Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005

Contents......Page 8
Preface......Page 6
Hyperbolic Geometry and the Moduli Space of Real Binary Sextics......Page 10
Gauss' Hypergeometric Function......Page 32
Moduli of K3 Surfaces and Complex Ball Quotients......Page 52
Macbeaths Infinite Series of Hurwitz Groups......Page 110
Relative Proportionality on Picard and Hilbert Modular Surfaces......Page 118
1. Preface......Page 120
2. Introduction......Page 124
3.1. Galois weights......Page 127
3.2. Orbital releases......Page 131
3.3. Homogeneous points......Page 135
3.4. Picard and Hilbert orbifaces......Page 138
3.5. Orbital arithmetic curves......Page 141
4. Neat Proportionality......Page 144
5.1. Ten rules for the construction of orbital heights and invariants......Page 150
5.2. Rational and integral self-intersections......Page 153
5.3. The decomposition laws......Page 155
5.5. Degree formula for smooth coverings......Page 157
5.6. The shift implications and orbital self-intersection......Page 158
5.7. Orbital Euler heights for curves......Page 161
5.8. Released weights......Page 164
6. Relative proportionality relations, explicit and general......Page 165
7. Orbital Heegner Invariants and Their Modular Dependence......Page 166
8. Appendix: Relevant Elliptic Modular Forms of Nebentypus......Page 169
References......Page 171
Hypergeometric Functions and Carlitz Differential Equations over Function Fields......Page 172
The Moduli Space of 5 Points on P[sup(1)] and K3 Surfaces......Page 197
Introduction......Page 215
1.1. Definition and first properties......Page 218
1.2. L-slits......Page 220
1.3. The rank of the Schwarz map......Page 222
1.4. When points coalesce......Page 223
Parabolic clustering......Page 224
1.5. Monodromy group and monodromy cover......Page 226
1.6. Invariant Hermitian forms......Page 227
1.7. Cohomological interpretation via local systems of rank one......Page 230
2.1. Monodromy defined by a simple Dehn twist......Page 232
2.2. Extension of the evaluation map......Page 233
2.3. The elliptic and parabolic cases......Page 234
3.1. A projective set-up......Page 237
3.2. Extending the range of applicability......Page 241
4.1. Cyclic covers of P[sup(1)]......Page 243
4.2. Arithmeticity......Page 244
4.3. Working over a ring of cyclotomic integers......Page 245
5.1. Higher dimensional integrals......Page 248
5.2. Geometric structures on arrangement complements......Page 249
References......Page 251
Invariant Functions with Respect to the Whitehead-Link......Page 253
1. Introduction......Page 254
2. The λ-function and the j-function......Page 255
3. Theta constants......Page 257
4. Theta functions on D......Page 261
5. A hyperbolic structure on the complement of the Whitehead link......Page 263
6. Discrete subgroups of GL[sub(2)](C), in particular Λ......Page 265
7. Symmetry of the Whitehead link......Page 268
8. Orbit spaces under W, SΓ[sub(0)](1 + i) and Λ......Page 269
9. Embedding of H[sup(3)] into D......Page 270
11. Invariant functions for Λ and an embedding of H[sup(3)]/Λ......Page 272
12. Invariant functions for W......Page 273
13. Embeddings of the quotient spaces......Page 274
References......Page 278
On the Construction of Class Fields by Picard Modular Forms......Page 280
Algebraic Values of Schwarz Triangle Functions......Page 293
1. Introduction......Page 319
2.1. Roots of polynomial equations......Page 324
2.2. Integral with polynomial integrand......Page 325
2.3. Integral with k-variable Laurent polynomial integrand......Page 326
2.5. General GKZ systems of differential equations......Page 327
2.6. Gauss's hypergeometric differential equation as a GKZ system......Page 328
2.7. Dimension of the solution space of a GKZ system......Page 329
3.1. The Γ-function......Page 330
3.2. Examples of Γ-series......Page 331
3.3. Growth of coefficients of Γ-series......Page 333
3.4. Γ-series and power series......Page 334
3.6. Γ-series and GKZ differential equations......Page 335
4. The Secondary Fan......Page 336
4.1. Construction of the secondary fan......Page 337
4.2. Alternative descriptions for secondary fan constructions......Page 339
4.3. The Secondary Polytope......Page 343
5.1. Construction of the toric variety for the secondary fan......Page 344
5.2. Convergence of Fourier Γ-series and the secondary fan......Page 347
5.3. Solutions of GKZ differential equations and the secondary fan......Page 348
6. Extreme resonance in GKZ systems......Page 350
7.1. Series, L, A and the primary polytope Δ[sub(A)]......Page 356
7.2. Integrals and differential equations for F[sub(D)]......Page 357
7.3. Triangulations of Δ[sub(A)], secondary polytope and fan for F[sub(D)]......Page 359
8.1. GKZ data from Calabi–Yau varieties......Page 361
8.2. The quintic in P[sub(4)]......Page 363
8.3. The intersection of two cubics in P[sub(5)]......Page 366
8.4. The hypersurface of degree (3, 3) in P[sub(2)] × P[sub(2)]......Page 368
8.5. The Schwarz map for some extended GKZ systems......Page 370
8.6. Manifestations of Mirror Symmetry......Page 374
References......Page 375
Orbifolds and Their Uniformization......Page 378
Introduction......Page 379
1. Branched Coverings......Page 380
1.1. Branched coverings of P[sup(1)]......Page 384
1.2. Fenchel's problem......Page 386
2.1. Transformation groups......Page 387
2.2. Transformation groups and branched coverings......Page 388
2.3. b-spaces and orbifolds......Page 389
2.4. A criterion for uniformizability......Page 391
2.5. Sub-orbifolds and orbifold coverings......Page 392
2.6. Covering relations among triangle orbifolds......Page 393
3. Orbifold Singularities......Page 395
3.1. Orbiface singularities......Page 396
3.2. Covering relations among orbiface germs......Page 397
3.3. Orbifaces with cusps......Page 398
4. Orbifaces......Page 400
4.1. Orbifaces (P[sup(2)],D) with an abelian uniformization......Page 401
4.2. Covering relations among orbifaces (P[sup(2)],D) uniformized by P[sup(2)]......Page 405
From the Power Function to the Hypergeometric Function......Page 412
1.2. Differential equations......Page 413
2. Power functions......Page 414
2.1. When a is real......Page 415
2.2. When a is purely imaginary......Page 416
3.1. At a regular point......Page 417
3.3. Around x = 1 and ∞......Page 418
4.2. A set of generators of the monodromy group......Page 419
5. The Schwarz map of the hypergeometric differential equation with real exponents......Page 420
5.1. Real but general exponents......Page 422
Problem Session......Page 435