This is the third Lecture Notes volume to be produced in the framework of the New York Number Theory Seminar. The papers contained here are mainly research papers. N

The goal of this research monograph is to derive the analytic continuation and functional equation of the "L"-functions attached by R.P. Langlands to automorphic representations of reductive algebraic groups. The first part of the book (by Piatetski-Shapiro and Rallis) deals with "L"-functions for the simple classical groups; the second part (by Gelbart and Piatetski-Shapiro) deals with non-simple groups of the form "G GL(n)," with "G" a quasi-split reductive group of split rank "n." The method of proof is to construct certain explicit zeta-integrals of Rankin-Selberg type which interpolate the relevant Langlands "L"-functions and can be analyzed via the theory of Eisenstein series and intertwining operators. This is the first time such an approach has been applied to such general classes of groups. The flavor of the local theory is decidedly representation theoretic, and the work should be of interest to researchers in group representation theory as well as number theory.

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 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 rep resentations) 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, includ ing numerical tables. The School was a NATO Advanced Study Institute, and the orga nizers 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.

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.

During the academic year 1916-1917 I had the good fortune to be a student of the great mathematician and distinguished teacher Adolf Hurwitz, and to attend his lectures on the Theory of Functions at the Polytechnic Institute of Zurich. After his death in 1919 there fell into my hands a set of notes on the Theory of numbers, which he had delivered at the Polytechnic Institute. This set of notes I revised and gave to Mrs. Ferentinou-Nicolacopoulou with a request that she read it and make relevant observations. This she did willingly and effectively. I now take advantage of these few lines to express to her my warmest thanks. Athens, November 1984 N. Kritikos About the Authors ADOLF HURWITZ was born in 1859 at Hildesheim, Germany, where he attended the Gymnasium. He studied Mathematics at the Munich Technical University and at the University of Berlin, where he took courses from Kummer, Weierstrass and Kronecker. Taking his Ph. D. under Felix Klein in Leipzig in 1880 with a thes i s on modul ar funct ions, he became Pri vatdozent at Gcitt i ngen in 1882 and became an extraordinary Professor at the University of Konigsberg, where he became acquainted with D. Hilbert and H. Minkowski, who remained lifelong friends. He was at Konigsberg until 1892 when he accepted Frobenius' chair at the Polytechnic Institute in Z rich (E. T. H. ) where he remained the rest of his 1 i fe.

The theory of arithmetical functions has always been one of the more active parts of the theory of numbers. The large number of papers in the bibliography, most of which were written in the last forty years, attests to its popularity. Most textbooks on the theory of numbers contain some information on arithmetical functions, usually results which are classical. My purpose is to carry the reader beyond the point at which the textbooks abandon the subject. In each chapter there are some results which can be described as contemporary, and in some chapters this is true of almost all the material. This is an introduction to the subject, not a treatise. It should not be expected that it covers every topic in the theory of arithmetical functions. The bibliography is a list of papers related to the topics that are covered, and it is at least a good approximation to a complete list within the limits I have set for myself. In the case of some of the topics omitted from or slighted in the book, I cite expository papers on those topics.

This is a volume of papers presented at the New York Number Theory Seminar. Since 1982, the Seminar has been meeting weekly during the academic year at the Graduate School and University Center of the City University of New York. This collection of papers covers a wide area of number theory, particularly modular functions, algebraic and diophantine geometry, and computational number theory.

This is Volume 3 of the Proceedings of the Interna- tional Summer School on "Modular functions of one variable and arithmetical applications" which took place at RUCA, Antwerp University, from July 17 to August 3, 1972. It contains papers by P.Cartier-[yen].Roy, B.Dwork, N.Katz, J-P.Serre and H.P.F.Swinnerton-Dyer on congruence proper- ties of modular forms, l-adic representations, p-adic modular forms and p-adic zeta functions. W.Kuyk J-P.Serre CONTENTS H.P.F. SWINNERTON-DYER On l-adic representations and congruences for coefficients 1 of modular forms B. DWORK The Up operator of Atkin on modular functions of level 2 57 with growth conditions N. KATZ p-adic properties of modular 69 schemes and modular forms J-P. SERRE Formes modulaires et fonctions 191 zeta p-adiques P. CARTIER-Y. ROY Certains calculs numeriques relatifs a l'interpolation 269 p-adique des series de Dirichlet Mailing addresses of authors 350 He~~n e.L. Siegel gewidmet ON l-ADIC REPRESENTATIONS AND CONGRUENCES FOR COEFFICIENTS OF MODULAR FORMS BY H.P.F. SWINNERTON-DYER International Summer School on Modular Functions Antwerp 1972 2 SwD-2 CONTENTS 1. Introduction. p.3 2. The possible images of Pl. p.l0 3. Modular forms mod l. p.1S 4. The exceptional primes. p.26 5. Congruences modulo powers of l. p.36 Appendix p.43 References p.ss 3 SwD-3 ON l-ADIC REPRESENTATIONS AND CONGRUENCES FOR COEFFICIENTS OF MODULAR FORMS * 1. Introduction.

A belian Varieties has been out of print for a while. Since it was written, the subject has made some great advances, and Mumford's book giving a scheme theoretic treatment has appeared (D. Mum- ford, Abelian Varieties, Tata Lecture Notes, Oxford University Press, London, 1970). However, some topics covered in my book were not covered in Mumford's; for instance, the construction of the Picard variety, the Albanese variety, some formulas concern- ing numerical questions, the reciprocity law for correspondences and its application to Kummer theory, Chow's theory for the K/k-trace and image, and others. Several people have told me they still found a number of sections of my book useful. There- fore I thank Springer-Verlag for the opportunity to keep the book in print. S. LANG v FOREWORD Pour des simplifications plus subs tan- tielles, Ie developpement futur de la geometrie algebrique ne saurait manquer sans do ute d' en faire apparaitre. It is with considerable pleasure that we have seen in recent years the simplifications expected by Weil realize themselves, and it has seemed timely to incorporate them into a new book. We treat exclusively abelian varieties, and do not pretend to write a treatise on algebraic groups. Hence we have summarized in a first chapter all the general results on algebraic groups that are used in the sequel. They are all foundational results.

The present book was conceived as an introduction for the user of universal algebra, rather than a handbook for the specialist, but when the first edition appeared in 1965, there were practically no other books entir ly devoted to the subject, whether introductory or specialized. Today the specialist in the field is well provided for, but there is still a demand for an introduction to the subject to suit the user, and this seemed to justify a reissue of the book. Naturally some changes have had to be made; in particular, I have corrected all errors that have been brought to my notice. Besides errors, some obscurities in the text have been removed and the references brought up to date. I should like to express my thanks to a number of correspondents for their help, in particular C. G. d'Ambly, W. Felscher, P. Goralcik, P. J. Higgins, H.-J. Hoehnke, J. R. Isbell, A. H. Kruse, E. J. Peake, D. Suter, J. S. Wilson. But lowe a special debt to G. M. Bergman, who has provided me with extensive comments. particularly on Chapter VII and the supplementary chapters. I have also con sulted reviews of the first edition, as well as the Italian and Russian translations."