EMS Series of Lectures in Mathematics Edited by Andrew Ranicki (University of Edinburgh, U.K.) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp, María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions
Koichiro Harada
“Moonshine” of Finite Groups
Author: Prof. em. Koichiro Harada Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210-1174 USA E-mail:
[email protected]
2010 Mathematics Subject Classification: 20B05, 11F03 Key words: Monster simple group, congruence groups, modular functions, eta function
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[email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Druckhaus Thomas Müntzer GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321
Preface This is an almost verbatim copy of the lecture notes from a two-quarter course I gave on Moonshine of Finite Groups at the Ohio State University in 1983–84. My original motivation was to understand the moonshine phenomena of the Monster simple group M. A quarter of a century later since, now around the year 2010, it does not seem that we understand them very well, although the main problem that existed then is now a theorem. Namely, the Conway–Norton conjecture has been solved by Richard Borcherds. The idea of his proof has created a new area of mathematics, now called the theory of vertex algebras. This algebraic structure has since become a flourishing area of mathematics for many researchers around the world. On the other hand, the interest generated by researchers in the original mystery of moonshine of the Monster simple group seems to