Riemann Surfaces

Riemann Surfaces by H. M. Farkas, I. Kra.

The present volume is the culmination often years' work separately and joint­ ly. The idea of writing this book began with a set of notes for a course given by one of the authors in 1970-1971 at the Hebrew University. The notes were refined serveral times and used as the basic content of course...

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Bibliographic Details
Main Authors: Farkas, H. M. (Author), Kra, I. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 1980.
Edition:1st ed. 1980.
Series:Graduate Texts in Mathematics, 71
Springer eBook Collection.
Subjects:
Mathematical analysis.
Analysis (Mathematics).
Algebraic geometry.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 0 An Overview
  • 0.1 Topological Aspects, Uniformization, and Fuchsian Groups
  • 0.2 Algebraic Functions
  • 0.3. Abelian Varieties
  • 0.4. More Analytic Aspects
  • I Riemann Surfaces
  • I.1. Definitions and Examples
  • I.2. Topology of Riemann Surfaces
  • I.3. Differential Forms
  • I.4. Integration Formulae
  • II Existence Theorems
  • II.1. Hilbert Space Theory—A Quick Review
  • II.2. Weyl’s Lemma
  • II.3. The Hilbert Space of Square Integrable Forms
  • II.4. Harmonic Differentials
  • II.5. Meromorphic Functions and Differentials
  • III Compact Riemann Surfaces
  • III.1. Intersection Theory on Compact Surfaces
  • III.2. Harmonic and Analytic Differentials on Compact Surfaces
  • III.3. Bilinear Relations
  • III.4. Divisors and the Riemann—Roch Theorem
  • III.5. Applications of the Riemann—Roch Theorem
  • III.6. Abel’s Theorem and the Jacobi Inversion Problem
  • III.7. Hyperelliptic Riemann Surfaces
  • III.8. Special Divisors on Compact Surfaces
  • III.9. Multivalued Functions
  • III.10. Projective Imbeddings
  • III.11. More on the Jacobian Variety
  • IV Uniformization
  • IV.1. More on Harmonic Functions (A Quick Review)
  • IV.2. Subharmonic Functions and Perron’s Method
  • IV.3. A Classification of Riemann Surfaces
  • IV.4. The Uniformization Theorem for Simply Connected Surfaces
  • IV.5. Uniformization of Arbitrary Riemann Surfaces
  • IV.6. The Exceptional Riemann Surfaces
  • IV.7. Two Problems on Moduli
  • IV.8. Riemannian Metrics
  • IV.9. Discontinuous Groups and Branched Coverings
  • IV.10. Riemann–Roch—An Alternate Approach
  • IV.11. Algebraic Function Fields in One Variable
  • V Automorphisms of Compact Surfaces Elementary Theory
  • V.1. Hurwitz’s Theorem
  • V.2. Representations of the Automorphism Group on Spaces of Differentials
  • V.3. Representations of Aut M on H>1(M)
  • V.4. The Exceptional Riemann Surfaces
  • VI Theta Functions
  • VI.1. The Riemann Theta Function
  • VI.2. The Theta Functions Associated with a Riemann Surface
  • VI.3. The Theta Divisor
  • VII Examples
  • VII.1. Hyperelliptic Surfaces (Once Again)
  • VII.2. Relations among Quadratic Differentials
  • VII.3. Examples of Non-hyperelliptic Surfaces
  • VII.4. Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods
  • VII.5. Examples of Prym Differentials.

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