6 1. The Classical Modular Forms

Exercise. Show that if /(x) is a distance function, then so is /(y); more

precisely, if

/(x)

=

5(W)

then

/(y) = Mlyl)

where h(y) is a Hankel type transform of g(x) given by

1 k f°° k

(1.12) h(y) =

27ryl~*

/ g(x)Jk_x{2nxy)x2dx

Jo

2

and Ji(x) is the Bessel function. Hence the Poisson summation formula

becomes

oo oo

(1.13) ^T rk(m)g(m) = ] T rk(m)h(m)

ra=0 m=Q

where rk(m) denotes the number of representations of m as the sum of k

squares;

(1.14) rk(m) = |{(m,..., nk) €

Zk

: n\ + • • • + n\ = m}\.

1.2, Elliptic functions

The next generation of periodic functions lives in the space of complex num-

bers X = C. This is not simply the former case in two dimensions. Here C is

considered as a Riemannian manifold with complex structure. Consequently

the theory of meromorphic functions can be employed, which is indeed a very

powerful tool. For instance, one can establish interesting relations, even of

arithmetic type, by examining the location of poles and zeros.

We shall use the standard complex number notation z = x + iy. The

group Z acts on C giving periodic functions in the horizontal direction, such

as

e(z) =

e27Tiz

=

e(x)e-2ny.

To produce periodicity in any two described directions, consider a group

A =

CJIZ

+ CJ2Z

where

UJ\,UJ2

are complex numbers linearly independent over K so that

C =

UJIR

+ o;2lR.