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CHAPTER 0 An Overview 1

0.1．Topological Aspects,Uniformization,and Fuchsian Groups 2

0.2．Algebraic Functions 4

0.3．Abelian Varieties 6

0.4．More Analytic Aspects 7

CHAPTER Ⅰ Riemann Surfaces 9

Ⅰ.1．Definitions and Examples 9

Ⅰ.2．Topology of Riemann Surfaces 13

Ⅰ.3．Differential Forms 22

Ⅰ.4．Integration Formulae 28

CHAPTER Ⅱ Existence Theorems 32

Ⅱ.1．Hilbert Space Theory—A Quick Review 32

Ⅱ.2．Weyl's Lemma 33

Ⅱ.3．The Hilbert Space of Square Integrable Forms 39

Ⅱ.4．Harmonic Differentials 45

Ⅱ.5．Meromorphic Functions and Differentials 50

CHAPTER Ⅲ Compact Riemann Surfaces 54

Ⅲ.1．Intersection Theory on Compact Surfaces 54

Ⅲ.2．Harmonic and Analytic Differentials on Compact Surfaces 56

Ⅲ.3．Bilinear Relations 64

Ⅲ.4．Divisors and the Riemann-Roch Theorem 69

Ⅲ.5．Applications of the Riemann-Roch Theorem 79

Ⅲ.6．Abel's Theorem and the Jacobi Inversion Problem 91

Ⅲ.7．Hyperelliptic Riemann Surfaces 99

Ⅲ.8．Special Divisors on Compact Surfaces 109

Ⅲ.9．Multivalued Functions 126

Ⅲ.10．Projective Imbeddings 136

Ⅲ.11．More on the Jacobian Variety 142

Ⅲ.12．Torelli's Theorem 161

CHAPTER Ⅳ Uniformization 166

Ⅳ.1．More on Harmonic Functions(A Quick Review) 166

Ⅳ.2．Subharmonic Functions and Perron's Method 171

Ⅳ.3．A Classification of Riemann Surfaces 178

Ⅳ.4．The Uniformization Theorem for Simply Connected Surfaces 194

Ⅳ.5．Uniformization of Arbitrary Riemann Surfaces 203

Ⅳ.6．The Exceptional Riemann Surfaces 207

Ⅳ.7．Two Problems on Moduli 211

Ⅳ.8．Riemannian Metrics 213

Ⅳ.9．Discontinuous Groups and Branched Coverings 220

Ⅳ.10．Riemann-Roch—An Alternate Approach 237

Ⅳ.11．Algebraic Function Fields in One Variable 241

CHAPTER Ⅴ Automorphisms of Compact Surfaces-Elementary Theory 257

Ⅴ.1．Hurwitz's Theorem 257

Ⅴ.2．Representations of the Automorphism Group on Spaces of Differentials 269

Ⅴ.3．Representation of Aut M on H1(M) 286

Ⅴ.4．The Exceptional Riemann Surfaces 293

CHAPTER Ⅵ Theta Functions 298

Ⅵ1.1．The Riemann Theta Function 298

Ⅵ.2．The Theta Functions Associated with a Riemann Surface 304

Ⅵ.3．The Theta Divisor 309

CHAPTER Ⅶ Examples 321

Ⅶ.1．Hyperelliptic Surfaces (Once Again) 321

Ⅶ.2．Relations Among Quadratic Differentials 333

Ⅶ.3．Examples of Non-hyperelliptic Surfaces 337

Ⅶ.4．Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods 348

Ⅶ.5．Examples of Prym Differentials 350

Ⅶ.6．The Trisecant Formula 351

Bibliography 356

Index 359