Automorphic Forms on SL2 (R)
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Part of Cambridge Tracts in Mathematics
 Author: Armand Borel
 Date Published: August 2008
 availability: Available
 format: Paperback
 isbn: 9780521072120
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This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upperhalf plane X, with respect to a discrete subgroup ^D*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 ^D*G\G and its relationship with the classical automorphic forms on X, Poincaré 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(^D*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.
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Reviews & endorsements
"...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
See more reviews"...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
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×Product details
 Date Published: August 2008
 format: Paperback
 isbn: 9780521072120
 length: 208 pages
 dimensions: 229 x 152 x 12 mm
 weight: 0.31kg
 availability: Available
Table of Contents
Part I. Basic Material On SL2(R), Discrete Subgroups and the UpperHalf Plane:
1. Prerequisites and notation
2. Review of SL2(R), differential operators, convolution
3. Action of G on X, discrete subgroups of G, fundamental domains
4. The unit disc model
Part II. Automorphic Forms and Cusp Forms:
5. Growth conditions, automorphic forms
6. Poincare series
7. Constant term:the fundamental estimate
8. Finite dimensionality of the space of automorphic forms of a given type
9. Convolution operators on cuspidal functions
Part III. Eisenstein Series:
10. Definition and convergence of Eisenstein series
11. Analytic continuation of the Eisenstein series
12. Eisenstein series and automorphic forms orthogonal to cusp forms
Part IV. Spectral Decomposition and Representations:
13.Spectral decomposition of L2(G\G)m with respect to C
14. Generalities on representations of G
15. Representations of SL2(R)
16. Spectral decomposition of L2(G\SL2(R)):the discrete spectrum
17. Spectral decomposition of L2(G\SL2(R)): the continuous spectrum
18. Concluding remarks.