Arakelov Geometry and Diophantine Applications (Lecture Notes in Mathematics)

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