Chapter

Introduction

The concept of an automorphic function (the name was given by F. Klein,

1890) is a natural generalization of a periodic function. Let X be a locally

compact space acted on discontinuously by a group T. Then a function

/ : X — • C is automorphic with respect to T if

(0.1) / (7X) = / (x) for all 7 e I\

In other words, / lives on the quotient space T \X (the space of orbits).

A typical habitat for automorphic functions is the homogeneous space X =

G/K of a Lie group G, where K is a closed subgroup. In this case X is

a Riemannian manifold, so the differential calculus on X is available. If

an automorphic function / is a common eigenfunction of the whole algebra

T of invariant differential operators on X (V is a commutative algebra; it

contains the Laplace-Beltrami operator),

(0.2) Df = Xf (D) f for all D e £,

then / is called an automorphic form.

More generally, one considers automorphic functions for which the trans-

formation rule (0.1) is modified by a differential factor and a suitable multi-

plier system. Those functions which live on X = SL/2(M) / S02(l&) and which

are complex-analytic (X is identified with the upper-half plane EI C C and

the differential equations (0.2) are the Cauchy-Riemann equations) are called

classical automorphic forms.

In this book our main interest will be in the classical automorphic forms

with respect to a congruence group, because of their importance for applica-

tions to arithmetic. On a few occasions we shall encounter the real-analytic

Eisenstein series, and for clarity, we introduce basic concepts in the context

of a general group acting discontinuously on the upper-half plane.

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http://dx.doi.org/10.1090/gsm/017/01