This short book gives an introduction to algebraic and abelian functions, withemphasis on the complex analytic point of view. It could be used for a course　or seminar addressed to second year graduate students.

The goal is the same as that of the first edition, although I have made a　number of additions. I have used the Weil proof of the Riemann-Roch the　orem since it is efficient and acquaints the reader with adeles, which are a very　useful tool pervading number theory.

Chapter Ⅰ The Riemann-Roch Theorem

1.Lemmas on Valuations

2.The Riemann-Roch Theorem

3.Remarks on Differential Forms

4.Residues in Power Series Fields

5.The Sum of the Residues

6.The Genus Formula of Hurwitz

7.Examples

8.Differentials of Second Kind

9.Function Fields and Curves

10.Divisor Classes

Chapter Ⅱ The Fermat Curve

1.The Genus

2.Differentials

3.Rational Images of the Fermat Curve

4.Decomposition of the Divisor Classes

Chapter Ⅲ The Riemann Surface

1.Topology and Analytic Structure

2.Integration on the Riemann Surface

Chapter Ⅳ The Theorem of Abel-Jacobi

1.Abelian Integrals

2.Abel's Theorem

3.Jacobi's Theorem

4.Riemann's Relations

5.Duality

Chapter Ⅴ Periods on the Fermat Curve

1.The Logarithm Symbol

2.Periods on the Universal Covering Space

3.Periods on the Fermat Curve

4.Periods on the Related Curves

Chapter Ⅵ Linear Theory of Theta Functions

1.Associated Linear Forms

2.Degenerate Theta Functions

3.Dimension of the Space of Theta Functions

4.Abelian Functions and Riemann-Roch Theorem on the Toru

5.Translations of Theta Functions

6.Projective Embedding

Chapter Ⅶ Homomorphisms and Duality

1.The Complex and Rational Representations

2.Rational and p-adic Representations

3.Homomorphisms

4.Complete Reducibility of Poincar

5.The Dual Abelian Manifold

6.Relations with Theta Functions

7.The Kummer Pairing

8.Periods and Homology

Chapter Ⅷ Riemann Matrices and Classical Theta Functions

1.Riemann Matrices

2.The Siegel Upper Half Space

3.Fundamental Theta Functions

ChapterⅨ

Involutions and Abelian Manifolds of Quaternion Type

1.Involutions

2.Special Gnerators

3.Orders

4.Lattices and Riemann Forms on Determined by Quaternion Algebras

5.Isomorphism Classes

ChapteⅩ

Theta Functions and Divisors

I.Positive Divisors

2.Arbitrary Divisors

3.Existence of a Riemann Form on an Abelian Variety

Bibliography

Index