Author(s): Masaaki Yoshida
Publisher: Vieweg
Year: 1997
Cover
Title page
Preface
Part 1 The Story of the Configuration Space X(2,4) of Four Points on the Projective Line
Chapter I. Configuration Spaces - The Simplest Case
1. Classifications and Equivalence Relations
2. Quotient Spaces
3. Realizations
4. The Exponential Function
5. The Logarithmic Function
6. Power Functions
7. Projective Spaces
8. Projective Transformations
9. Configuration Space of 4 Points on the Projective Line
10. An Easy-going Realization (Cross Ratio)
11. A Democratic Realization
12. Configuration Space of n Points on Projective Spaces
13. The Grassmann Isomorphism X(k,n) and X(n-k,n)
14. Configuration Spaces of Unlabeled Point Sets
14.1. X{4} in Terms of a Cross Ratio
14.2. X{4} in Terms of the Democratic Realization
Chapter II. Elliptic Curves
1. Lattices in C
2. Elliptic Curves as Quotients of C by Lattices
3. Isomorphism Classes of Elliptic Curves
4. Realization in Terms of the Weierstrass P Function
4.1. Elliptic Functions in General
4.2. The Weierstrass P Function
4.3. The Algebraic Relation between P and P'
4.4. A Realization
4.5. Cubic Plane Curves
4.6. Elliptic Curves as Double Covers of the Line
4.7. The Lambda Function - A Realization of H/Γ(2)
4.8. The J-invariant - A Realization of H/SL(2,Z)
5. A Realization in Terms of the Theta Functions
5.1. Coffee Break? - Pencils of Quadrics
5.2. Theta Functions
5.3. Number of Zeros of Theta Functions
5.4. Position of Zeros of Theta Functions
5.5. A Projective Embedding
5.6. How Does the Image Look?
5.7. Invariants of the Space Curves in Question
5.8. The Values of the Theta Functions at 0,1 and ∞
Chapter III. Modular Interpretations of X(2,4)
1. The Hypergeometric Series
2. The Hypergeometric Differentiai Equation
3. Another Solution around the Origin
4. Symmetries of the Hypergeometric Equation
5. Time to Pay
6. The Schwarz Map and Schwarz Triangles
7. Schwarz's Reflection Principle
8. Modular Interpretations
8.1. l/p + l/q + l/r > 1
8.2. l/p + l/q + l/r = 1
8.3. l/p + l/q + l/r < 1
Chapter IV. Hypergeometric Integrals and Loaded Cycles
1. Hypergeometric Integrais
2. Paths of Integration
2.1. The Segment (0,1)
2.2. A Double Contour Loop around 0 and 1
2.3. The Euler Transformation
2.4. The Derivation ∇ Acting on Rational Forms
2.5. The Relation between the Two Kinds of Paths
3. Loaded Paths
4. Relations among Loaded Cycles
5. Monodromy of Loaded Cycles and of Hypergeometric Functions
6. Invariant Hermitian Forms
7. Intersections of Loaded Cycles
8. A Review of the Modular Interpretation of X(4)
9. The Relation between s(μ) and the Hermitian form H(α)
10. Periods of Curves
11. Excuse for My Using Many Kinds of Parameters
12. Toward Generalizations
Part 2 The Story of the Configuration Space X(2,n) of n Points on the Projective Line
Chapter V. The Configuration Space X(2,5)
1. Juzu Sequences
2. Blowing Up and Down
3. The Democratic Compactification ̅X_R
4. The Democratic Compactification ̅X
5. The Orbifold ̅X /(c)
6. The Graph G
7. A Presentation of the Fundamental Group of X
Chapter VI. Modular Interpretation of the Configuration Space X(2,n)
1. Admissible Sequences
2. Families of Curves and Their Periods
3. Typical Examples Modeled after B_n
Part 3 The Story of the Configuration Space X(3, 6) of Six Lines on the Projective Plane
Chapter VII. The Configuration Space X(3,6)
1. One Attempt to Make a Democratic Projective Embedding
2. A Non-democratic Embedding of X
3. The Involution *
4. A Democratie Embedding
5. Degenerate Arrangements
6. A Democratie Compactification ̅X(3,6)
7. Intersection Pattern of the Divisors ̅X₃^abc} and ̅Q in ̅X
8. The Structure of X_R 3,6) ⊂ ̅X_R (3,6)
8.1. Four Types of Arrangements
8.2. Adjacency of Chambers
8.3. Intersections of the Closures of Adjacent Chambers
8.4. The Shapes of Chambers
8.5. The Action of the Weyl Group W(E₆ )
Chapter VIII. Hypergeometric Functions of Type (3,6)
1. Hypergeometric Integrals of Type (3,6)
2. Domains of Integration, Loaded Cycles
2.1. The Submanifold Q of X and a Base Arrangement
2.2. Loaded Cycles
2.3. Regularizations
3. Intersections of Loaded Cycles and the Invariant Form
3.1. The Intersection Matrix and the Invariant Form H
3.2. Deformations of Loaded Cycles
3.3. Intersections of Loaded Cycles
3.4. Intersection Numbers for D_{12} , ... , D_{34}
3.5. Higher Dimensional Pochhammer Loops
4. Monodromy of Loaded Cycles
4.1. The Circuit Matrix M(l,...,r+1;α)
4.2. The Circuit Matrix M(123;α) as a Quasi-reflection
4.3. Circuit Matrices M (j_l,..., j_{r+l};α)
4.4. Monodromy of the Loaded Cycles
5. The Hypergeometric System E(k,n;α)
6. Local Properties of E(3,6;α)
6.1. Transforming the System into a Pfaffian Form
6.2. Expanding Solutions in Power Series
7. The Duality of E(3,6;α)
Chapter IX. Modular Interpretation of the Configuration Space X(3,6)
1. A Family of K3 Surfaces
2. The Riemann Equality and the Riemann Inequality
3. The Monodromy Group MG as a Reflection Group
3.1. A Cosmetic Treatment
3.2. The Geometrie Meaning of the Reflections
4. The Monodromy Group as a Congruence Subgroup on D
5. The Map Φ : X -> D and Its Extension to ̅X'
6. Boundary Components
7. The Map Φ along the Strata X_{2α} and φ : ̅X ->̅D/Γ_A(2)
8. The Relation between the Involution * on ̅X and the Map φ
9. The Symmetric Domain H₂
9.1. The Isomorphism D ->H₂
9.2. The Isomorphism ι_l : D-> H₂
9.3. The Isomorphism ι₂ : D ->D
9.4. The Isomorphism ι : H₂ ->D
10. The Final Form of the Modular Interpretation
10.1. The Monodromy Group as a Congruence Subgroup on H₂
10.2. A Paraphrase of Theorem 7.3
11. The Structure of the Cusps
11.1. Linear Parabolic Parts
11.2. Reflection Groups
11.3. Coxeter Graphs and Weyl Chambers
12. Theta Functions on H₂ Giving the Inverse of ψ : X ->H₂
12.1. Theta Functions Θ on H₂
12.2. Relations between Θ and Riemann's Theta Functions
12.3. Transformation Formulae
12.4. Quadratic Relations among the Theta Functions
12.5. Coding the Theta Functions
12.6. Modular Forms on H₂
12.7. Inverse of the Map ψ : X/<*> -> H₂/Γ_T(1+i)
Bibliography
