Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions

521 Pages·2011·2.09 MB·English

by  Eberhard Freitag (auth.)

Universitext
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Eberhard Freitag
C
Complex Analysis 2
Riemann Surfaces, Several Complex Variables,
Abelian Functions, Higher Modular Functions
123
Eberhard Freitag
Universität Heidelberg
Mathematisches Institut
Im Neuenheimer Feld 288
69120 Heidelberg
Germany
[email protected]
Editorialboard:
SheldonAxler,SanFranciscoStateUniversity
VincenzoCapasso,UniversitàdegliStudidiMilano
CarlesCasacuberta,UniversitatdeBarcelona
AngusJ.MacIntyre,QueenMary,UniversityofLondon
KennethRibet,UniversityofCalifornia,Berkeley
ClaudeSabbah,CNRS,ÉcolePolytechnique
EndreSüli,UniversityofOxford
WojborA.Woyczynski,CaseWesternReserveUniversity
ISSN 0172-5939 e-ISSN 2191-6675
ISBN978-3-642-20553-8 e-ISBN978-3-642-20554-5
DOI10.1007/978-3-642-20554-5
SpringerHeidelbergDordrechtLondonNewYork
Library of Congress Control Number: 2011930840
Mathematics Subject Classification (2010): 30, 31, 32
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Contents
Chapter I. Riemann Surfaces 1
0 Basic Topological Notions 3
1 The Notion of a Riemann Surface 13
2 The Analytisches Gebilde 27
3 The Riemann Surface of an Algebraic Function 35
Appendix A. A Special Case of Covering Theory 46
Appendix B. A Theorem of Implicit functions 51
Chapter II. Harmonic Functions on Riemann Surfaces 54
1 The Poisson Integral Formula 56
2 Stability of Harmonic Functions on Taking Limits 60
3 The Boundary Value Problem for Disks 63
4 The Formulation of the Boundary Value Problem on Riemann Surfaces
and the Uniqueness of the Solution 68
5 Solution of the Boundary Value Problem by Means of the
Schwarz Alternating Method 74
6 The Normalized Solution of the External Space Problem 80
Appendix. Countability of Riemann Surfaces 88
7 Construction of Harmonic Functions with Prescribed
Singularities: The Bordered Case 91
8 Construction of Harmonic Functions with a Logarithmic
Singularity: The Green’s Function 95
9 Construction of Harmonic Functions with a
Prescribed Singularity: The Case of a Positive Boundary 99
10 A Lemma of Nevanlinna 103
11 Construction of Harmonic Functions with a
Prescribed Singularity: The Case of a Zero Boundary 113
12 The Most Important Cases of the Existence Theorems 118
13 Appendix to Chapter II. Stokes’s Theorem 120
Chapter III. Uniformization 141
1 The Uniformization Theorem 143
2 A Rough Classification of Riemann Surfaces 152
VIII Contents
3 Picard’s Theorems 159
4 Appendix A. The Fundamental Group 163
5 Appendix B. The Universal Covering 170
6 Appendix C. The Monodromy Theorem 180
Chapter IV. Compact Riemann Surfaces 184
1 Meromorphic Differentials 184
2 Compact Riemann Surfaces and Algebraic Functions 192
3 The Triangulation of a Compact Riemann Surface 204
Appendix. The Riemann-Hurwitz Ramification Formula 208
4 Combinatorial Schemes 211
5 Gluing of Boundary Edges 218
6 The Normal Form of Compact Riemann Surfaces 222
7 Differentials of the First Kind 231
Appendix. The Polyhedron Theorem 237
8 Some Period Relations 238
Appendix. Piecewise smoothness 243
9 The Riemann–Roch Theorem 245
10 More Period Relations 254
11 Abel’s Theorem 261
12 The Jacobi Inversion Problem 270
Appendix. Continuity of Roots 278
Appendices to Chapter IV 279
13 Multicanonical Forms 279
14 Dimensions of Vector Spaces of Modular Forms 285
15 Dimensions of Vector Spaces of Modular Forms
with Multiplier Systems 295
Chapter V. Analytic Functions of Several Complex Variables 300
1 Elementary Properties of Analytic Functions
of Several Variables 300
2 Power Series in Several Variables 302
3 Analytic Maps 311
4 The Weierstrass Preparation Theorem 317
5 Representation of Meromorphic Functions as Quotients
of Analytic Functions 326
6 Alternating Differential Forms 337
Contents IX
Chapter VI. Abelian Functions 347
1 Lattices and Tori 347
2 Hodge Theory of the Real Torus 352
3 Hodge Theory of a Complex Torus 355
4 Automorphy Summands 357
5 Quasi-Hermitian Forms on Lattices 364
6 Riemannian Forms 372
7 Canonical Lattice Bases 378
8 Theta Series (Construction of the Spaces [Q,l,E]) 384
Appendix. Complex Fourier Series 392
9 Graded Rings of Theta Series 393
10 A Nondegenerateness Theorem 396
11 The Field of Abelian Functions 403
12 Polarized Abelian Manifolds 408
13 The Limits of Classical Complex Analysis 412
Chapter VII. Modular Forms of Several Variables 427
1 Siegel’s Modular Group 427
2 The Notion of a Modular Form of Degree n 431
3 Koecher’s Principle 435
4 Specialization of Modular Forms 438
5 Generators for Some Modular Groups 441
6 Computation of Some Indices 449
7 Theta series 452
8 Group-Theoretic Considerations 459
9 Igusa’s Congruence Subgroups 462
10 The Fundamental Domain of the Modular Group of Degree Two 466
11 The Zeros of the Theta Series of Degree two 469
12 A Ring of Modular Forms 475
Chapter VIII. Appendix: Algebraic Tools 483
1 Divisibility 483
2 Factorial Rings (UFD rings) 485
3 The Discriminant 488
4 Algebraic Function Fields 490
References 494
Index 499