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.
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