Modular Functions of One Variable I Proceedings International Summer
An international Summer School on: "Modular functions of one variable and arithmetical applications" took place at RUCA, Antwerp University, from July 17 to - gust 3, 1972. This book is the first volume (in a series of four) of the Proceedings of the Summ
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320 Modular Functions of One Variable I Proceedings International Summer School University of Antwerp, RUCA July 17-August 3, 1972
Edited by W Kuijk
Springer-Verlag Berlin Heidelberg New York Tokyo
EdItor
Willem Kuijk Rijksuniversitair Centrum Antwerpen, Leerstoel Algebra Groenenborgerlaan 171,2020 Antwerpen, Belgium
1st Edition 1973 2nd Corrected Printing 1986 Mathematics Subject Classification (1970): 10005, 10025, 10C 15, 14K22, 14K25 ISBN 3-540-06219-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN o-387-06219-X Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-U$8 of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in dsta banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to 'Verwertungagesellschaft Wort", Munich.
C> by Springer-Verlag Berlin Heidelberg '973 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Preface An international Summer School on: "Modular functions of one variable and arithmetical applications" took place at RUCA, Antwerp University, from July 17 to August 3, 1972.
This book is the first volume (in a series of four) of the Proceedings of the Summer School.
It includes the basic
course given by A. Ogg, and several other papers with a strong analyt~c
flavour.
Volume 2 contains the courses of R. P. Langlands (l-adic representations) and P. Deligne (modular schemes - representations of GL ) and papers on related topics. 2 Volume 3 is devoted to p-adic properties of modular forms and applications to l-adic representations and zeta functions. Volume 4 collects various material on elliptic curves, including numerical tables.
The School was a NATO Advanced Study Institute, and the organizers want to thank NATO for its major subvention.
Further
support, in various forms, was received from IBM Belgium, the Coca-Cola Co. of Belgium, Rank Xerox Belgium, the Fort Food Co. of Belgium, and NSF Washington, D.C ••
We extend our warm-
est thanks to all of them, as well as to RUCA and the local staff (not forgetting hostesses and secretaries!) who did such an excellent job. P. Deligne W. Kuyk G. Poitou J-P. Serre
CONTENTS
Survey of Modular Functions of One Variable (Notes by F. Van Oystaeyen)
1
G. SHIMURA,
Complex Multiplication (Notes by Willem Kuyk
37
G. SHIMURA,
Modular Forms of Half Integral Weight
57
M. EICHLER,
The Basis Problem for Modular Forms and the Traces of the Hecke Operators
75
H.M. STARK,
Class-Numbers of Complex Quadratic Fields
153
B. BIRCH,
Some Calculations of Modular Relations
175
J. LEHNER,
On the A (f) C B q q Conjecture
187
A. OGG,
(f)
Mailing addresses of authors and writers
195
SURVEY OF MODVLAR FUNCTIONS OF ONE VARIABLE
by Andrew Ogg
(Notes by F. Van Oystaeyen)
International Summer School on Modular Functions Ant
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