Chapter 1

The Classical

Modular Forms

We begin by giving a glimpse of how automorphic forms emerged on the

hyperbolic plane as descendants of those living on flat spaces.

1.1. Periodic functions

Take X = R and V = Z, so R/Z is the circle. A function / : R -+ C is

periodic of period 1 if

f(x + n) = f(x) for all n € Z.

To construct a periodic function one can simply take any / on the seg-

ment [0,1) and extend its values uniquely to R by requiring periodicity. For

example

{x} = x — [x] (the fractional part of x)

is obtained from f(x) = x on [0,1). Another construction uses the averaging

method. Let / : R — C be any function of rapid decay at ±oc so that the

series

(1.1) ?(*)

=

]T/(*

+ n)

neZ

converges absolutely. Clearly the function g is periodic of period 1.

The exponential function

(1.2) e(x) =

e2*ix

= cos

2TTX

+ i sin

2-KX

3

http://dx.doi.org/10.1090/gsm/017/02