Automorphic Forms on SL2 (R)

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

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.

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(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.

An introduction to the analytic theory of automorphic forms, limited to the case of fuchsian groups, but from the point of view of the general theory, ending with an introduction to unitary infinite dimensional representations. The main prerequisites are familiarity with functional analysis and elementary Lie theory.