Introduction The concept of an automorphic function is a natural generalization of that of a periodic function. Furthermore, an automorphic form is a generalization of the exponential function e(z) = e2πiz. To define an automorphic function in an abstract setting, one needs a group Γ acting discontinuously on a locally compact space X the functions on X which are invariant under the group action are called automorphic functions (the name was given by F. Klein in 1890). A typical case is the homogeneous space X = G/K of a Lie group G, where K is a closed subgroup. In this case the differential calculus is available, since X is a riemannian manifold. The automorphic functions which are eigenfunctions of all invariant differ- ential operators (these include the Laplace operator) are called automorphic forms. The main goal of harmonic analysis on the quotient space Γ\X is to decompose every automorphic function satisfying suitable growth conditions into automorphic forms. In these lectures we shall present the basic theory for Fuchsian groups acting on the hyperbolic plane. When the group Γ is arithmetic, there are interesting consequences for number theory. What makes a group arithmetic is the existence of a large family (commutative algebra) of certain invariant, self-adjoint operators, the Hecke operators. We shall get into this territory only briefly in Sections 8.4 and 13.3 to demonstrate its tremendous potential. Many important topics lie beyond the scope of these lectures for instance, the theory of automorphic L-functions is omitted entirely. A few traditional applications are included without straining for the best results. For more recent applications the reader is advised to see the original sources (see the surveys [Iw 1, 2] and the book [Sa 3]). 1
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