Modular Functions of One Variable II Proceedings International Summe
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349 Modular Functions of One Variable II Proceedings International Summer School University of Antwerp, RUCA July 17-August 3, 1972
Edited by P. Deligne and W Kuijk
Springer-Verlag Berlin Heidelberg New York Tokyo
Editors Pierre Oeligne Institute for Advanced Study, School of Mathematics Princeton, NJ 08540, USA Willem Kuijk Rijksuniversitair Centrum Antwerpen, Leerstoel Algebra Groenenborgerlaan 171, 2020 Antwerpen, Belgium
1st Edition 1973 2nd Corrected Printing 1986 Mathematics Subject Classification (1970): 10-02, 10015, 10Hl0, 14Gl0, 14G99, 14H 10, 14L99 ISBN 3-540-06558-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06558-X Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1973 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140,543210
PREFACE
This is Volume 2 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.
It contains papers by W. Casselman, P. Deligne, R. Langlands and 1.1. Piateckii-Shapiro.
Its theme is the
interplay between modular schemes for elliptic curves, and representations of GL(2).
P. Deligne
W. Kuyk
CONTENTS
W. CASSELMAN
An assortment of results on represen-
1
tations of GL (k) 2
P. DELIGNE
Formes mOdulaires et representations
55
de GL(2)
W. CASSELMAN
On representations of GL
and the arith-
107
Les schemas de modules de courbes ellip-
143
metic of modular curves
P. DELIGNE - M. RAPOPORT
2
tiques
1.1. PIATECKII-SHAPIRO
Zeta-functions of modular curves
317
R.P. LANGLANDS
Modular forms and t-adic representations
361
P. DELIGNE
Les constantes des equations fonctionnelles 501 des fonctions L
Addresses of authors
598
AN ASSORTMENT OF RESULTS ON REPRESENTATIONS OF GL (k) 2
by W. Casselman~
International Summer School on Modular Functions} Antwerp
~ The author's travel expences for this conference were paid for by a grant from the National Research Council of Canada.
1972
-2-
Cas-2
Contents
Introduction
3
Notations §1.
Generalities
5
Appendix
§2.
The principal series representations of
15
GL (O) and the associated Hecke algebras 2 §3.
The principal series of G
29
§4.
Absolutely cuspidal representations
43
References
53
-3-
Cas-3
Introduction
Let k be a p-adic field, 0 its integers.
My aim in these notes has been
to collect in one convenient place most of the results I need in [5] concerning the representations of GL (k). 2
My choice of which results
to include
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