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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 2 metic of modular curves P. DELIGNE - M. RAPOPORT Les schemas de modules de courbes ellip 143 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 1972 ~ The author's travel expences for this conference were paid for by a grant from the National Research Council of Canada. -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.
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.