Riemann surfaces, hershel m farkas, irwin kra 1

Graduate Texts in Mathematics 71 Editorial Board F W Gehring c C Moore P R Halmos (Managing Editor) H M Farkas I Kra Riemann Surfaces With 27 Figures Springer Science+Business Media, LLC Hershel M Farkas lrwin Kra Department of Mathematics The Hebrew University of Jerusalem Jerusalem Israel Department of Mathematics S.U.N.Y at Stony Brook Stony Brook, NY 11794 USA Editorial Board P R Halmos F W Gehring Managing Editor Indiana University Department of Mathematics Bloomington, Indiana 47401 USA University of Michigan Department of Mathematics Ann Arbor, Michigan 48104 USA c C Moore University of California Department of Mathematics Berkeley, California 94720 USA AMS Classification (1980): 30Fxx, 14H15, 20H1O Library of Congress Cataloging in Publication Data Farkas, Hershel M Riemann surfaces (Graduate texts in mathematics; v 71) Bibliography: p lncludes index Riemann surfaces Kra, Irwin, joint authoL II Title III Series QA333.F37 515'.223 79-24385 AII rights reserved No part of this book may be translated or reprodueed in any form without written permission from Springer-Verlag © 1980 by Springer Seience+Business Media New York Originally publishcd bySpringer-Verlag New York lne in 1980 Softcover reprint ofthe hardcover 1st edition 1980 87654 32 ISBN 978-1-4684-9932-2 ISBN 978-1-4684-9930-8 (eBook) DOI 10.1007/978-1-4684-9930-8 To Eleanor Sara Preface The present volume is the culmination often years' work separately and jointly 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 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 of the 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, 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 Commonly Used Symbols xi CHAPTER 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 1.1 1.2 1.3 IA Definitions and Examples Topology of Riemann Surfaces Differential Forms Integration Formulae 9 13 19 26 CHAPTER II Existence Theorems 30 11.1 11.2 11.3 11.4 11.5 30 Hilbert Space Theory-A Quick Review Weyl's Lemma The Hilbert Space of Square Integrable Forms Harmonic Differentials Meromorphic Functions and Differentials 31 37 43 48 CHAPTER III Compact Riemann Surfaces 111.1 111.2 I1I.3 IlIA I1I.5 111.6 I1I.7 111.8 111.9 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 52 52 54 62 67 76 86 93 103 119 x Contents IILlO Projective Imbeddings IIUL More on the Jacobian Variety 129 132 CHAPTER IV Uniformization lSI IV.l IV.2 IV.3 IVA IV.5 IV.6 IV.7 IV.8 IV.9 IV.IO IV.Il lSI 156 163 179 188 192 196 198 205 222 226 More on Harmonic Functions (A Quick Review) Subharmonic Functions and Perron's Method A Classification of Riemann Surfaces The Uniformization Theorem for Simply Connected Surfaces Uniformization of Arbitrary Riemann Surfaces The Exceptional Riemann Surfaces Two Problems on Moduli Riemannian Metrics Discontinuous Groups and Branched Coverings Riemann~Roch-An Alternate Approach 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 Representations of Aut M on H (M) VA The Exceptional Riemann Surfaces 241 241 252 269 276 CHAPTER VI Theta Functions 280 VI.l The Riemann Theta Function VI.2 The Theta Functions Associated with a Riemann Surface VU The Theta Divisor 280 286 291 CHAPTER VII Examples 301 VII I VII.2 VIU VIlA Hyperelliptic Surfaces (Once Again) Relations among Quadratic Differentials Examples of Non-hyperelliptic Surfaces Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods VII.5 Examples of Prym Differentials 301 311 315 Bibliography 330 Index 333 326 329 Commonly Used Symbols IRn en Re 1m 1·1 Yfq(M) ff(M) deg L(D) r(D) Q(D) i(D) [] c(D) ordpf nl(M) H1(M) J(M) n 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 holm orphic q-differentials 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 Jacobian variety of M period matrix of M integral divisors of degree n on M {D E Mn;r(D-1):::o-: r + I} image of Mn in J(M) image of M~ in J(M) canonical divisor vector of Riemann constants (usually) transpose of the matrix X (vectors are usually written as columns; thus for x E IR n, tx is a row vector) CHAPTER An Overview The theory of Riemann surfaces lies in the intersection of many important areas of mathematics Aside from being an important field of study in its own right, it has long been a source of inspiration, intuition, and examples for many branches of mathematics These include complex manifolds, Lie groups, algebraic number theory, harmonic analysis, abelian varieties, algebraic topology The development of the theory of Riemann surfaces consists of at least three parts: a topological part, an algebraic part, and an analytic part In this chapter, we shall try to outline how Riemann surfaces appear quite naturally in different guises, list some of the most important problems to be treated in this book, and discuss the solutions As the title indicates, this chapter is a survey of results Many of the statements are major theorems We have indicated at the end of most paragraphs a reference to subsequent chapters where the theorem in question is proven or a fuller discussion of the given topic may be found For some easily verifiable claims a (kind of) proof has been supplied This chapter has been written for the reader who wishes to get an idea of the scope of the book before entering into details It can be skipped, since it is independent of the formal development of the material This chapter is intended primarily for the mathematician who knows other areas of mathematics and is interested in finding out what the theory of Riemann surfaces contains The graduate student who is familiar only with first year courses in algebra, analysis (real and complex), and algebraic topology should probably skip most of this chapter and periodically return to it We, of course, begin with a definition: A Riemann surface is a complex I-dimensional connected (analytic) manifold 325 VII.3 Examples of Non-hyperelliptic Surfaces VII.3.10 The Riemann surface of (3.9.1) actually has 168 = 84(3 - 1) automorphisms, which shows that the maximum number (of Hurwitz's theorem, V.l.3) is achieved It is easy to exhibit an automorphism of period 7: (Z,W) I -> (Z,eW), e = exp (72ni) To exhibit other automorphisms of M we must use some algebraic geometry The abelian differentials of the first kind provide an embedding of Minto [p2 (II1.10) Thus setting we see from (3.9.l) or (3.9.2) that a projective equation for our curve is X Y+ y Z+Z X=0 Hence we see that we have another automorphism of M (of order 3) given by the permutation (X, Y,Z) I -> (Y,Z,X) We will not proceed with the above line of thought (The interested reader should consult the work of A Kuribayashi and K Komiya for the complete classification of automorphism groups and Weierstrass points for surfaces of genus 3.) (The authors have a preprint of their forthcoming article, and hence cannot supply a reference to the literature.) If we are willing to use the fact that M has 168 automorphisms, we can conclude without calculation that each Weierstrass point is simple We have seen in the proof of Hurwitz's theorem, that the maximum number of automorphisms occur only if M/Aut M :;:; IC U {oo} and the canonical projection n:M-> lCu {oo} is branched over three points with ramification numbers 2, 3, Thus these are the only possible orders of the stability subgroups of points We conclude that each orbit under Aut M must contain at least 1~8 = 24 points In particular, the Weierstrass points must be the fixed points of the elements of order 7, and there must be 24 such points Hence they must all be simple VII.3.lt For our last example we consider the Riemann surface of which is a hyperelliptic surface of genus In addition to the hyperelliptic involution this surface permits an automorphism of period (Z,W) I -> (eZ,e 3W) with [; = expC;i} Let us denote this automorphism by T and observe that the automorphism JT with J the hyperelliptic involution is of order 10 The automorphism 326 VII Examples JT is given by (z,w) f-+ (8Z, -1: w), so that the only possible fixed points of JT are PI = Z-I(O) or Qlo Q2 the two points lying over CXJ Since T is of prime order 5, the Riemann-Hurwitz formula gives = 5(29 - 2) + 4v(T), with v(T) as usual the number of fixed points of T The only way this can be satisfied is with = and v(T) = Hence T fixes Plo QI, and Q2' It is therefore obvious that JT fixes Pj, but cannot fix either Ql or Q2, because JTQl = JQl = Q2 and JTQ2 = JQ2 = QI' The reader should now recall Theorem V.2.1l ° VIl.4 Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods In IV.7, we solved two elementary moduli problems In this section we will describe one way of obtaining moduli for hyperelliptic surfaces; another elementary case VII.4.I We return once again to the concrete representation of a hyperelliptic surface M of genus g 2: We will now assume that our function z of degree two is branched over 0, 1, and CXJ, and hence represent M by 29- w2 = z(z - 1) TI (4.1.1) (z - ;'k), k= where )'10"', ;'29- are distinct points in C\{O,l} We are using a slight variation of (1.0.1) To fix notation, we set PI P j + = Z-l(J.), = Z-I(O), j = P2 = z-I(1), P 29 + = Z-I(CfJ) 1, , 2g - 1, We have seen in VII 1.2, that using PI as a base point for cp: M vector K of Riemann constants is given by > J(M), the l: -IJ l~J) K=21 ( IT: +1:, where IT is the period matrix for the canonical homology basis constructed in VII.l.1 We shall show that the Aj are holomorphic functions of the entries 7rkl of n We will accomplish this by expressing the function z in terms of 8-functions The function z E %(M) is characterized (up to a constant multiple) by the property that it has a double pole at P 29 + 2' a double zero at PI' and is 327 VIl.4 Branch Points of Hyperelliptic Surfaces holomorphic on M\{P 2g + 2} We will produce such a function in terms of 8-functions One main tool will be the Riemann vanishing theorem, and our explicit knowledge of the images in J(M) of the Weierstrass points on M VII.4.2 We shall see that there are many ways to proceed We begin with the point of order ({J(PIP SP ", P 2g + l ) + K = ({J(P I P ) = ({J(P ), by virtue of (1.2.1) and the fact that ({J(P 2) = for every Weierstrass point P on M We have computed ({J(P 3) in VII.l.1: ({J(P 3) = ~(1!(l) + e(l) + e(2») Similarly, ({J(P2g+2PSP7 P 2g + l ) + K = ((J(P 2g + 2P ) = ~(1!(l) + e(2») We also observe that i(P I P S P '" P 2g + l ) = = i(P 2g +2 PSP7'" P 2g + l ) (compare with VILl.4) We now consider the multiplicative function o ~}({J(P),lI) P~ [ o o OJ o (({J(P),lI) (4.2.1) According to Theorem VI.2.4, the numerator vanishes precisely (it does not vanish identically by Theorem VI.3.3) at PI' P s, P 7,· , P 2g + and the denominator at P 2g + 2, P s , P 7, , P 2g + • In particular, the function (4.2.1) vanishes to first order at P I and has a simple pole at P 2g + 2' Examining the multiplicative behavior of the above function, we see that [1 82 0 ~J (({J( P),lI) f( P) = !=;-1_1_0_ _ ""; _ _ 82[~ ~ ~ ~}({J(P),lI) is a merom orphic function on M with divisor PiP2/+2' Hence f= cz, The constant c is evaluated by f(P 2) CE = C\{O} cz(P 2) = c Thus we see that 328 VII Examples and z(P) = ~J Gn(1),JI )8ZG 0 0 0 8z[1 8Z[~ ~J Gn(l),JI)8z[~ 0 0 0 82[~ G 8Z 1 ~J82G ~J82[~ 0 0 0 0 0 ~}