THE UNIVERSITY SERIES IN MATHEMATICS Series Editor: Joseph J. Kohn
Princeton University INTRODUCTION TO PSEUDODIFFERENT...
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THE UNIVERSITY SERIES IN MATHEMATICS Series Editor: Joseph J. Kohn
Princeton University INTRODUCTION TO PSEUDODIFFERENTIAL AND FOURIER INTEGRAL OPERATORS Fran~ois
Treves
VOLUME 1: PSEUDODIFFERENTIAL OPERATORS VOLUME 2: FOURIER INTEGRAL OPERATORS A SCRAPBOOK OF COMPLEX CURVE THEORY C. Herbert Clemens
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Complex Curve 'Ihcory C. I-Icrhcrt (]crncns I -/111'('
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Ck 1cns, Charles Herbert, 1939;crapbook of complex curve theory. University series in mathematics) P illiography: p. I .:ludes index. Curves, Algebraic. 2. Functions, Theta. 3. Jacobi varieties. l. Title. II. Series: Un crsity series in mathematics (New York, 1980-. ) Q,o ·h5.C55 516.3'5 80-20214 lSI -! 0-306-40536-9 (' he
@ 1980 Plenum Press, New York A Division of Plenum Publishing Corporation 22~ West 17th Street, New York, N.Y. 10011
All rights reserved
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Part of this book may be reproduced, stored in a retrieval system, or transmitted, h a.ny form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher Printed in the United States of America
Preface
This is a book of "impressions ol a journey through the theory of con'· plex algebraic curves. It is neither self-contained. balanced, nor particularly tightly organized. As with any notebook made nn a journey, what appe::~.r~ is that which strikes the writer's fancy. Some topiCS ar,Jear because of their compelling intrinsic beauty. Others are left out hccause, for all their impor .. tance, the traveler found them boring or was too dull or lazy to give them their due. Looking back at the end of the journey. one can see that a common theme in fact does emerge, as is so often thr case: that theme is the thcorv of theta functions. In fact very much of the material in the book is prepara tion for our study of the final topic, the so-calkd Schottky problem. More than once, in fact, we tear ourselves away frum interesting topics lead ir_1~· elsewhere and return to our main route. -- -·Some of the subjects are extremely elementary. In fact, we begin wtth some musings in the vicinity of secondary school algebra. Later, on occasion and without much warning, we _jump into some fairly deep water. Our intent is to struggle with some deep topics in much the same way that a beginning researcher might, using whatever tools we have at hand or can grab somehow or other. Sometimes we use no background material and do everything in detail; sometimes we use some of the heaviest of modern machinery. We hope to motivate further study or, preferably, further discussion with an expert in the fkld. In short, our aim is to motivate and stimulate mathematical activity rather than to present a finished product. and our point of view is romantic rather than rigorous. The material treated here was originally brought together for a Summer Course of the Italian National Research Council held in Cortona, Italy. in 1976. It comes from so many sources that adequate acknowledgment would be difficult. The treatment of real two-dimensional geometries of
Preface
constant curvawr~ com.:s from Cartan's classic text on Riemanqian geometry; ;(·vera! items concerning the arithmetic of curves are borrowed from Serre's lovely book, A Course in Arithmetic; Manin's beautiful I heorem on rational points of elliptic curves given in Chapter Two was cxpla inel1 to the author hy A. Beau ville; some of the theta identities in Chapt~.::1 Thn~~~ an: lifted from the famous analysis text of Whittaker and Watson, and the construction of the level-two moduli space for elliptic curves was motivated by David Mumford'~ way of viewing the moduli space of curves of a Jlxed genus. The discussion of the Jacobian variety in Chapter Four leans heavily on work of Joseph Lewittes, and the discussion of the Schottky problem comes from work of Accola, Farkas, Igusa, and Rauch. But perhaps the author's greatest debt is to Phillip Griffiths, through whom he came to enjoy this subject. The author also wishes to thank Sylvia M. Morris, Mathematics Department of the University of Utah, for preparing the manuscript, and.Toni W Bunke-r, of. the-same department, for preparing the figures. Herbert Clemens Salt Lake City, Utah
Contents Notation
ix
Chapter One • Conics 1.1. Hyperbola Shadows 1.2. Real Projective Space, The "Unifier" 1.3. Complex Projective Space, The Great "Unifier" 1.4. Linear Families of Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. The Mystic Hexagon... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. The Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 1.7. Cayley's Way of Doing Geometries of ~onstant Curvature......... 1.8. Through the Looking Glass 1.9. The Polar· Curve ............................_._.,.................... 1.10. Perpendiculars in Hyperbolic Space 1.11. Circles in the K-Geometry 1.12. Rational Points on Conics
I 5 7 9 11 13 17 20 22 26 30 33
Chapter Two Cubics 2.1. Inflection Points 2.2. Normal Form for a Cubic 2.3. Cubics as Topological Groups .................................... . 2.4. The Group of Rational Points gn a Cubic ........................ . 2.5. A Thought about Complex Cofijugation 2.6. Some Meromorphic Functions on Cubics ........................ . 2.7. Cross Ratio Revisited, A Moduli Space for Cubics .•....•.......... 2.8. The Abelian Differential on a Cubic ............ ; ..... _............ . 2.9. The Elliptic Integral 2.10. The Picard-Fuchs Equation ....................................... . 2.11. Rational Points on Cubics over IFP ............................... . 2.12. Manin's Result: The Unity of Mathematics ...................... . 2.13. Some Remarks on Serre Duality ......................... ·........ .
37 39
42 45
50 51 52 53
55 58 62 ~s
69
Contents
Chapter l'hree .\.1 J.2. 3.3. .l.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15.
Chapter Four 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.
4. 7. 4.8.
4.9. 4.10. 4.11. 4.12.
Theta
Fmu~tions
Hack to th~: Uroup Law on Cubics You Can't Parann~trize a Smooth Cubic Algebraically Meromorpluc Fun.:tiuns on Elliptic Curves Meromorphic Functions on Plane Cubics The Weierstra~s p-hmction Theta-Null Values Give Moduli of Elliptic Curves The Moduli Space of" Level-Two Structures" on Elliptic Curves Automorphisms of Elliptic Curves . . . . . . . . . . . . . . . . The Moduli Space of Elliptic Curves And So, By the Way, We Get Picard's Theorem The Complex Structure of..// The j-Invariant of an Elliptic Curve Theta-Nulls as Modular Forms A Fundamental Domain for r 2 Jacobi's Identity
95 96 98
100 102 106 109 111
• The Jacobian Variety
Cohomology of a Complex Curve Duality The Chern Class of a Holomorphic Line Bundle Abel's Theorem for Curves The Classical Version of Abei's Theorem The Jacobi Inversion Theorem Back to Theta Functions The Basic Computation Riemann's Th~:orem Linear Systems of D..:grec f/ Riemann's Constant Riemann's Singulariti~::s Theorem
Chapter Five
113 116 118 122 127 131 132 134 136 138 139 142
Quarries and Quintics
5.1. Topology of Plane Quartics .~.-"fhe Twenty-Eight Bitangents ,-- 5.3. Where Are the Hyperellipti,: Curves of Genus 3? 5.4. Quintics
Chapter Six
73 75 78 82 85 89 92
•
147 150 155 158
The Schottky Relation
6.1. Prym Varieties ..
161
6.2. Riemann's Theta Rclauon 6 3. Products of Pairs of Theta Functwus 6.4. A Proportionality Theorem Rdating Jat:obians and Pryms 6.5. The Proponionality Thwrem of Sd:utlky .lung 6.6. The Schollk:, Relation
164 167 168
173 174
HEI·I·.IU:NCES
181
1"--nE\
18.~
Notation Most of the notation used in this book is quite standard, for example. 1L = ring of integers,
Q =field of rational numbe~, IR = field of real numbers,
C =field of complex numbers. Each of the six chapters is divided into sections, for instance, Chu lLer Three has Sections 3.1, 3.2, etc. Equations are numbered consecut i 'ely within chapters-(3.1), (3.2), etc.-as are the figures. Square brackets will be used to enclose matrices and are also sed later in the book in expressions involving theta functions with chara, .eristic, for example, O[l](u; r) When there are complicated exponents, the exp form of the expo .:!ntial is used with the convention exp{x} = e".
CHAPTER ONE
Conics
1.1 HyperbOla Shadows Let's start our study of curves in a very elementary way, with the ct th