6 Varia
6 A Mathematical Model of the Thermodynamic Formalism
6.1 Measure Spaces with a Hamiltonian: Basic Definitions
6.2 Main Theorem
6.3 Relation with Statistical Physics
6.4 Gaussian Integrals and Maxwell's Kinetic Gas Model
6.5 Application to Euclidean Lattices: Proof of Theorem 5.4.3
7 Proof of the Main Theorem
7.1 The Functions , U and S
7.2 The Convergence of (1/n) logAn(E)
7.3 The Zero Temperature Limit
8 Complements
8.1 The Main Theorem When (T, H) = (R+, IdR+)
8.2 Chernoff's Bounds and Rankin's Method
8.3 Lanford's Estimates
8.4 Products and Thermal Equilibrium
9 The Approaches of Poincaré and of Darwin-Fowler
9.1 Preliminaries
9.2 Asymptotics of An(E) by the Saddle-Point Method
9.3 Approximation Arguments
References
Part B Distribution of Rational Points and Dynamics
Chapter V: Beyond Heights: Slopes and Distribution of RationalPoints
1 Introduction
2 Norms and Heights
2.1 Adelic Metric
2.2 Arakelov Heights
3 Accumulation and Equidistribution
3.1 Sandbox Example: The Projective Space
3.2 Adelic Measure
3.3 Weak Approximation
3.4 Accumulating Subsets
The Plane Blown Up in One Point
The Principle of Manin
The Counterexample of V. V. Batyrev and Y. Tschinkel
The Example of C. Le Rudulier
4 All the Heights
4.1 Heights Systems
4.2 Compatibility with the Product
4.3 Lifting to Versal Torsors
Versal and Universal Torsors
Structures on Versal Torsors
Lifting of the Asymptotic Formula
4.4 Varieties of Picard Rank One
5 Geometric Analogue
5.1 The Ring of Motivic Integration
5.2 A Sandbox Example: The Projective Space
5.3 Equidistribution in the Geometric Setting
5.4 Crash Course about Obstruction Theory
6 Slopes à la Bost
6.1 Definition
Slopes of an Adelic Vector Bundle Over Spec(K)
Slopes on Varieties, Freeness
6.2 Properties
6.3 Explicit Computations
In the Projective Space
Products of Lines
6.4 Accumulating Subsets and Freeness
Rational Curves of Low Degree
Fibrations
6.5 Combining Freeness and Heights
7 Local Accumulation
7.1 Local Distribution
8 Another Description of the Slopes
9 Conclusion and Perspectives
References
Chapter VI: On the Determinant Method and Geometric Invariant Theory
1 Introduction
2 Chow Forms and Chow Weights
3 Hilbert Polynomials and Hilbert Weights
4 Estimates of Some Determinants
5 The Determinant Method
References
Chapter VII: Arakelov Geometry, Heights, Equidistribution, and the Bogomolov Conjecture
1 Introduction
2 Arithmetic Intersection Numbers
3 The Height of a Variety
4 Adelic Metrics
5 Arithmetic Ampleness
6 Measures
7 Volumes
8 Equidistribution
9 The Bogomolov Conjecture
References
Chapter VIII: Autour du théorème de Fekete-Szegő
1 Introduction
2 Théorie du potentiel sur C
3 Lien avec l'intersection arithmétique
4 Théorème de Fekete
5 Théorème de Fekete-Szegő
6 Théorie du potentiel sur les courbes
7 Points entiers
8 Théorèmes de Rumely
9 Équidistribution dans le cas critique
Références
Chapter IX: Some Problems of Arithmetic Origin in RationalDynamics
1 Introduction
PART IX.A: Basic Holomorphic and Arithmetic Dynamics on P1
2 A Few Useful Geometric Tools
2.1 Uniformization
2.2 The Hyperbolic Metric
3 Review of Rational Dynamics on P1(C)
3.1 Fatou–Julia Dichotomy
3.2 What Does J(f) Look Like?
3.3 Periodic Points
3.4 Fatou Dynamics
4 Equilibrium Measure
4.1 Definition and Main Properties
4.2 The Case of Polynomials
5 Non-archimedean Dynamical Green Function
5.1 Vocabulary of Valued Fields
5.2 Dynamical Green Function
6 Logarithmic Height
6.1 Definition and Basic Properties
6.2 Action Under Rational Maps
7 Consequences of Arithmetic Equidistribution
7.1 Equidistribution of Preperiodic Points
7.2 Rigidity
PART IX.B: Parameter Space Questions
8 The Quadratic Family
8.1 Connectivity of J
8.2 Aside: Active and Passive Critical Points
8.3 Post-Critically Finite Parameters in the Quadratic Family
9 Higher Degree Polynomials and Equidistribution
9.1 Special Points
9.2 Equidistribution of Special Points
10 Special Subvarieties
10.1 Prologue
10.2 Classification of Special Curves
References
Part C Shimura Varieties
Chapter X: Arakelov Theory on Shimura Varieties
1 Introduction
2 Hermitian Symmetric Spaces
3 Connected Shimura Varieties
4 Equivariant Vector Bundles and Invariant Metrics
5 Log-Singular Metrics and Log–log Forms
6 Arakelov Geometry with Log–log Forms
References
Chapter XI: The Arithmetic Riemann–Roch Theorem and the Jacquet–Langlands Correspondence
1 Introduction
2 Riemann–Roch Theorem for Arithmetic Surfaces and Hermitian Line Bundles
2.1 Riemann–Roch Formulae in Low Dimensions
2.2 Arithmetic Intersections on Arithmetic Surfaces
2.3 The Determinant of Cohomology and the Quillen Metric
2.4 The Arithmetic Riemann–Roch Theorem of Gillet–Soulé
3 An Arithmetic Riemann–Roch Formula for Modular Curves
3.1 The Setting
3.2 Renormalized Metrics (Wolpert Metrics)
3.3 A Quillen Type Metric
3.4 An Arithmetic Riemann–Roch Formula
4 Modular and Shimura Curves
4.1 Modular Curves
4.2 Modular Forms
4.3 Shimura Curves and Quaternionic Modular Curves
5 The Jacquet–Langlands Correspondence and the Arithmetic Riemann–Roch Theorem
5.1 On the Jacquet–Langlands Correspondence for Weight 2 Forms
5.2 The Jacquet–Langlands Correspondence for Maass Forms
5.3 Relating Arithmetic Intersection Numbers
References
Chapter XII: The Height of CM Points on Orthogonal Shimura Varieties and Colmez's Conjecture
1 Introduction
2 The Averaged Colmez's Conjecture
2.1 Faltings' Height
2.2 Colmez's Theorem
2.3 Colmez's Conjecture
2.4 Some Consequences and Reduction Steps
3 Shimura Varieties of Orthogonal Type and CM Cycles
3.1 GSpin Shimura Varieties
3.2 Examples of GSpin Groups
Example I: The Case n = 0
Example II: The Case n = 1
3.3 Hermitian Symmetric Spaces
3.4 GSpin-Shimura Varieties
4 Extra Structures on GSpin-Shimura Varieties
4.1 Example A: The Representation V
4.2 Example B: The Kuga–Satake Abelian Scheme
5 The Big CM Points
5.1 The Total Reflex Algebra
5.2 Extra Structure on the Big CM Cycle
6 Integral Models
6.1 Integral Models of GSpin-Shimura Varieties
6.2 Integral Models of Big CM Cycles
7 The Bruinier, Kudla, Yang Conjecture
7.1 Special Divisors
7.2 Integral Models of Special Divisors
7.3 Special Divisors and ω
7.4 Arithmetic Intersection and Special Values
References
Glossary
Index