Modular Functions of One Variable IV Proceedings of the Internationa
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476 Modular Functions of One Variable IV Proceedings of the International Summer School, University of Antwerp, RUCA, July 17 - August 3, 1972
Edited by B. J. Birch and W. Kuyk
Spri nger-Verlag Berlin· Heidelberg· New York 1975
Editors
Dr. Bryan J. Birch Mathematical Institute University of Oxford 24-29 8t. Giles Oxford OX1 3 LB/England Prof. Willem Kuyk Leerstoel Algebra Rijksuniversitair Centrum Antwerpen 2020 Antwerpen Middelheimlaan 1 Holland
Library of Congress Cataloging in Publication Data
(Revised)
Main entry under title:
Modular functions of one variable. (Lecture notes in mathematics 320, 349-350, 476) A NATO Advanced Study Institute; held University of Antwerp, RUCA, July 17-August 3. 1972. Includes bibliographies. 1. Functions, Modular--Congresses. I. Kuyk, Willem, ed. -II. NATO Advanced Study Institute. III. Series: Lecture notes in mathematics (Berlin) 320. [etc.] QA3.L28 [QA343] no. 320, etc. 510'.8s [515'.9] 73-78427
AMS Subject Classifications (1970): lOC15, 10 D05, lOD25, 14K22, 14K25 ISBN 3-540·07392-2 Springer·Verlag Berlin' Heidelberg· New York ISBN 0-387-07392-2 Springer-Verlag New York' Heidelberg· Berlin 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-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1975 Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Preface
This is Volume 4 of the Proceedings of the International Summer School on
"Modular functions of one variable and arithmetical applications"
which took place at RUCA, Antwerp University, from July 17 till August 3, 1972.
The preceding three volumes have
been published as issues 320, 349 and 350 of the Lecture Notes Series. This final volume contains papers by B. Birch, P. Deligne and H.P.F. Swinnerton-Dyer, a letter from Tate to Cassels, and several numerical tables.
Its overall theme is the
arithmetic of elliptic curves. For the acknowledgements regarding financial and material support, the reader is referred to the preface of volume 320.
Once more, the editors wish to thank their major
financial sponsors, NATO, IBM Belgium and Rank Xerox Belgium for their support.
B.J. BIRCH
W. KUYK
CONTENTS
1.
Foreword
1
2.
H.P.F. SWINNERTON-DYER~B.J. BIRCH, Elliptic Curves and Modular Functions
2
3.
J. TATE, Algorithm for determining the Type of a Singular Fiber in an Elliptic Pencil
33
4.
P. DELIGNE, Courbes Elliptiques: Formulaire (d'apres J. Tate)
53
5.
Sources and reliability of the tables Remarks on isogenies
78
6.
Table 1
81
7.
Table 2
114
8.
Table 3
116
9.
Table 4
123
10.
Table 5
135
11-
Table 6
142
12.
Corrigenda to Volumes I-III
145
13.
List of Adresses of Authors
15
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