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 functionsChapter 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 Greens 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. Stokess TheoremChapter III. Uniformization 1 The Uniformization Theorem 2 A Rough Classification of Riemann Surfaces 3 Picards Theorems 4 Appendix A. The Fundamental Group 5 Appendix B. The Universal Covering 6 Appendix C. The Monodromy TheoremChapter 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 Abels 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 SystemsChapter 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 ContentsChapter 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 AnalysisChapter VII. Modular Forms of Several Variables 1 Siegels Modular Group 2 The Notion of a Modular Form of Degree n 3 Koechers 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 Igusas 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 FormsChapter VIII. Appendix: Algebraic Tools 1 Divisibility 2 Factorial Rings (UFD rings) 3 The Discriminant 4 Algebraic Function FieldsReferencesIndex