http://dx.doi.org/10.1090/pspum/049.2

PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S Volum e 49 , Par t 2

Thet a Function s Bowdoi n 198 7

Leo n Ehrenprei s an d Rober t C . Gunning , Editor s

AMERICA N MATHEMATICA L SOCIET Y PROVIDENCE , RHOD E ISLAN D PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON THETA FUNCTIONS HELD AT BOWDOIN COLLEGE BRUNSWICK, MAINE JULY 6-24, 1987

with support from the National Science Foundation, Grant DMS-8611435

1980 Mathematics Subject Classification (1985 Revision). Primary 00A11.

Library of Congress Cataloging-in-Publication Data Theta functions, Bowdoin 1987/Leon Ehrenpreis and Robert C. Gunning, editors. p. cm. - (Proceedings of symposia in pure mathematics, ISSN 0082-0717; v. 49) Proceedings of the thirty fifth Summer Research Institute on Theta Functions held at Bow• doin College, Brunswick, Me., July 6-24, 1987; sponsored by the American Mathematical Soci• ety. Includes bibliographies. 1. Functions, Theta-Congresses. I. Ehrenpreis, Leon. II. Gunning, R. C. (Robert Clifford), 1931-. III. Summer Research Institute on Theta Functions (35th: 1987: Brunswick, Me.) IV. American Mathematical Society. V. Series. QA345.T47 1989 515'.984-dc20 89-6723 ISBN 0-8218-1483-4 (part 1) CIP ISBN 0-8218-1484-2 (part 2) ISBN 0-8218-1485-0 (set) (alk. paper)

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Copyright ©1989 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @ This publication was typeset using A^S-T^X, the American Mathematical Society's TgX macro system. Contents

Preface ix PART 1

Infinite Analysis

Systems of linear differential equations of infinite order: an aspect of infinite analysis TAKAHIRO KAWAI 3 A particular partition of unity: an auxiliary tool in Hodge theory MASAKI KASHIWARA AND TAKAHIRO KAWAI 19 Is there an infinite-dimensional algebraic geometry? Hints from KDV H. P. MCKEAN 27 A correspondence between an infinite Grassmannian and arbitrary vec• tor bundles on algebraic curves MOTOHICO MULASE 39 The KP hierarchy and infinite-dimensional Grassmann manifolds MIKIO SATO 51 Integrable Systems

Some geometrical techniques in integrable systems MARK ADLER 69 Generalized theta functions and homoclinic varieties N. ERCOLANI 87 Explicit equations for the KP and MKP hierarchies SHIRONG LU 101 Introduction to algebraic integrable systems and their Painleve analysis PIERRE VAN MOERBEKE 107 Polynomial r-functions for the AKNS hierarchy ROBERT L. SACHS 133 Integrable systems as deformations of ^-modules KANEHISA TAKASAKI 143 IV CONTENTS

Kac-Moody Algebras The infinite wedge representation and the reciprocity law for algebraic curves E. ARBARELLO, C. DE CONCINI, AND V. G. KAC 171 Exceptional hierarchies of soliton equations VICTOR G. KAC AND MINORU WAKIMOTO 191 Unitary representations of some infinite-dimensional Lie algebras ANDREW PRESSLEY AND VYJAYANTHI CHARI 239 On highest weight and Fock space representations of the Virasoro algebra A. ROCHA-CARIDI 259 Modular forms, strings, and ghosts GREGG J. ZUCKERMAN 273 Lattice Models

Solution of Hirota's discrete-time Toda lattice equation and the critical correlations in the Z-invariant Ising model HELEN AU-YANG AND JACQUES H. H. PERK 287 Solvable lattice models ETSURO DATE, MICHIO JIMBO, TETSUJI MIWA, AND MASATO OKADO 295 Theta function identities in a series of solvable lattice models ATSUO KUNIBA 333 Star-triangle equations, quantum Lax pairs, and higher genus curves JACQUES H. H. PERK 341 Introduction to exactly solvable models in statistical mechanics CRAIG A. TRACY 355 Physics

Introduction to holonomic quantum fields for mathematicians MICHIO JIMBO 379 Quantum field theory: a guide for mathematicians EMIL MARTINEC 3 91 A conformal field theory on Riemann surfaces realized as quantized moduli theory of Riemann surfaces YUKIHIKO NAMIKAWA 413 The physics of string theory A. NEVEU 445 CONTENTS v Jacobian Varieties Schottky-Jung theory HERSHEL M. FARKAS 459 Schottky relations on \{C - C) JOHN FAY 485 Analytic identities for theta functions R. C. GUNNING 503 Translation manifolds and the Schottky problem JOHN B. LITTLE 517 Mappings of closed Riemann surfaces HENRIK H. MARTENS 531 Geometric characterization of Jacobians and the Schottky equations JOSE MA MUNOZ PORRAS 541 Vector bundles over curves and solutions of the KP equations EMMA PREVIATO AND GEORGE WILSON 553 Deformations of singular points on theta divisors ROY SMITH AND ROBERT VARLEY 571 Uniformization, local index theorem, and geometry of the moduli spaces of Riemann surfaces and vector bundles L. A. TAKHTAJAN 581 Explicit actions of the theta groups for theta divisors on Jacobian sur• faces NORIAKI YAMAWAKI 597 Prym Varieties Prym varieties: a survey ARNAUD BEAUVILLE 607 The trisecant conjecture for Pryms OLIVIER DEBARRE 621 Spectral curves, simple Lie algebras, and Prym-Tjurin varieties VASSIL KANEV 627 Algebraic Geometry A new look for thetas IACOPO BARSOTTI 649 The cube structure on the determinant bundle L. BREEN 663 Unramified Abelian extensions of Galois covers MICHAEL D. FRIED AND HELMUT VOLKLEIN 675 VI CONTENTS

On twisted Legendre equations and Kummer surfaces WILLIAM L. HOYT 695 On embedded tangent and secant varieties of projective algebraic varieties A. B. JIJTCHENKO 709

PART 2

Modular Forms

Siegel modular forms and theta series SIEGFRIED BOCHERER 3 Arithmetic compactification of the Siegel moduli space CHING-LI CHAI 19 Fourier analysis, partial differential equations, and automorphic functions LEON EHRENPREIS 45 Problems on theta functions JUN-ICHI IGUSA 101 On the Fourier coefficients of cusp forms having small positive weight MARVIN I. KNOPP 111 Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables JOHN J. MILLSON 129 Vector-valued modular forms of weight s+{~{ RICCARDO SALVATI MANNI 143 Some nonzero harmonic forms and their geometric duals YUE LIN LAWRENCE TONG 151 Number Theory

Transcendental methods and theta-functions D. V. CHUDNOVSKY AND G. V. CHUDNOVSKY 167 Singular moduli and modular equations for Fricke's cases HARVEY COHN 233 The sums of the Kloosterman sums N. V. KUZNETSOV 251 The existence of Maass cusp forms and Kloosterman sums 1.1. PIATETSKI-SHAPIRO 267 CONTENTS vii

Combinatorics On the complex Selberg integral K. AOMOTO 279 Mock theta functions GEORGE E. ANDREWS 283 Orthogonal polynomials and theta functions RICHARD ASKEY 299 The multidimensional i*Fi sum and Macdonald identities for A^ S. C. MILNE 323 List of Participants 363 Preface

m Theta functions apparently first appeared in the forms X^L0 " , [ 2n n+{ ml/2n{n+3) in the work £~0 m / ( \ T,Zo of Jakob Bernoulli. In his work on partition theory, Euler introduced a second variable £ and studied func• tions of the form n^Li(l ~ Qn0~l- For Euler, the primary objects were partition functions such as f](l ~ Qn)> but the function Yi(l - QnQ~x was considered as a function of £ with q occurring as a parameter; after deriving identities for the function of £ he then set £ = 1. Jacobi made two important notational changes that turned out to be crucial for the modern development. He replaced q by enix and £ by e2lz; thus was born the theta function in its present form d{T,z) = J2enin2z+2inz.

The change from q to r allowed him to formulate the "imaginary transforma• tion" r —• — 1/T, which together with the obvious transformation r —• r + 2 leads to the modular group and eventually to the modern theory of modular forms and their ramifications. (The formulation of the modular group in the variable q is complicated; see the paper by Ehrenpreis in this volume.) In addition, Jacobi studied 0(T, Z) as a function of z in its own right. The quasi double periodicity under z —• z + n and z —• z + nz enabled him to relate theta functions as functions of z to elliptic function theory. For Jacobi as for Euler the primary working variable was z. Of course, this theory has had far reaching generalizations to higher genera Riemann surfaces, abelian varieties, etc. Surprisingly, theta functions made their appearance in another case of nineteenth century mathematics, namely mechanics. It was discovered by Carl Neumann and Jacobi that certain mechanical (Hamiltonian) systems could be explicitly integrated by means of theta functions. These ideas could have formed the foundation of some of the modern ideas on KdV, KP, and integrable systems in general, but the modern viewpoint seems to have been discovered without knowledge of the eighteenth century results. When the organizing committee met to discuss the possibility of a confer• ence on theta functions, we saw how perfectly the notation 0(T, Z) fit into a three week conference: one week for T, one week for z, and one week for the

IX x PREFACE comma. (This conforms to the above described three aspects of theta func• tions that appeared in the nineteenth century.) The conference was thereby organized accordingly. The first week was devoted to the comma, that is, to the interplay of r and z. The sections on infinite analysis, integrable systems, Kac-Moody algebras, lattice models, and physics are, roughly speaking, de• voted to this interplay; the sections on Jacobi varieties, Prym varieties, and algebraic geometry emphasize the z variable. These sections form Part 1 of Volume 49. The sections on modular forms, number theory, and combina• torics emphasize the T variable. They comprise Part 2 of Volume 49. It was our hope in organizing the conference that the presentation of a cross section of modern work on theta functions would enable mathematicians to see where we stand now and in what directions we should go in the future. Leon Ehrenpreis Robert C. Gunning List of Participant s List of Participants

Organizing committee: Enrico Arbarello, David Chudnovsky, Gregory Chudnovsky, Takahiro Kawai, Henry McKean, and co-chairmen Leon Ehrenpreis and Robert C. Gunning. ABLOWITZ, Mark J. Clarkson University, Potsdam, NY ACCOLA, Robert D. M. Brown University, Providence, RI ADDINGTON, Susan L. Harvard University, Cambridge, MA ADLER, Mark Brandeis University, Waltham, MA ANDREWS, George E. Pennsylvania State University, University Park, PA AOKI, Kenichiro Princeton University, Princeton, NJ AOMOTO, Kazuhiko Nagoya University, Nagoya, Japan ARBARELLO, Enrico University of Rome, Rome, Italy ASKEY, Richard A. University of Wisconsin, Madison, WI AU-YANG, Helen State University of New York, Stony Brook, NY BEAUVILLE, Arnaud University of Paris, le Pecq, France BERNSTEIN, Joseph N. Harvard University, Cambridge, MA BLUHER> Antonia Princeton University, Fine Hall, Princeton, NJ BLUHER, Grigory Princeton University, Fine Hall, Princeton, NJ BOCHERER, Siegfreid Math. Instut der Univ, Freiberg, West Germany BORWEIN, Jonathan Dalhousie University, Halifax, Nova Scotia, Canada BORWEIN, Peter Dalhousie University, Halifax, Nova Scotia, Canada BREEN, Lawrence Centre de Mathematiques, de l'Ecole Polytechnique, Palaiseau, France BUONOCORE, Michael H. University of California, Sacramento, CA CANDILERA, Maurizio Universita degli Studi, Padova, Italy CHAI, Ching-Li Princeton University, Princeton, NJ CHARI, Vyjayanthi Rutgers University, New Brunswick, NJ CHUDNOVSKY, David Columbia University, New York, NY CHUDNOVSKY, Gregory Columbia University, New York, NY COGDEL, James W. Oklahoma State University, Stillwater, OK COHN, Harvey City College, City University of New York, New York, NY COLLINO, Alberto Verzuolo (CN), Italy CRISTANTE, Valentino Universita degli Studi, Padova, Italy CUKIERMAN, Fernando University of California, Los Angeles, CA D'HOKER, Eric Princeton University, Princeton, NJ DATE, Etsuro Max Planck Institute, Bonn, West Germany DEBARRE, Olivier Universite de Paris-SUD, Orsay, France DUCROT, Francois Universite D'Angers, Angers Cedex, France DUKE, William Rutgers University, New Brunswick, NJ DUNCAN, Tyrone E. The University of Kansas, Lawrence, KS EARLE, Clifford J. Cornell University, Ithaca, NY EHRENPREIS, Leon Temple University, Philadelphia, PA ERCOLANI, Nicholas Courant Institute, New York, NY

363 364 LIST OF PARTICIPANTS

FARKAS, Hershel Hebrew University, Jerusalem, Israel FAY, John D. Haverford College, Haverford, PA FEINGOLD, Alex J. State University of New York, Binghamton, NY FLASCHKA, Hermann University of Arizona, Tucson, AZ FRENKEL, Igor Yale University, New Haven, CT FRIED, Michael D. University of Florida, Gainsville, FL GAETA, Federico Universitas Complutense, Madrid, Spain GARVAN, Francis University of Minnesota, Minneapolis, MN GEL'FAND, Israil Moscow University, Moscow, USSR GILBERT, Gerald CalTech, Pasadena, CA GOMEZ-MONT, Carlos University of Mexico/UN AM, Mexico, D. F., Mexico GRANT, David University of Michigan, Ann Arbor, MI GRUNBAUM, F. Alberto University of California, Berkeley, CA GUNNING, Robert C. Princeton University, Princeton, NJ GURARIE, David Case Western Reserve University, Case Institute of Technology, Cleveland, OH GUSTAFSON, Robert Texas A & M, College Station, TX HAINE, Luc University of Arizona, Tuscon, AZ HAMMOND, William F. State University of New York, Albany, NY HELMINCK, G. F. University of Twente, AE Enschede, the Netherlands HENDON, M. D. Athens, GA HOYT, William L. Rutgers University, New Brunswick, NJ HSIEH, June Academia Sinica, Nankang, Taipei, Taiwan, Republic of China IGUSA, Jun-Ichi The Johns Hopkins University, Baltimore, MD JABLOW, Eric State University of New York, Stony Brook, NY JIJTCHENKO, Alexei B. Academy of Sciences USSR, Moscow, USSR JIMBO, Michio Kyoto University, Kyoto, Japan JORGENSON, Jay. A Stanford University, Stanford, CA JUN, Sung-Tae University of Rochester, Rochester, NY KAC, Victor Massachusetts Institute of Technology, Cambridge, MA KANEV, Vassil Bulgarian Academy of Sciences, Sofia, Bulgaria KAREL, Martin L. Wilmington, DE KASHIWARA, Masaki Kyoto University, Kyoto, Japan KASTOR, David University of Chicago, Chicago, IL KATZ, Talbot Michael Hunter College, New York, NY KAWAI, Takahiro Kyoto University, Kyoto, Japan KAZHDAN, David Harvard University, Cambridge, MA KENT, Adrian University of Chicago, Chicago, IL KNOPP, Marvin Temple University, Philadelphia, PA KRUSEMEYER, Mark Carleton College, Northfield, MN KUZNETSOV, Nikolai V. ul P. Komarouva, Khabarousk, USSR KUNIBA, Atsuo University of Tokyo, Tokyo, Japan LAGARIAS, Jeffery AT&T Bell Laboratories, Murray Hill, NJ LANDESMAN, Peter New York, NY LATHAM, Geoff University of California, Berkeley, CA LEE, Ronnie Yale University, New Haven, CT LEHNER, Joseph Jamesburg, NJ LEWIS, Adrian Dalhousie University, Halifax, Nova Scotia, Canada LIM, Chjan Chin University of Michigan, Ann Arbor, MI LITTLE, John B. College of the Holy Cross, Worcester, MA LU, Shirong Massachusetts Institute of Technology, Cambridge, MA MARTENS, Henrick H. Norges Tekniske Hogskole, Trondheim, Norway MARTINEC, Emil University of Chicago, Chicago, IL MAYER, Alan L. Brandeis University, Waltham, MA McCOY, Barry M. State University of New York, Stony Brook, NY McDANIEL, Andrew George Mason University, Fairfax, VA McKEAN, Henry New York University, New York, NY LIST OF PARTICIPANTS 365

MENEZES, Maria Lucia Pontifica Universidade Catolica, Rio de Janeiro, Brazil MILNE, Stephen C. University of Kentucky, Lexington, KY MIWA, Tetsuji Kyoto University, Kyoto, Japan MOLL, Victor H. Tulane University, New Orleans, LA MORENO, Carlos J. City University of New York, New York, NY MULASE, Motohico University of California, Los Angeles, CA MUNOZ-PORRAS, J. M. Universidad de Salamanca, Salamanca, Spain MURTY, V. Kumar University of Toronto, Toronto, Ontario, Canada NAMIKAWA, Yukihiko Max Planck Institute, Bonn, West Germany NARANJO del val, Juan C. Universidad de Barcelona, Barcelona, Spain NARASIMHAN, Mudumbai S. Tata Institute for Fundamental Research, Bombay, India NEMCHENOK, Jacob Dartmouth College, Hanover, NH NEVEU, Andre USTL, France NORMAN, Peter University of Massachusetts, Amherst, MA NOURI-MOGHADM, M. R. King's College, University of London, London, England OKADO, Masato Kyoto University, Kyoto, Japan PALMER, John University of Arizona, Tuscon, AZ PARKER, Phillip E. Wichita State University, Wichita, KA PERIWAL, Vipul Princeton University, Princeton, NJ PERK, Jacques H. H. State University of New York, Stony Brook, NY PERLINE, Ronald K. Drexel University, Philadelphia, PA PETRI, Monica Massachusetts Institute of Technology, Cambridge, MA PHONG, Duong Columbia University, New York, NY PIATETSKI-SHAPIRO, I. I. Tel-Aviv University, Ramat Aviv, Israel PILA, Jonathan Stanford University, Stanford, CA PIOVAN, Luis Brandeis University, Waltham, MA PIROLA, Gian Pietro University di Pavia, Pavia, Italy PIZER, Arnold K. University of Rochester, Rochester, NY POOR, Chris Princeton University, Princeton, NJ PRESSLEY, Andrew N. Rutgers University, New Brunswick, NJ PREVIATO, Emma Imperial College, London, England RAMANAN, Sundararaman Tata Inst, of Fund. Research, Bombay, India REACH, Michael Berkeley, CA RHODES, John A. Harvard University, Cambridge, MA RIBET, Kenneth IMES, Bures sur Yvette, France RIES, John F. X. State University of New York, Binghamton, NY RIGGS, Harold University of Chicago, Chicago, IL ROCHA-CARIDI, Alvany Baruch College, City University of New York, New York, NY ROHM, Ryan M. California Institute of Technology, Pasadena, CA ROSS, Shepley L., II Bates College, Lewiston, ME SACHS, Robert L. Pennsylvania State University, University Park, PA SALVATI-MANNI, R. University de Roma, Rome, Italy SATO, Mikio Kyoto University, Kyoto, Japan SCHILLING, Randolph J. Louisiana State University, Baton Rouge, LA SCHLICHENMAIER, Martin Universitat Mannheim, Mannheim, West Germany SEILER, Wolfgang K. Universitat Mannheim, Mannheim, West Germany SEKIGUCHI, Tsutomu Chuo University, Tokyo, Japan SHEMANSKE, Thomas R. Dartmouth College, Hanover, NH SHIOTA, Takahiro Institute for Advanced Study, Princeton, NJ SINCLAIR, Paul SINNOU, David Witchita State University, Witchita, KS SIPE, Patricia L. University of Paris VI, Paris, France SMITH, Roy Smith College, Northampton, MA STANTON, Dennis University of Georgia, Athens, GA SULLIVAN, John M. University of Minnesota, Minneapolis, MN TAI, Y.-S. Princeton University, Princeton, NJ TAKASAKI, Kanehisa Haverford College, Haverford, PA Kyoto University, Kyoto, Japan 366 LIST OF PARTICIPANTS

TAKHTAJAN, Leon A. Steklov Mathematical Institute, Leningrad, USSR TAMANOI, Hirotaka The Johns Hopkins University, Baltimore, MD TAUBES, Clifford Harvard University, Cambridge, MA TONG, Yue-Lin Purdue University, West Lafayette, IN TRACY, Craig A. University of California, Davis, CA TREGER, Robert Queen's University, Kingston, Ontario, Canada TRETKOFF, Marvin Stevens Inst, of Technology, Princeton, NJ VAKILIAN, Ramin University of California, Davis, CA van GEEMEN, Bert Math. Inst. Utrecht, Holland van MOERBEKE, Pierre University of Louvain, Louvain-la-Neuve, Belgium van MULBREGT, Paul Massachusetts Institute of Technology, Cambridge, MA VARLEY, Robert University of Georgia, Athens, GA WAKIMOTO, Minoru Massachusetts Institute of Technology, Cambridge, MA WASHNITZER, Gerard Princeton University, Princeton, NJ WELTERS, Gerald E. Universidad de Barcelona, Barcelona, Spain WILSON, George Imperial College, London University, London, England, U. K. WODZICKI, Mariusz Institute for Advanced Study, Princeton, NJ WRIGHT, Paul E. University of California, Berkeley, CA WU, F. Y. Northeastern University, Boston, MA YAMAWAKI, Noriaki Waseda University, Tokyo, Japan YASUDA, Osamu CALTECH, Pasadena, CA YOU, Yu-Ching Massachusetts Institute of Technology, Cambridge, MA YUEN, David Princeton University, Princeton, NJ ZUBELLI, Jorge P. University of California, Berkeley, CA ZUCKERMAN, Gregg J. Yale University, New Haven, CT

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