Automorphic Forms on SL2 (R)

An introduction to the analytic theory of automorphic forms in the case of fuchsian groups.

Armand Borel (Author)

9780521580496, Cambridge University Press

Hardback, published 28 August 1997

208 pages
23.4 x 15.9 x 2.1 cm, 0.44 kg

Review of the hardback: '… carefully and concisely written … Clearly every mathematical library should have this book.' Zentralblatt

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 GG 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 (GG). 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.

Part I. Basic Material On SL2(R), Discrete Subgroups and the Upper-Half 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(GG)m with respect to C
14. Generalities on representations of G
15. Representations of SL2(R)
16. Spectral decomposition of L2(GSL2(R)):the discrete spectrum
17. Spectral decomposition of L2(GSL2(R)): the continuous spectrum
18. Concluding remarks.

Subject Areas: Number theory [PBH]