Graduate Texts in Mathematics 71 Editorial Board

S. Axler F.w. Gehring K.A. Ribet

Springer Science+Business Media, LLC

Graduate Texts in Mathematics

T AKEUTJIZARING. Introduction to 33 HIRSCH. Differential Topology. Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk.

2 OXTOBY. Measure and Category. 2nd ed. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 35 ALEXANDERIWERMER. Several Complex 4 HILTON/STAMMBACH. A Course in Variables and Banach Algebras. 3rd ed.

Homological Algebra. 2nd ed. 36 KELLEy/NAMIOKA et aL Linear

5 MAC LANE. Categories for the Working Topological Spaces. Mathematician. 2nd ed. 37 MONK. Mathematical Logic.

6 H UGHES/PiPER. Projective Planes. 38 GRAUERT/FRITZSCHE. Several Complex 7 SERRE. A Course in Arithmetic . Variables. 8 TAKEUTJIZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to C"-Algebras. 9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELL/KNAPP. Denumerable

and Representation Theory. Markov Chains. 2nd ed. 10 COHEN. A Course in Simple Homotopy 41 APOSTOL. Modular Functions and

Theory. Dirichlet Series in Number Theory. 11 CONWAY. Functions of One Complex 2nd ed.

Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite 12 BEALS. Advanced Mathematical Analysis . Groups. 13 ANDERSONIFULLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous

of Modules. 2nd ed. Functions. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry.

and Their Singularities. 45 LOEVE. Probability Theory I. 4th ed. IS BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed.

Analysis and Operator Theory. 47 MOISE. Geometric Topology in 16 WINTER. The Structure of Fields. Dimensions 2 and 3. 17 ROSENBLATT. Random Processes. 2nd ed. 48 SACHslWu. General Relativity for 18 HALMos. Measure Theory. Mathematicians. 19 HALMos. A Hilbert Space Problem Book. 49 GRUENBERGIWEIR. Linear Geometry.

2nd ed. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem.

21 HUMPHREYS. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential 22 BARNES/MACK. An Algebraic Introduction Geometry.

to Mathematical Logic. 5T HARTSHORNE. Algebraic Geometry. 23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic. 24 HOLMES. Geometric Functional Analysis 54 GRAVER/WATKINS. Combinatorics with

and Its Applications. Emphasis on the Theory of Graphs. 25 HEWITT/STROMBERG. Real and Abstract 55 BROWNIPEARCY. Introduction to Operator

Analysis. Theory I: Elements of Functional 26 MANES. Algebraic Theories. Analysis. 27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An 28 ZARISKJiSAMUEL. Commutative Algebra. Introduction.

VoL I. 57 CROWELL/Fox. Introduction to Knot 29 ZARISKJiSAMUEL. Commutative Algebra. Theory.

VoL II. 58 KOBLITZ. p -adic Numbers, p-adic 30 JACOBSON. Lectures in Abstract Algebra I. Analysis, and Zeta-Functions. 2nd ed.

Basic Concepts. 59 LANG. Cyclotomic Fields. 31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in

II. Linear Algebra. Classical Mechanics. 2nd ed. 32 JACOBSON. Lectures in Abstract Algebra

Ill . Theory of Fields and Galois Theory. continued after index

H. M. Farkas 1. Kra

Riemann Surfaces

Second Edition

With 27 Figures

Springer

Irwin Kra Hershel M. Farkas Department of Mathematics Hebrew University 1erusalem 91904 Israel

Department of Mathematics State University of New York Stony Brook, NY 11794-3651 USA

Editorial Board

S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

F.w. Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA

Mathematics Subject Classification (1991): 30FIO, 32ClO

Library of Congress Cataloging-in-Publication Data Farkas, Hershel M.

Riemann surfaces / H.M. Farkas, 1. Kra. - 2nd ed. p. em. - (Graduate teltts in mathematics: 71)

Includes bibliographical references and indelt.

K.A. Ribet Department of

Mathematics University of California

at Berkeley Berkeley, CA 94720-3840 USA

ISBN 978-1-4612-7391-2 ISBN 978-1-4612-2034-3 (eBook) DOI 10.1007/978-1-4612-2034-3 1. Riemann surfaees. 1. Kra, Irwin. II. Title. III. Series.

QA333.F37 1991 SI5'.223-dc20 91-30662

Printed on acid-free paper.

© 1992 Springer Science+Business Media New York Originally published by Springer-Verlag New York Tnc. in 1980, 1992 Softcover reprint of the hardcover 2nd edition 1992

All rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merehandise Marks Act, may accordingly be used freely byanyone.

Production coordinated by Brian Howe and managed by Francine Sikorski: manufacturing supervised by Robert Paella. Typeset by Asco Trade Typesetting Ltd., Hong Kong.

987 6 S 4 3

ISBN 978-1-4612-7391-2

To Eleanor Sara

Preface to the Second Edition

It is gratifying to learn that there is new life in an old field that has been at the center of one's existence for over a quarter of a century. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generation of mathematicians from (newly) adjacent fields (for example, those interested in hyperbolic manifolds and iterations of rational maps) and young physicists who have been convinced (certainly not by mathematicians) that compact Riemann surfaces may play an important role in their (string) universe. We hope that non-mathematicians as well as mathematicians (working in nearby areas to the central topic of this book) will also learn part of this subject for the sheer beauty and elegance of the material (work of Weierstrass, Jacobi, Riemann, Hilbert, Weyl) and as healthy exposure to the way (some) mathematicians write about mathematics.

We had intended a more comprehensive revision, including a fuller treatment of moduli problems and theta functions. Pressure of other commitments would have substantially delayed (by years) the appearance of the book we wanted to produce. We have chosen instead to make a few modest additions and to correct a number of errors. We are grateful to the readers who pointed out some of our mistakes in the first edition; the responsibility for the remaining mistakes carried over from the first edition and for any new ones introduced into the second edition remains with the authors.

June 1991 Jerusalem and Stony Brook

H.M. FARKAS

and I. KRA

Preface to the First Edition

The present volume is the culmination of ten years' work separately and joint­Iy. 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 several times and used as the basic content of courses given sub­sequently by each of the authors at the State University of New York at Stony Brook and the Hebrew University.

In this book we present the theory of Riemann surfaces and its many dif­ferent facets. We begin from the most elementary aspects and try to bring the reader up to the frontier of present-day research. We treat both open and closed surfaces in this book, but our main emphasis is on the compact case. In fact, Chapters III, V, VI, and VII deal exclusively with compact surfaces. Chapters I and II are preparatory, and Chapter IV deals with uniformization.

All works on Riemann surfaces go back to the fundamental results of Rie­mann, Jacobi, Abel, Weierstrass, etc. Our book is no exception . In addition to our debt to these mathematicians of a previous era, the present work has been influenced by many contemporary mathematicians.

At the outset we record our indebtedness to our teachers Lipman Bers and Harry Ernest Rauch, who taught us a great deal of what we know about this subject, and who along with Lars V. Ahlfors are responsible for the modern rebirth of the theory of Riemann surfaces. Second, we record our gratitude to our colleagues whose theorems we have freely written down without attri­bution. In particular, some of the material in Chapter III is the work of Henrik H. Martens, and some ofthe material in Chapters V and VI ultimately goes back to Robert D. M. Accola and Joseph Lewittes.

We thank several colleagues who have read and criticized earlier versions of the manuscript and made many helpful suggestions: Bernard Maskit,

x Preface to the First Edition

Henry Laufer, Uri Srebro, Albert Marden, and Frederick P. Gardiner. The errors in the final version are, however, due only to the authors. We also thank the secretaries who typed the various versions: Carole Alberghine and Estella Shivers.

August 1979 H . M. FARKAS I. KRA

Contents

Preface to the Second Edition Preface to the First Edition Commonly Used Symbols

CHAPTER 0

An Overview

0.1. Topological Aspects, Uniformization, and Fuchsian Groups 0.2. Algebraic Functions 0.3. Abelian Varieties 0.4. More Analytic Aspects

CHAPTER I

Riemann Surfaces

I.I. Definitions and Examples 1.2. Topology of Riemann Surfaces 1.3. Differential Forms 1.4. Integration Formulae

CHAPTER II

Existence Theorems

1I.1. Hilbert Space Theory- A Quick Review 11.2. Weyl's Lemma II.3. The Hilbert Space of Square Integrable Forms lI.4. Harmonic Differentials 11.5. Meromorphic Functions and Differentials

Vll

IX

xv

2 4 6 7

9

9 13 22 28

32

32 33 39 45 50

Xli

CHAPTER III

Compact Riemann Surfaces

III. I. III.2. IIU. 1Il.4. III.5 . III.6. IIU. III.S. III.9. III.IO. HI.l1. III. 12.

Intersection Theory on Compact Surfaces Harmonic and Analytic Differentials on Compact Surfaces Bilinear Relations Divisors and the Riemann-Roch Theorem Applications of the Riemann-Roch Theorem Abel's Theorem and the Jacobi Inversion Problem Hyperelliptic Riemann Surfaces Special Divisors on Compact Surfaces Multivalued Functions Projective Imbeddings More on the Jacobian Variety Torelli's Theorem

CHAPTER 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.S. Riemannian Metrics IV.9. Discontinuous Groups and Branched Coverings IV.lO. Riemann-Roch- An Alternate Approach IV.II. Algebraic Function Fields in One Variable

CHAPTER V Automorphisms of Compact Surfaces- Elementary Theory

V.l. Hurwitz's Theorem V.2. Representations of the Automorphism Group on

Spaces of Differentials V.3. Representation of Aut M on HI (M) V.4. The Exceptional Riemann Surfaces

CHAPTER VI Theta Functions

Vr.I. The Riemann Theta Function VI.2. The Theta Functions Associated with a Riemann Surface VI.3. The Theta Divisor

CHAPTER VII

Examples

VII. I. Hyperelliptic Surfaces (Once Again) VII.2. Relations Among Quadratic Differentials

Contents

54

54 56 64 69 79 91 99

109 126 136 142 161

166

166 171 17S 194 203 207 211 213 220 237 241

257

257

269 286 293

298

298 304 309

321

321 333

Contents

VII.3. Examples of Non-hyperelliptic Surfaces VIlA. Branch Points of Hyperelliptic Surfaces as Holomorphic

Functions of the Periods VII.5. Examples of Prym Differentials VII.6. The Trisecant Formula

Bibliography

Index

XIII

337

348 350 351

356

359

Commonly Used Symbols

IR"

C" Re

1m

1·1 coo J{'q(M)

%(M)

deg

L(D)

reD) Q(D)

i(D)

[ ] c(D)

ordpf

1t 1(M)

H1(M)

integers

rationals

real numbers

n-dimensional real Euclidean spaces

n-dimensional complex Euclidean spaces

real part

imaginary part

absolute value

infinitely differentiable (function or differential)

linear space of holomorphic q-difTerentials on M

field of meromorphic functions on M

degree of divisor or map

linear space of the divisor D

dim L(D) = dimension of D

space of meromorphic abelian differentials of the divisor D

dim Q(D) = index of specialty of D

greatest integer in

Clifford index of D

order of f at P fundamental group of M

first (integral) homology group of M

XVI

J(M)

n

'x

Jacobian variety of M

period matrix of M

integral divisors of degree n on M

{D E Mn;r(D - 1 ) ~ r + 1} image of M n in J(M)

image of M~ in J(M)

canonical divisor

vector of Riemann constants (usually)

Commonly Used Symbols

transpose of the matrix x (vectors are usually written as columns; thus for x E IRn, 'x is a row vector)