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
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