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.
General
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