This book consists of two parts. The first is devoted to the theory of curves, which are treated from both the analytic and algebraic points of view. Starting with the basic notions of the theory of Riemann surfaces the reader is lead into an exposition covering the Riemann-Roch theorem, Riemann's fundamental existence theorem.uniformization and automorphic functions. The algebraic material also treats algebraic curves over an arbitrary field and the connection between algebraic curves and Abeli

Introduction by I.R.Shafaxevich
Chapter 1.Riemann Surfaces
§1.Basic Notions
1.1.Complex Chart；Complex Coordinates
1.2.Complex Analytic Atlas
1.3.Complex Analytic Manifolds
1.4.Mappings of Complex Manifolds
1.5.Dimension of a Complex Manifold
1.6.Riemann Surfaces
1.7.Di6erentiable Manifolds
§2.Mappings of Riemann Surfaces
2.1.Nonconstant Mappings of Riemann Surfaces axe Discrete
2.2.Meromorphic Functions on a Pdemann Surface
2.3.Meromorphic Functions With Prescribed Behaviour at Poles
2.4.Multiplicity of a Mapping；Order of a Function
2.5.Topological Properties of Mappings of Riemann Surfaces
2.6.Divisors on Riemann Surfaces
2.7.Finite Mappings of Riemann Surfaces
2.8.Unramified Coverings of Pdemann Surfaces
2.9.The Universal Covering
2.10.COntinuation of Mappings
2.n.The Riemann Surface of al2 Algebraic Function
§3.Topology of Riemann Surfaces
3.1.Orientability
3.2.Triangulability
3.3.Development；Topological Genus
3.4.Structure of the Fundamental Group
3.5.The Euler Characteristic
3.6.The Hurwitz Formulae
3.7.Homology and Cohomology；Betti Numbers
3.8.Intersection Product；PoincareDUalitV
§4.Calculus on Riemann Surfaces
4.1.Tangent Vectors；Differentiations
4.2.Differential Forms
4.3.Exterior Differentiations；de Rham Cohomology
4.4.Kihler and Riemann Metrics
4.5.Integration of Exterior Differentials；Gteen，s Formula
4.6.Periods；Rham Isomorphism
4.7.Holomorphic Diffentials；Geometric Genus；Riemann，S Bilinear Delations
4.8.Meromorphic Differentials；Canonical Divisors
4.9.Meromorphic Differentials with Prescribed Behaviour at P0les；Residues
4.10.Periods of Meromorphic Differentials
4.11.Harmonic Differentials
4.12.Hilbert Space of Differentials；Harmonic Projection
4.13.Hodge Decomposition
4.14.Existence of Meromorphic Differentials and Functions
4.15.Dirichlet’S Principle
§5.Classification of njemann Surfaces
5.1.Canonical Regions
5.2.Uniformization
5.3.Types of Riemann Surfaces
5.4.Automorphisms ofCanonical Regions
5.5.Pdemann Surfaces of Elliptic Type
5.6.Riemann Surfaces of Parabolic Type
5.7.Riemann Surfaces ofHyperbolic Type
5.8.Automorphic Forms；Poincar6 Series
5.9.Quotient Riemann Surfaces；the Absolute Invariant
5.10.Moduli of Riemann Surfaces
§6.Algebraic Nature of Compact Riemann Surfaces
6.1.Function Spaces and Mappings Associated with Divisors
6.2.Riemann.RDch Formula；Reciprocity Law for Differentialsof the First and Second Kind
6.3.Applications of the Riemann—nDch Formula to Problems0f Existence of Meromorphic Functions and Differentials
6.4.Compact Riemann Surfaces are Projective
6.5.Algebraic Nature of Projective Models；Arithmetic Riemann Surfaces
6.6.Models of Riemann Surfaces of Genus l
Chapter 2.Algebraic Curves
Chapter 3.Jaclbians and Abelian Varieties
References