Riemann surfaces, hershel m farkas, irwin kra

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index H M Farkas Kra Riemann Surfaces Second Edition With 27 Figures Springer Hershel M Farkas Department of Mathematics Hebrew University 1erusalem 91904 Israel Irwin Kra 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 K.A Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (1991): 30FIO, 32ClO Library of Congress Cataloging-in-Publication Data Farkas, Hershel M Riemann surfaces / H.M Farkas, Kra - 2nd ed p em - (Graduate teltts in mathematics: 71) Includes bibliographical references and indelt ISBN 978-1-4612-7391-2 ISBN 978-1-4612-2034-3 (eBook) DOI 10.1007/978-1-4612-2034-3 Riemann surfaees 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 S 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 Vll IX xv 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 I.I 1.2 1.3 1.4 Definitions and Examples Topology of Riemann Surfaces Differential Forms Integration Formulae 9 13 22 28 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 32 32 33 39 45 50 351 VII.6 The Trisecant Formula On M we can construct (locally) the function y given by y2 = z(z - 1) n (z - 2g - Ak ), k=l and (locally) the differentials dz dz g _ dz z - , z y y' y Continuation of these differentials along the curves ai' , ag , b l , , bg _ I of Figure VII.l (interpreted correctly) leaves them invariant However, continuation along bg leads to a change of sign We have hence constructed a basis for the Prym differentials with characteristic [8 ::: ?J as defined in III.9 The fascinating relation between the lifts of these differentials to a smooth two-sheeted cover and the theory of moduli will have to be pursued elsewhere VII.6 The Trisecant Formula Although a purist might object that we are no longer considering examples, we will proceed nevertheless The formula in the title of this section has turned out to have remarkable applications VII.6.l The notation is the same as in Section VII.2 Let rx be a nonsingular element of the theta divisor, e (thus 8(rx) = and (o8j ozj )(rx) -# for at least one integer j, ~ j ~ g) For any such rx, and any four points PI' , P4 on the Riemann surface consider the following expression: 8(rx 8(rx + f~~)8(rx + f~~) + J~:)8(rx + f~~) = A(PI'P2 ,P3 ,P4 ), In the above formula f~ is used as an abbreviation for cp(P) - cp(Q) to emphasize the independence of this difference on the base point for the map cp In the above expression for A, we may think of the four points Pi as being fixed It is more useful, however, to think of the last three points as fixed and the point PI as a variable point on the surface M We hence view A as a multi valued meromorphic function of Pion the surface M When the point PI is continued over cycles the expression picks up multiples by an exponential We leave for the reader the task of computing the multipliers The multi valued function A has a zero at PI = Pz and a pole at PI = P4 If we restrict PI (and choose Pz , P3 , and P4 also) to lie in a fundamental polygon for the surface (or in a simply connected region whose closure is the surface), then the value of the function (now single-valued) at PI = P3 is one 352 VII Examples The above properties suggest calling A(PI,PZ 'P3 ,P4 ) the generalized cross in the specified order) We (should) quickly remark that it is possible for the expression not to be well defined (the cases resulting in either the numerator or denominator of the formula defining A vanishing) for a given rJ and a given set of four points (without the use of some limiting procedure) However, if we are given rJ and a set of three generic (we will not define this term here) points Pz , P3 , P4 , on the surface M, there are at most g - points Pion the surface where A is not well defined It is therefore not too hard to show that the generalized cross ratio of the four distinct points can always be defined This expression would not be of much use if it were to depend on the point rJ Our next result shows that this is not the case ratio oj the Jour points (taken The multivalued Junction A(PI 'P2 ,P3 ,P4 ) is independent oj the point used to define it Theorem rJ PROOF Let rJ and fl be two distinct elements in the nonsingular set of the theta divisor Consider the quotient of the two expressions, one formed with rJ and the other with fl This quotient is easily seen to be a (single-valued) meromorphic function of Pion the surface It however has no zeros or poles and is therefore a constant The constant is one since at PI = P3 each of the two expressions has value one Consider now the meromorphic function G on the Jacobian variety J(M) of the surface M given by (we are now fixing the four points PI, P2 , P3 , P4) The reader will easily confirm that G(z) is actually a meromorphic function on the torus J(M) and that therefore the function Z E C9 is also such a function which however vanishes on the theta divisor (by the previous theorem) If we now multiply h by the denominator of G we find e(z + f:")e(z + f:")h(Z) = e(z + f:")e(z + f:") - A(PI,PZ ,P3 ,P4 )e(z + f:")e(z + f:")' In particular, we find that the right-hand side vanishes for z E e It thus 353 VII.6 The Trisecant Formula follows that O(z + S~~)O(z + S~!) - A(P1 ,P2 ,P3 ,P4 )O(Z O(z) + S~!)O(z + S~~) is a holomorphic function of z on ICg but not on the torus If we check, its behavior on the torus though we find that it has the same multiplicative character as the function O(z + S~~ + S~!) and therefore must be a constant multiple of this function This last point follows from the fact (which we have not proved here) that the dimension of the space of holomorphic functions on ICg with the given multiplicative behavior is one It thus follows that = cO(Z)O(z + f:" + f:' J ) for some constant c The preceding is an identity for all values of z In particular, if we set z = ex - S~~ we find that _ - A(P1 ,P2 ,P3 ,P4 ) - c O(a - + S~!) S~~ + S~!)O(a - S~~ + S~~)" O(a - S~~)O(a The evenness of the theta function and the fact that the expression defining A is independent of the point ex in the theta divisor gives us -A(P1 ,P2 ,P3 ,P4 ) = d(P1 ,P2 ,P4 ,P3 )· We have therefore proved the following Theorem Let Pj' i = 1, , be four distinct points on a compact Riemann surface of positive genus Then the following identity holds: =- A(P1,P2,P3,P4)O(Z)O(z A(P1 ,P2 ,P4 ,P3 ) + f P' + fPJ) P2 p In order to make use of this result we need to undertake a careful study of the possible cross ratios of four points; in other words, how the order of the points affects the cross ratio If we denote the expression A(Pj,P.i,PbP1) by Ajjk1 , it follows immediately from the definitions that In addition, we also see that if we denote A1234 by A and finally A1423 by v, then A1432 = l /A, A1243 = 1111, and A1324 }' 1342 = Il v by 11 and 354 VII Examples The reader can now easily check that A v = e(ex /1 e(ex + H!)e(ex + J~~)e(ex + J~~) + J~~)e((X + J~~)e(ex + J~;) and this is clearly equal to e(ex + J~!)e(ex + J~~)e(ex + J~~) e(ex - H!)e(ex - J~~)e(ex - J~~) Let us now consider the case of ex, a nonsingular point of order two, in the theta divisor and write ex = 1(e'/2) + r(e/2) By the exercise preceding Section VI.2.6 and the fact that J~! + J~~ + J~; = (when all integrations take place along paths lying inside the fundamental polygon) we get that the above expression is equal to - The same argument, in fact, shows that for ex as above, We have therefore just proved the following: Lemma With A, /1, and v defined as above, we have AflV = - Let us now return to our theorem which gave us the fundamental identity In view of the lemma we have just proved we can rewrite the theorem and replace the quotient -A1234 /A I243 which appears there by - Afl This by the lemma is simply I/ v, which is )01324 For the convenience of the reader we rewrite the theorem in this new notation as The above formula is known as the trisecant formula It has been shown to be a very useful formula Here we content ourselves with an application to the case of genus with period r in the upper half-plane We can clearly choose a torus and four points on it so that J~~ = ~, J~~ = r/2, and such that J~~ = (I + r)/2 This is the usual choice of the four points of order two in the period parallelogram If we now substitute this into the above formula and use the identity in the exercise preceding Section VI.2.6 we find that it reduces to the well-known elliptic theta VII.6 The Trisecant Formula 355 identity e2[~}O)e2[~}z) = e2[~}O)e2[~}Z) + e2[~}0)82[~}Z) By setting z = this now becomes a theta constant identity We are now at the beginning of another story; material for another book Bibliography I Accola, R D M.: Riemann surfaces, theta functions, and abelian automorphisms groups Lecture Notes in Mathematics vol 483 Springer : Berlin, Heidelberg, New York 1975 Ahlfors, L V.: Lectures on quasiconformal mappings Van Nostrand: New York 1966 Reprinted Brooks-Cole, 1987 Ahlfors, L V.: Conformal invariants: topics in geometric function theory McGraw Hill: New York 1973 Ahlfors, L V Sario, L.: Riemann surfaces Princeton Univ Press: Princeton, New Jersey 1960 Alling, N L , Greenleaf, N.: Foundations of the theory of Klein surfaces Lecture Notes in Mathematics vol 219 Springer: Berlin, Heidelberg, New York 1971 Appel, P , Goursat, E : Theorie des fonctions algebriques et de leurs integrales: I Etude des fonctions analytiques sur une surface de Riemann Gauthier- Villars : Paris 1929 Baker, H : Abel's theorem and the allied theory including the theory of theta functions Cambridge Univ Press: Cambridge 1897 Behnke, H , Sommer, F.: Theorie der analytischen Funktionen einer komplexen Veranderlichen (second edition) Springer: Berlin, G6ttingen, Heidelberg 1962 Bers, L.: Riemann surfaces Lecture Notes, New York University, Institute of Mathematical Science Lecture Notes, 1957- 58 10 Chevalley, c : Introduction to the theory of algebraic functions of one variable American Mathematical Society, Providence, Rhode Island 1951 II Conforto, F.: Abelsche Funktionen und algebraische Geometrie Springer: Berlin, G6ttingen, Heidelberg 1956 12 Fay, J D.: Theta functions on Riemann surfaces Lecture Notes in Mathematics vol 352 Springer: Berlin, Heidelberg, New York 1973 13 Ford, L.: Automorphic functions (second edition) Chelsea : New York 1951 14 Fricke, R., Klein, F : Vorlesunger tiber die Theorie der automorphen Funktionen: I Die gruppentheoretischen Grundlagen, Teubner: Leipzig 1897 II Die Funktionen theoretischen Ausftihrungen und die Anwendungen; a) Engere Theorie der automorphen Funktionen, 1901 b) Kontinuitatsbetrachtungen im Gebiete der Hauptkreisgruppen, 1911 Bibliography 357 15 Fuchs, W H J :Topics in the theory of functions of one complex variable Van Nostrand: Princeton, New Jersey 1967 16 Griffiths J P., Harris, J.: Principles of algebraic geometry Wiley: New York 1978 17 Gunning, R c.: Lectures on Riemann surfaces Mathematical Notes Princeton Univ Press : Princeton New Jersey 1967 18 Gunning, R c.: Lectures on vector bundles qver Riemann surfaces Mathematical Notes Princeton Univ Press : Princeton, New Jersey 1967 19 Gunning R c.: Lectures on Riemann surfaces: Jacobi varieties Mathematical Notes.Princeton Univ Press: Princeton, New Jersey 1972 20 Gunning, R c.: Riemann surfaces and generalized theta functions Springer: New York, Heidelberg Berlin 1976 21 Hensel, K., Landsberg, G : Theorie der algebraischen Funktionen einer Variabeln Teubner: Leipzig 1902 Reprinted Chelsea, 1965 22 Hurwitz, A , Courant, R : Vorlesungen tiber allgemeine Funktionen theorie und elliptische Funktionen Springer: Berlin 1929 23 Igusa J.-I.: Theta functions Springer: New York , Heidelberg, Berlin 1972 24 Klein , F.: Uber Riemann's Theorie der algebraischen Funktionen und ihrer Integrale Teubner : Leipzig 1882 25 Kra I.: Automorphic forms and Kleinian groups Benjamin: Reading Massachusetts 1972 26 Krazer, A.: Lehrbuch der Thetafunktionen Teubner: Leipzig 1903 Reprinted Chelsea, 1970 27 Krushkal , S L.: Quasiconformal mappings and Riemann surfaces Winston & Sons: Washington D.C 1979 28 Kunzi H P.: Quasikonforme Abbildungen: Springer: Berlin Gi:ittingen, Heidelberg 1960 29 Lang, S.: Introduction to algebraic and abelian functions Addison- Wesley: Reading, Massachusetts 1972 30 Lang, S.: Elliptic functions Addison - Wesley: Reading, Massachusetts 1973 31 Lehner J : Discontinuous groups and automorphic functions Mathematical Surveys, Number VIII American Mathematical Society: Providence, Rhode Island 1964 32 Lehner J : A short course in automorphic functions Holt: New York 1966 33 Lehto, Virtanen, K I : Quasiconformal mappings in the plane Springer: New York, Heidelberg, Berlin 1973 34 Magnus J W : Noneuclidean tesselations and their groups Academic Press: New York 1974 35 Mumford D : Curves and their Jacobians The Univ of Michigan Press: Ann Arbor 1975 36 Nevanlinna R : Uniformisierung Springer: Berlin Gi:ittingen, Heidelberg 1953 37 Pfluger A.: Theorie der Riemannschen Flachen Springer: Berlin Gi:ittingen Heidelberg 1957 38 Rauch H E., Farkas H M : Theta functions with applications to Riemann surfaces Williams & Wilkins: Baltimore Maryland 1974 39 Rauch H E , Lebowitz A.: Elliptic functions, theta funct ions, and Riemann surfaces Williams & Wilkins: Baltimore Maryland 1973 40 Riemann B.: Gesammelte Mathematische Werke Dover: New York 1953 41 Schiffer, M , Spencer, D c.: Functionals of finite Riemann surfaces Princeton Univ Press: Princeton, New Jersey 1954 Second edition, Chelsea 42 Springer, G : Introduction to Riemann surfaces Addison Wesley: Reading Massachusetts 1957 43 Siegel, C L.: Topics in complex function theory Wiley- Interscience, New York Vol I, 1969, Elliptic functions and uniformization theory Vol II , 1971, Automorphic functions and abelian integrals Vol III, 1973, Abelian functions and modular functions of several variables 358 Bibliography 44 Swinnerton- Dyer, H P F : Analytic theory of abelian varieties Cambridge Univ Press: Cambridge 1974 45 Walker , R.: Algebraic curves Springer: New York , Heidelberg, Berlin 1978 46 Weyl, H : Die Idee der Riemannschen Flache Teubner : Berlin 1923 (Reprint, Chelsea, 1947) English Translation : The concept of a Riemann surface Addison Wesley: Reading, Massachusetts 1955 47 Wirtinger, W.: Untersuchungen iiber Thetafunktionen Teubner : Leipzig 1895 48 Zieschang, H , vogt, E , Coldewey, H D.: FJachen und ebene diskontinuierliche Gruppen Lecture Notes in Mathematics voL 122 Springer : Berlin, Heidelberg, New York 1970 Index abelian differential 51, 65, 241 Abel- Jacobi embedding 92 Abel's theorem 93 for multivalued functions 134 Accola 267 additive multivalued function (belonging to a character) 135 algebraic function algebraic function field 6, 250 algebraically essential singularity 242 analytic configuration 245 continuation along a curve 127, 128 coordinate chart II mapping II annulus 212 barrier (at a point) 176 normalized 176 base point of linear series 79 basis adapted to a point 84 Betti number 15 bilinear relation 64 branch 12 number (of mapping) 21 point 15, 348 branched covering 234 canonical class 71,76 divisor 71 homology basis 6, 56 Cauchy's integral formula 36 chain 14 character (on the fundamental group) 127 inessential 129 normalized 127 characteristic 221 Clifford index (of a divisor) 109 Clifford's theorem 112, 156 closed surface 10 cohomology class 44 complementary divisor 110 complex sphere 75 complex torus 142 conformal disc 171 mapping 11 conjugation operator * 25 constant mapping 11 coordinate chart 10 coordinate disc II covering branched 234 group 206 manifold 15 map 15 transformation 16 curvature (of metric) 214, 216 cusp 218 degree of mapping 13 360 Dehn twist 19 de Rham cohomology 44, 88 diagonalization procedure 189, 193 differentiable manifold 13 differential (form) abelian 51, 65, 241 of the first kind 65 of the second kind 66 of the third kind 66 associated with closed curve 41 closed 26 co-closed 26 co-exact 26 exact 26 G-invariant form 271 harmonic 42,45, 57, 241 holomorphic 4,27,61,62 measurable 29 meromorphic 51 multiplicative 129 Prym 128 q-canonical 71 differential operators d, a, a 25 * 25,61 dimension of divisor 72 Dirichlet principle 49, 187 problem 161 , 176 disc (corresponding to a puncture) 229 discontinuous group 2, 203, 222 discrete group 204 divisor 70 canonical 71 class group 70 Clifford index of 109 complementary 110 dimension of 72 equivalence of 70 free points in 125 greatest common divisor 110 group 70 index of specialty of 72 integral 71 least common multiple 111 multiple of 71 polar 71 principal 70 ramification 271 special 95, 109 strictly integral 71 zero 71 double (of a Riemann surface) 49 Index Earle 292 Eichler trace formula 281 elementary group 205 elliptic function Mobius transformation 208 surface 100, 179 equivalent divisors 70 essential analytic singularity 243 Euclidean metric 215 Euler (- Poincare) characteristic 6, 21 exceptional Riemann surface 207, 293 extended complex plane 10 finite sheeted cover 13 finite type 221 free points (in a divisor) 125 Fuchsian equivalent 222 Fuchsian group 204 of first kind 207 of second kind 207 function additive 135 algebraic element 104, 126 elliptic field 2, 250 group 204, 222 harmonic 26, 166 holomorphic 11 meromorphic 11 multiplicative 7, 126, 301 subharmonic 171 superharmonic 171 fundamental domain 221 fundamental group 14,209 gap 81 Gauss- Bonnet formula 220 genus (of a surface) 6, 17, 221 geodesic 215, 217 germ (of a merom orphic function) 126,241 greatest common divisor (of two divisors) 110 Green's function 181 group covering 17, 206 discontinuous 2, 203, 221 discrete 204 divisor class 70 elementary 205 Fuchsian 204 361 Index Jacobian variety function 204, 222 fundamental 14, 209 homology 15 Kleinian 204 of divisors 70 symplectic 287 unimodular 213 Guerrero 274 g-hyperelliptic 341 G-invariant differential 271 Kleinian group 204 Komiya 347 Kuribayashi 347 half-period 304 half plane (corresponding to a puncture) 229 harmonic differential 42,45, 57, 241 form 26 function 26, 166 measure 180, 182 harmonization 172 Harnack's inequality 168 principle 169 Hodge theorem 44 holomorphic differential 4,61,62 form 27 function II mapping II q-differential 80 homology group 14 Hurwitz's theorem 258, 260, 346 hyperbolic Mobius transformation 208 Riemann surface 179 hyperelliptic involution 107 surface 88, 100, 138, 250, 266, 322, 350 inessential character 129 integer characteristics 133 integral divisor 71 integration by parts 28 intersection number (of two curves) invariant metric 216 irreducible subvariety ISS isometry 214 Jacobi inversion theorem 314 6, 97, 134, 8,92, 105, 146 55 least common multiple (of two divisors) III Lefschetz fixed point formula 282 limit set 203 linear series 78, 79 local coordinate t parameter II loxodromic Mobius transformation 208 manifold differentiable 13 universal covering unlimited covering 15 mapping analytic t conformal 11 constant 11 holomorphic 11 measurable I-form 29 meromorphic differential 51 function 11 q-differential 51 metric Euclidean 215 invariant 216 non-Euclidean 217 Poincare 217 Mobius transformation 208 elliptic 208 hyperbolic 208 loxodromic 208 parabolic 208 moduli 3, 211, 348 monodromy theorem 16 multiple of a divisor 71 multiplicative differential (belonging to a character) 128 multiplicative (holomorphic) function 7, 127,301 multiplicative multi valued function (belonging to a character) 127 multiplicity (of mapping) 12 362 Noether gap theorem 81 Noether's theorem 159 non-Euclidean area 217 metric (= Poincare metric) 217 non-gap 80 normal form 17 normalized barrier 176 character 127 nth-integer characteristic 133 open surface 10 order of a differential 51 of divisor 78 orthogonal complement 32 decomposition 40 parabolic Mobius transformation 207 Riemann surface 179 parametric disc 11 period matrix 143, 182 Perron family 174 Perron's principle 174 Picard group 204 Poincare metric (= non-Euclidean metric) 232 Poisson kernel 167 polar divisor 71 principal divisor 71 part 135 projective space 136 prolongable Riemann surface 203 proper solution for Dirichlet problem 176 Prym differential 128, 350 Puiseaux series 234, 342 puncture 221 q-canonical class 71 divisor 71 mapping 136 ramification divisor 271 q- Weierstrass point 87 Index ramification number (of mapping) 12 ramification point 15 region of discontinuity 203 regular point for Dirichlet problem 176 representation of Aut M 270, 288 residue (of a differential) 51 Riemann- Hurwitz formula 21 , 77, 237 Riemann inequality 74, 237 Riemann mapping theorem 197 Riemann- Roch theorem 7, 73, 76, 77,239 for multivalued functions 133 Riemann sphere 10, 213 Riemann surface 1,9 closed 10 double of 47 elliptic 100, 179 exceptional 207, 293 genus(of) 18 hyperbolic 179 of finite type 221 open 10 parabolic 179 prolongable 207 Riemann theta function 7, 298 singular point of 328 si ngular set 319 zero set of 319 Riemannian metric 213 Schwarz 258 Serre 292 sheaf of germs 195 sheet interchange 107 Siegel upper half space 298 signature 221 simple Weierstrass point 88 simplex 14 singular point of the {I-function 328 singular set of the {I-function 319 singularity algebraically essential 242 essential analytic 243 special divisor 95, 110 sphere 18 stereographic projection 214 Stokes' theorem 24 strictly integral divisor 71 subharmonic function 171 super harmonic function 171 surface closed 10 363 Index genus of 6, 18, 221 open 10 triangulability 53 symbol of polygon 17 symplectic group 287 theta function 298 associated with a Riemann surface 305 of first order 302 with integral characteristic 302 Torelli's theorem 163 torus 3, 19,142,212 total branching number 19 transition function 10 triangulability of surfaces 53 uniformization uniformizing disc 11 variable 11 unimodular group 213 matrix 287 91, 97, 195, 206 unit 129 universal covering manifold unlimited covering manifold 15 valuation (on a field) 251 Vandermonde matrix 324 vector of Riemann constants 317 Weierstrass gap sequence 90, 332, 338 gap theorem 81 fJ-function 4, 98, 106 point 8, 87, 258, 345 simple 88 weight of a point 84 Weyl's lemma 33, 38, 39 Wronskian 84, 346 zero divisor 71 set of B-function 318 308, Graduate Texts in Mathematics contiroud from page ii 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 WHITEHEAD Elements of Homotopy Theory KARGAPOLOV/MERl ZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory FARKAS/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 2nd ed HOCHSCHILD 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