The book contains a complete self-contained introduction to highlights of classical complex analysis. New proofs and some new results are included. All needed notions are developed within the book: with the exception of some basic facts which can be found in the ¯rst volume. There is no comparable treatment in the literature.

Review

“The book under review is the second volume of the textbook Complex analysis, consisting of 8 chapters. It provides an approach to the theory of Riemann surfaces from complex analysis. … The book is self-contained and, moreover, some notions which might be unfamiliar for the reader are explained in appendices of chapters. … this book is an excellent textbook on Riemann surfaces, especially for graduate students who have taken the first course of complex analysis.” (Hiroshige Shiga, Mathematical Reviews, Issue 2012 f)

“The book under review is largely self-contained, pleasantly down-to-earth, remarkably versatile, and highly educating simultaneously. No doubt, this fine textbook provides an excellent source for the further study of more advanced and topical themes in the theory of Riemann surfaces, their Jacobians and moduli spaces, and in the general theory of complex Abelian varieties and modular forms likewise. It is very welcome that the English translation of the German original has been made available so quickly!” (Werner Kleinert, Zentralblatt MATH, Vol. 1234, 2012)

“The author provides a (very brief) introduction to fundamental notions of topology, but develops fully the theory of surfaces and covering spaces he needs. … the book includes a proof of the classification of compact orientable surfaces by their genus. … this one is definitely a graduate text. … There is a lot of mathematics in this book, presented efficiently and well. … It is a book I am glad to have, and that I will certainly refer to in the future.” (Fernando Q. Gouvêa, The Mathematical Association of America, May, 2012)

Table of Contents

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

Exercises for Sect. 1.3

II. Harmonic Functio ns 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

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. Construction 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

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

Exercises for the Appendices to Chap. Ill

IV. Compact Riemann Surfaces

1. Meromorphic Differentials

2. Compact Riemann Surfaces and Algebraic Functions

3. The Triangulation of a Compact Riemann Surface

4. Combinatorial Schemes

5. Gluing of Boundary Edges

6. The Normal Form of Compact Riemann Surfaces

7. Differentials of the First Kind

8. Some Period Relations

9. The Riemann- Roch Theorem

10. More Period Relations

11. Abel's Theorem

12. The Jacobi Inversion Problem

Appendices to Chapter IV. Dimension Formulae for Spaces of Modular Forms

13. Multicanonical Forms

14. Dimensions of Vector Spaces of Modular Forms

15. Dimensions of Vector Spaces of Modular Forms with Multiplier Systems

V. Analytic Functio ns 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

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,1,E))

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

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. lgusa'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

VIII. Appendix: Algebraic Tools

1. Divisibility

2. Factorial Rings (UFD rings)

3. The Discriminant

4. Algebraic Function Fields