Mordell–Weil Lattices

Preface
Acknowledgements
Contents
1 Introduction
2 Lattices
	2.1 Generalities on Lattices
	2.2 Sphere Packings
	2.3 Root Lattices and Their Dual Lattices
		2.3.1 Auxiliary Lattice Lr,s
		2.3.2 Ar and Arvee
		2.3.3 Dr and Drvee
		2.3.4 Er and Ervee (r=6,7,8)
		2.3.5 Discriminant Groups
		2.3.6 Invariant Theory of the Weyl Groups, I
		2.3.7 Invariant Theory of the Weyl Groups, II
3 Elliptic Curves
	3.1 Elliptic Curves
	3.2 The Group Law
	3.3 Mordell–Weil Theorem
	3.4 Degenerate Plane Cubics
		3.4.1 Case (a)
		3.4.2 Case (b)
		3.4.3 Further Degenerations
		3.4.4 The First Elliptic Surface
		3.4.5 An Elliptic Surface with Constant Moduli
4 Algebraic Surfaces
	4.1 Divisors and Picard Group
	4.2 Néron–Severi Group
	4.3 Intersection Theory
	4.4 Hodge Index Theorem
	4.5 Blow-Ups
	4.6 Minimal Models
	4.7 Invariants of Surfaces
		4.7.1 Relation with Picard Number
	4.8 Enriques–Kodaira Classification
	4.9 Castelnuovo's Criterion
	4.10 Non-smooth Surfaces
5 Elliptic Surfaces
	5.1 Definition of an Elliptic Surface
	5.2 Sections Versus Rational Points
	5.3 Examples
	5.4 Singular Fibres
	5.5 Connection with Dynkin Diagrams
	5.6 The Kodaira–Néron Model
		5.6.1 Group Structure
	5.7 The Weierstrass Model
	5.8 Tate's Algorithm
		5.8.1 Multiplicative Reduction
		5.8.2 Additive Reduction
	5.9 Singular Fibres Versus Discriminant and j-Invariant
		5.9.1 Quadratic Twists
		5.9.2 Base Change
		5.9.3 Dokchitsers' Refinement
	5.10 Minimal Weierstrass Models
	5.11 Canonical Divisor
	5.12 Euler Characteristic and Euler Number
	5.13 Differential Forms
6 Mordell–Weil Lattices
	6.1 The Trivial Lattice
	6.2 The Néron–Severi Lattice
	6.3 Horizontal and Vertical Divisors
	6.4 Essential Lattice and Frame
	6.5 Mordell–Weil Lattices
	6.6 Torsion Sections
		6.6.1 Determinant Formula
		6.6.2 Frame Versus Singular Fibres
	6.7 Narrow Mordell–Weil Lattice
	6.8 The Case of a Unimodular Néron–Severi Lattice
	6.9 Functorial Properties
7 Rational Elliptic Surfaces
	7.1 Preliminaries
	7.2 Basic Properties
	7.3 The Case of High Rank
	7.4 Weierstrass Form
	7.5 Cubic Pencils
	7.6 General Cubic Pencil
8 Rational Elliptic Surfaces and E8-Hierarchy
	8.1 Singular Fibres
	8.2 Structure of Mordell–Weil Lattice
	8.3 Classification
	8.4 Torsion Sections on Rational Elliptic Surfaces
	8.5 Proof of Classification Theorem 8.8
		8.5.1 r 4 (Nos. 1–14)
		8.5.2 r=3 (Nos. 15–24)
		8.5.3 r=2 (Nos. 25–42)
		8.5.4 r=1 (Nos. 43–61)
		8.5.5 r=0 (Nos. 62–74)
	8.6 Corollaries
	8.7 Integral Sections
	8.8 Extremal Rational Elliptic Surfaces
	8.9 Existence of Types of Rational Elliptic Surfaces
	8.10 Maximal Singular Fibres
		8.10.1 T=A8
		8.10.2 T=D8
		8.10.3 T=E8
		8.10.4 Comment on K3 Surfaces and Beyond
		8.10.5 Connection with Polynomial Sections
9 Galois Representations and Algebraic Equations
	9.1 Galois Representations Arising From Mordell–Weil Lattices
	9.2 Algebraic Equations Arising From Mordell–Weil Lattices
	9.3 The Specialization Maps
	9.4 Galois Representation and Algebraic Equation of Type E6, E7 or E8
		9.4.1 Galois Representations on Hexagonal Elliptic Curves
		9.4.2 Set-Up for the Additive Type Er  (r=6,7,8)
		9.4.3 Generic Galois Representation and Algebraic Equation of Additive Type Er  (r=6,7,8)
	9.5 Application to Number Theory
		9.5.1 Exceptional Type Er (r=6,7,8) Versus Classical Type An-1
		9.5.2 Systematic Construction of Elliptic Curves over mathbbQ(t) of Rank r=6,7,8, with Prescribed Generators
		9.5.3 Non-degeneracy Condition
		9.5.4 Galois Extensions over mathbbQ with Galois Group W(Er)  (r=6,7,8)
	9.6 Appendix: Non-perfect Fields
10 Applications to Classical Topics
	10.1 Multiplicative Excellent Families
		10.1.1 Excellent Families
		10.1.2 Excellent Families for the Hexagonal Case
	10.2 Multiplicative Excellent Families of Type E6, E7 or E8
		10.2.1 Set-Up for the Multiplicative Type Er (r=6,7,8)
		10.2.2 Generic Galois Representation and Algebraic Equation of Multiplicative Type Er  (r=6,7,8)
		10.2.3 Preparation: Standard Generators of E6vee
		10.2.4 Proof of Theorem10.8 in the Case r=6
		10.2.5 Reverse Engineering of Linear Sections
		10.2.6 Formulation of an Algorithm (M)
		10.2.7 Variant: A Refined Algorithm (M')
	10.3 Non-degeneracy Condition and Vanishing Roots
		10.3.1 Excellent Families and Non-degeneracy Condition
		10.3.2 Vanishing Roots
	10.4 Examples and Applications
		10.4.1 Big Galois over mathbbQ
		10.4.2 Small Galois over mathbbQ
		10.4.3 Degenerations via Vanishing Roots
		10.4.4 Application to Polynomial Sections
	10.5 Applications to Classical Topics
		10.5.1 The 27 Lines on a Cubic Surface
		10.5.2 Zeta Functions and Cubic Surfaces
		10.5.3 Relation with Del Pezzo Surfaces
		10.5.4 Cubic Surfaces as the Plane Blown up in Six Points
		10.5.5 The 28 Bitangents to a Plane Quartic
		10.5.6 Degenerations of Cubic Surfaces
11 Elliptic K3 Surfaces—Basics
	11.1 Definition and First Examples
	11.2 Elliptic K3 Surfaces
		11.2.1 Kummer Surfaces of Product Type
		11.2.2 First Alternative Elliptic Fibration
		11.2.3 Second Alternative Elliptic Fibration
	11.3 Lattice Theory for K3 Surfaces
		11.3.1 K3 Lattice
		11.3.2 Polarized K3 Surfaces
		11.3.3 Hodge Theory and Transcendental Lattice
		11.3.4 Lattice Polarized K3 Surfaces
		11.3.5 Examples
		11.3.6 (-2)-Curves on K3 Surfaces
		11.3.7 Positive Characteristic
	11.4 Elliptic Fibrations
		11.4.1 Explicit Divisors and Elliptic Fibrations
	11.5 Classification of Complex Elliptic K3 Surfaces
12 Elliptic K3 Surfaces—Special Topics
	12.1 Tate Conjecture
	12.2 Isogeny Structures Between K3 Surfaces
		12.2.1 Sandwich Structure
		12.2.2 Shioda–Inose Structure
		12.2.3 Inose's Fibration
		12.2.4 Picard Numbers
		12.2.5 Isogeny Notion
		12.2.6 Explicit Mordell–Weil Ranks
		12.2.7 Mordell–Weil Rank 15
	12.3 Elliptic fibrations on a given K3 Surface
		12.3.1 Elliptic Fibrations on a Kummer Surface of Product Type
		12.3.2 Kneser–Nishiyama Method
		12.3.3 Partner Lattice
		12.3.4 Elliptic Fibrations from the Partner Lattice
		12.3.5 Applications to Complex K3 Surfaces
	12.4 Supersingular K3 Surfaces
		12.4.1 Two Notions of Supersingularity
		12.4.2 Artin Invariant
		12.4.3 Explicit Generators of NS(X)
		12.4.4 The Supersingular K3 Surface with (p,σ)=(2,1)
		12.4.5 Duality of Néron–Severi Lattices
13 Ranks and Sphere Packings
	13.1 Rank Problem over mathbbQ and mathbbQ(t)
		13.1.1 An Effective Version of Néron's Method
		13.1.2 Mestre's Method
		13.1.3 Elkies' Use of Elliptic K3 Surfaces
		13.1.4 Mordell–Weil Rank 17 over mathbbQ
	13.2 Rank Problem over mathbbC(t)
		13.2.1 Delsarte Surfaces
		13.2.2 Elliptic Delsarte Surfaces and Rank Record over mathbbC(t)
		13.2.3 Exceptional Hodge Classes
	13.3 Rank Problem in Positive Characteristic
		13.3.1 Tate–Shafarevich Approach
		13.3.2 Approach Through Delsarte Surfaces
		13.3.3 Delsarte Approach over mathbbFp(t)
	13.4 Application to Sphere Packings
		13.4.1 Exemplary Result
		13.4.2 Invariants of the Elliptic Surface
		13.4.3 Elkies' Results
Appendix  Notation
Appendix  References
Index