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.

The proceedings of the conference are being published in two parts, and the present volume is mostly algebraic (congruence properties of modular forms, modular curves and their rational points, etc.), whereas the second volume will be more analytic and also include some papers on modular forms in several variables.

The proceedings of the conference are being published in two parts, and the present volume is mostly algebraic (congruence properties of modular forms, modular curves and their rational points, etc.), whereas the second volume will be more analytic and also include some papers on modular forms in several variables.

This book is a fully detailed introduction to the theory of modular functions of a single variable. I hope that it will fill gaps which in view ofthe lively development ofthis theory have often been an obstacle to the students' progress. The study of the book requires an elementary knowledge of algebra, number theory and topology and a deeper knowledge of the theory of functions. An extensive discussion of the modular group SL(2, Z) is followed by the introduction to the theory of automorphic functions and auto morphic forms of integral dimensions belonging to SL(2,Z). The theory is developed first via the Riemann mapping theorem and then again with the help of Eisenstein series. An investigation of the subgroups of SL(2, Z) and the introduction of automorphic functions and forms belonging to these groups folIows. Special attention is given to the subgroups of finite index in SL (2, Z) and, among these, to the so-called congruence groups. The decisive role in this setting is assumed by the Riemann-Roch theorem. Since its proof may be found in the literature, only the pertinent basic concepts are outlined. For the extension of the theory, special fields of modular functions in particular the transformation fields of order n-are studied. Eisen stein series of higher level are introduced which, in case of the dimension - 2, allow the construction of integrals of the 3 rd kind. The properties of these integrals are discussed at length.

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.

This is an advanced book on modular forms. While there are many books published about modular forms, they are written at an elementary level, and not so interesting from the viewpoint of a reader who already knows the basics. This book offers something new, which may satisfy the desire of such a reader. However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types. Thus the book is presented so that the reader can learn such theories systematically.

This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory. --John E. Cremona, University of Nottingham William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics.

Based on courses given at Eötvös Loránd University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable — systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student’s mathematical intuition. The book provides a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of the theoretical concepts treated. In the spirit of learning-by-doing, Real Analysis includes more than 500 engaging exercises for the student keen on mastering the basics of analysis. The wealth of material, and modular organization, of the book make it adaptable as a textbook for courses of various levels; the hints and solutions provided for the more challenging exercises make it ideal for independent study.