Chapter I. Riemann Surfaces
0 Basic Topological Notions
1 The Notion of a Riemann Surface
2 The Analytisches Gebilde
3 The Riemann Surface of an Algebraic Function
Appendix A. A Special Case of Covering Theory
Appendix B. A Theorem of Implicit functions
Chapter II. Harmonic Functions on Riemann Surfaces
1 The Poisson Integral Formula
2 Stability of Harmonic Functions on Taking Limits
3 The Boundary Value Problem for Disks
4 The Formulation of the Boundary Value Problem on Riemann Surfaces
and the Uniqueness of the Solution
5 Solution of the Boundary Value Problem by Means of the
Schwarz Alternating Method
6 The Normalized Solution of the External Space Problem
Appendix. Countability of Riemann Surfaces
7 Construction of Harmonic Functions with Prescribed
Singularities: The Bordered Case
8 Construction of Harmonic Functions with a Logarithmic
Singularity: The Green's Function
9 Constrtwtion of Harmonic Functions with a
Prescribed Singularity: The Case of a Positive Boundary
10 A Lemma of Nevanlinna
11 Construction of Harmonic Functions with a Prescribed
Singularity: The Case of a Zero Boundary
12 The Most Important Cases of the Existence Theorems
13 Appendix to Chapter II. Stokes's Theorem
Chapter III. Uniformization
1 The Uniformization Theorem
2 A Rough Classification of Riemann Surfaces
3 Picard's Theorems
4 Appendix A. The Fundamental Group
5 Appendix B. The Universal Covering
6 Appendix C. The Monodromy Theorem
Chapter IV. Compact Riemann Surfaces
1 Meromorphic Differentials
2 Compact Riemann Surfaces and Algebraic Functions
3 The Triangulation of a Compact Riemann Surface Appendix. The Riemann-Hurwitz Ramification Formula
4 Combinatorial Schemes
5 Gluing of Boundary Edges
6 The Normal Form of Compact Riemann Surfaces
7 Differentials of the First Kind Appendix. The Polyhedron Theorem
8 Some Period Relations Appendix. Piecewise smoothness
9 The Riemann-Roch Theorem
10 More Period Relations
11 Abel's Theorem
12 The Jacobi Inversion Problem Appendix. Continuity of Roots
Appendices to Chapter IV
13 Multicanonical Forms
14 Dimensions of Vector Spaces of Modular Forms
15 Dimensions of Vector Spaces of Modular Forms with Multiplier Systems
Chapter V. Analytic Functions of Several Complex Variables
1 Elementary Properties of Analytic Functions
of Several Variables
2 Power Series in Several Variables
3 Analytic Maps
4 The Weierstrass Preparation Theorem
5 Representation of Meromorphic Functions as Quotients of Analytic Functions
6 Alternating Differential Forms
Contents
Chapter VI. Abelian Functions
1 Lattices and Tori
2 Hodge Theory of the Real Torus
3 Hodge Theory of a Complex Torus
4 Automorphy Summands
5 Quasi-Hermitian Forms on Lattices
6 Riemannian Forms
7 Canonical Lattice Bases
8 Theta Series (Construction of the Spaces [Q, l, E]) Appendix. Complex Fourier Series
9 Graded Rings of Theta Series
10 A Nondegenerateness Theorem
11 The Field of Abelian Functions
12 Polarized Abelian Manifolds
13 The Limits of Classical Complex Analysis
Chapter VII. Modular Forms of Several Variables
1 Siegel's Modular Group
2 The Notion of a Modular Form of Degree n
3 Koecher's Principle
4 Specialization of Modular Forms
5 Generators for Some Modular Groups
6 Computation of Some Indices
7 Theta series
8 Group-Theoretic Considerations
9 Igusa's Congruence Subgroups
10 The Fundamental Domain of the Modular Group of Degree Two
11 The Zeros of the Theta Series of Degree two
12 A Ring of Modular Forms
Chapter VIII. Appendix: Algebraic Tools
1 Divisibility
2 Factorial Rings (UFD rings)
3 The Discriminant
4 Algebraic Function Fields
References
Index
