Hypergeometric functions, my love: modular interpretations of configuration spaces
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Braunschweig [u.a.]
Vieweg
1997
|
| Series: | [Aspects of mathematics / E]
32 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 292 S. |
| ISBN: | 3528069252 |
Staff View
MARC
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| 100 | 1 | |a Yoshida, Masaaki |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Hypergeometric functions, my love |b modular interpretations of configuration spaces |c Masaaki Yoshida |
| 264 | 1 | |a Braunschweig [u.a.] |b Vieweg |c 1997 | |
| 300 | |a XVI, 292 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 4 | |a Fonctions hypergéométriques | |
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| 650 | 4 | |a Hypergeometric functions | |
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Record in the Search Index
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| adam_text | Contents
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 3
1. Classifications and Equivalence Relations 3
2. Quotient Spaces 5
3. Realizations 6
4. The Exponential Function 7
5. The Logarithmic Function 11
6. Power Functions 13
7. Projective Spaces 13
8. Projective Transformations 14
9. Configuration Space of 4 Points on the Projective Line 16
10. An Easy going Realization (Cross Ratio) 20
11. A Democratic Realization 21
12. Configuration Space of n Points on Projective Spaces 23
13. The Grassmann Isomorphism X(k,n) and X(n — k,n) 24
14. Configuration Spaces of Unlabeled Point Sets 25
14.1. A {4} in Terms of a Cross Ratio 25
14.2. X{4} in Terms of the Democratic Realization 27
xi
xu CONTENTS
Chapter II. Elliptic Curves 29
1. Lattices in C 30
2. Elliptic Curves as Quotients of C by Lattices 30
3. Isomorphism Classes of Elliptic Curves 31
4. Realization in Terms of the Weierstrass p Function 34
4.1. Elliptic Functions in General 35
4.2. The Weierstrass p Function 36
4.3. The Algebraic Relation between p and p 38
4.4. A Realization 39
4.5. Cubic Plane Curves 40
4.6. Elliptic Curves as Double Covers of the Line 41
4.7. The Lambda Function A Realization of H/T(2) 41
4.8. The ./ invariant A Realization of E/SL(2,Z) 42
5. A Realization in Terms of the Theta Functions 43
5.1. Coffee Break? Pencils of Quadrics 43
5.2. Theta Functions 50
5.3. Number of Zeros of Theta Functions 51
5.4. Position of Zeros of Theta Functions 52
5.5. A Projective Embedding 53
5.6. How Does the Image Look? 54
5.7. Invariants of the Space Curves in Question 55
5.8. The Values of the Theta Functions at 0,1 and oo 56
Chapter III. Modular Interpretations of X(2,4) 59
1. The Hypergeometric Series 59
2. The Hypergeometric Differential Equation 60
3. Another Solution around the Origin 62
4. Symmetries of the Hypergeometric Equation 64
5. Time to Pay 65
6. The Schwarz Map and Schwarz Triangles 67
7. Schwarz s Reflection Principle 69
CONTENTS xiii
8. Modular Interpretations 72
8.1. l/p + l/q + l/r l 72
8.2. l/p+l/q+l/r = l 75
8.3. l/p+l/q+l/r l 76
Chapter IV. Hypergeometric Integrals and Loaded Cycles 81
1. Hypergeometric Integrals 81
2. Paths of Integration 83
2.1. The Segment (0,1) 84
2.2. A Double Contour Loop around 0 and 1 85
2.3. The Euler Transformation 86
2.4. The Derivation V Acting on Rational Forms 88
2.5. The Relation between the Two Kinds of Paths 90
3. Loaded Paths 91
4. Relations among Loaded Cycles 93
5. Monodromy of Loaded Cycles and of Hypergeometric
Functions 95
6. Invariant Hermitian Forms 98
7. Intersections of Loaded Cycles 100
8. A Review of the Modular Interpretation of X(4) 103
9. The Relation between s(/x) and the Hermitian form H(a) 105
10. Periods of Curves 107
11. Excuse for My Using Many Kinds of Parameters 108
12. Toward Generalizations 109
Part 2
The Story of the Configuration Space X(2, n) of n Points on the
Projective Line
Chapter V. The Configuration Space A (2,5) 113
1. Juzu Sequences 114
2. Blowing Up and Down 116
3. The Democratic Compactification XK 118
4. The Democratic Compactification X~ 120
xiv CONTENTS
5. The Orbifold X/(c) 123
6. The Graph G 129
7. A Presentation of the Fundamental Group of X 130
Chapter VI. Modular Interpretation of the Configuration
Space X(2, n) 135
1. Admissible Sequences 136
2. Families of Curves and Their Periods 141
3. Typical Examples Modeled after ln 143
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) 149
1. One Attempt to Make a Democratic Projective Embed¬
ding 150
2. A Non democratic Embedding of X 151
3. The Involution * 154
4. A Democratic Embedding 157
5. Degenerate Arrangements 162
6. A Democratic Compactification X(3,6) 163
7. Intersection Pattern of the Divisors 3^ ° and Q in X 166
8. The Structure of XR(3,6) C XR(3,6) 169
8.1. Four Types of Arrangements 169
8.2. Adjacency of Chambers 170
8.3. Intersections of the Closures of Adjacent Chambers 173
8.4. The Shapes of Chambers 175
8.5. The Action of the Weyl Group W(E6) 178
Chapter VIII. Hypergeometric Functions of Type (3,6) 181
1. Hypergeometric Integrals of Type (3,6) 181
2. Domains of Integration, Loaded Cycles 183
2.1. The Submanifold Q of X and a Base Arrangement 184
2.2. Loaded Cycles 186
2.3. Regularizations 190
CONTENTS xv
3. Intersections of Loaded Cycles and the Invariant Form 191
3.1. The Intersection Matrix and the Invariant Form H 191
3.2. Deformations of Loaded Cycles 193
3.3. Intersections of Loaded Cycles 196
3.4. Intersection Numbers for D^,..., Z)34 200
3.5. Higher Dimensional Pochhammer Loops 202
4. Monodromy of Loaded Cycles 205
4.1. The Circuit Matrix M(l,...,r + 1;a) 205
4.2. The Circuit Matrix M(123;a) as a Quasi reflection 209
4.3. Circuit Matrices M(j ,... ,jr+i , a) 211
4.4. Monodromy of the Loaded Cycles 214
5. The Hypergeometric System E(k, n; a) 215
6. Local Properties of E(3,6; a) 220
6.1. Transforming the System into a Pfaffian Form 222
6.2. Expanding Solutions in Power Series 223
7. The Duality of £(3,6; a) 224
Chapter IX. Modular Interpretation of the Configuration
Space X(3,6) 229
1. A Family of K3 Surfaces 230
2. The Riemann Equality and the Riemann Inequality 232
3. The Monodromy Group MG as a Reflection Group 234
3.1. A Cosmetic Treatment 234
3.2. The Geometric Meaning of the Reflections 238
4. The Monodromy Group as a Congruence Subgroup on
D 241
5. The Map f : X »D and Its Extension to It 243
6. Boundary Components 245
7. The Map j along the Strata X2a and p : ~X D/TA{2) 247
8. The Relation between the Involution * on X and the
Map ip 251
xvi CONTENTS
9. The Symmetric Domain H2 253
9.1. The Isomorphism D^H2 253
9.2. The Isomorphism ix : D^U2 254
9.3. The Isomorphism t2 : D ^B 256
9.4. The Isomorphism i : H2 ^D 258
10. The Final Form of the Modular Interpretation 258
10.1. The Monodromy Group as a Congruence Subgroup on H2 258
10.2. A Paraphrase of Theorem 7.3 260
11. The Structure of the Cusps 262
11.1. Linear Parabolic Parts 262
11.2. Reflection Groups 263
11.3. Coxeter Graphs and Weyl Chambers 265
12. Theta Functions on H2 Giving the Inverse of V : X H2269
12.1. Theta Functions 0 on H2 270
12.2. Relations between 0 and Riemann s Theta Functions 271
12.3. Transformation Formulae 272
12.4. Quadratic Relations among the Theta Functions 272
12.5. Coding the Theta Functions 273
12.6. Modular Forms on H2 274
12.7. Inverse of the Map V : X/(*) » H2/rr(l + i) 275
Bibliography 277
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| indexdate | 2024-07-09T18:06:38Z |
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| spelling | Yoshida, Masaaki Verfasser aut Hypergeometric functions, my love modular interpretations of configuration spaces Masaaki Yoshida Braunschweig [u.a.] Vieweg 1997 XVI, 292 S. txt rdacontent n rdamedia nc rdacarrier [Aspects of mathematics / E] 32 Espace de configuration Espace de configuration ram Fonctions hypergéométriques Fonctions hypergéométriques ram Hypergeometrische functies gtt Configuration space Hypergeometric functions Hypergeometrische Reihe (DE-588)4161061-1 gnd rswk-swf Konfigurationsraum (DE-588)4199256-8 gnd rswk-swf Konfigurationsraum (DE-588)4199256-8 s Hypergeometrische Reihe (DE-588)4161061-1 s DE-604 E] [Aspects of mathematics 32 (DE-604)BV000018737 32 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007555389&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yoshida, Masaaki Hypergeometric functions, my love modular interpretations of configuration spaces Espace de configuration Espace de configuration ram Fonctions hypergéométriques Fonctions hypergéométriques ram Hypergeometrische functies gtt Configuration space Hypergeometric functions Hypergeometrische Reihe (DE-588)4161061-1 gnd Konfigurationsraum (DE-588)4199256-8 gnd |
| subject_GND | (DE-588)4161061-1 (DE-588)4199256-8 |
| title | Hypergeometric functions, my love modular interpretations of configuration spaces |
| title_auth | Hypergeometric functions, my love modular interpretations of configuration spaces |
| title_exact_search | Hypergeometric functions, my love modular interpretations of configuration spaces |
| title_full | Hypergeometric functions, my love modular interpretations of configuration spaces Masaaki Yoshida |
| title_fullStr | Hypergeometric functions, my love modular interpretations of configuration spaces Masaaki Yoshida |
| title_full_unstemmed | Hypergeometric functions, my love modular interpretations of configuration spaces Masaaki Yoshida |
| title_short | Hypergeometric functions, my love |
| title_sort | hypergeometric functions my love modular interpretations of configuration spaces |
| title_sub | modular interpretations of configuration spaces |
| topic | Espace de configuration Espace de configuration ram Fonctions hypergéométriques Fonctions hypergéométriques ram Hypergeometrische functies gtt Configuration space Hypergeometric functions Hypergeometrische Reihe (DE-588)4161061-1 gnd Konfigurationsraum (DE-588)4199256-8 gnd |
| topic_facet | Espace de configuration Fonctions hypergéométriques Hypergeometrische functies Configuration space Hypergeometric functions Hypergeometrische Reihe Konfigurationsraum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007555389&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018737 |
| work_keys_str_mv | AT yoshidamasaaki hypergeometricfunctionsmylovemodularinterpretationsofconfigurationspaces |