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.