Overview

This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on G\G and its relationship with the classical automorphic forms on X, Poincare series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2 (G\G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras. Graduate students and researchers in analytic number theory will find much to interest them in this book.

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9780511896064

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Review of the hardback: 'This text will serve as an admirable introduction to harmonic analysis as it appears in contemporary number theory and algebraic geometry.' Victor Snaith, Bulletin of the London Mathematical Society Review of the hardback: '... carefully and concisely written ... Clearly every mathematical library should have this book.' Zentralblatt ...is a very carefully written book, which is accessible to the graduate student. ... Its size of 184 pages makes it even more possible to complete the material in one semester. Thus, it can be used as preparation for learning the `Langlands program' since the case of SL2(R) is concrete enough not to be lost in the maze of generality and typical enough to exemplify the general case. The topic is now a dominant theme in both algebraic and analytic number theory as well as algebraic geometry, and so the monograph is timely. Mathematical Reviews Clippings ...I would recommend [Bu] to graduate students or anyone else who wants to get into the subject. There is a great deal of foundational material here. Any effort put into reading it and working exercises will be rewarded with a good understanding of the basics. Bulletin of the American Mathematical Society

Review of the hardback: 'This text will serve as an admirable introduction to harmonic analysis as it appears in contemporary number theory and algebraic geometry.' Victor Snaith, Bulletin of the London Mathematical Society Review of the hardback: '... carefully and concisely written ... Clearly every mathematical library should have this book.' Zentralblatt

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