When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics, and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian functions.Felix Klein [21]
Felix Klein's lament from a hundred years ago has an uncomfortable timelessness to it. Sadly, it is now possible never to see what Bochner referred to as ``the miracle of the theta functions'' in an entire university mathematics program. A small piece of this miracle is required here [6], [11], [28]. First some standard notations. The complete elliptic integrals of the first and second kind, respectively,
and
The second integral arises in the rectification of the ellipse, hence the name
elliptic integrals. The complementary modulus is
and the complementary integrals 
and 
are defined by
The first remarkable identity is
Legendre's relation
namely
(for 
, which is pivotal in relating these quantities to 
. We also need
to define two Jacobian theta functions
and
These are in fact specializations with 
of the general theta functions. More
generally
with similar extensions of 
. In Jacobi's approach these general theta
functions provide the basic building blocks for elliptic functions, as functions of
t (see [11], [39]).
The complete elliptic integrals and the special theta functions are related as
follows. For |q| <1
and
where
and
The modular function 
is defined by
where
We wish to make a few comments about modular functions in general before restricting
our attention to the particular modular function 
. Modular functions
are functions which are meromorphic in H, the upper half of the complex plane, and
which are invariant under a group of linear fractional transformations, G, in the
sense that
[Additional growth conditions on f at certain points of the associated fundamental
region (see below) are also demanded.] We restrict G to be a subgroup of the
modular group 
where 
is the set of all transformations w of the
form
with a,b,c,d integers and ad - bc =1. Observe that 
is a group under
composition. A
fundamental region
is a set in H with the property that
any element in H is uniquely the image of some element in 
under the action of
G. Thus the behaviour of a modular function is uniquely determined by its
behaviour on a fundamental region.
Modular functions are, in a sense, an extension of elliptic (or doubly periodic)
functions --- functions such as sn which are invariant under linear transformations
and which arise naturally in the inversion of elliptic integrals.
The definitions we have given above are not complete. We will be more precise in our
discussion of 
. One might bear in mind that much of the theory for 
holds in considerably greater generality.
The fundamental region F we associate with 
is the set of complex
numbers
The 
-group (or theta-subgroup) is the set of linear fractional
transformations w satisfying
where a,b,c,d are integers and ad - bc =1, while in addition a and d are odd
and b and c are even. Thus the corresponding matrices are unimodular. What makes
a 
-modular function is the fact that 
is meromorphic in
and that
for all w in the 
-group, plus the fact that
tends to a definite limit (possibly infinite) as t tend to a vertex of
the fundamental region (one of the three points 
). Here
we only allow convergence from within the fundamental region.
Now some of the miracle of modular functions can be described. Largely because every
point in the upper half plane is the image of a point in F under an element of the
-group, one can deduce that any 
-modular function that is bounded
on F is constant. Slightly further into the theory, but relying on the above, is
the result that any two modular functions are algebraically related, and resting on
this, but further again into the field, is the following remarkable result. Recall
that q is given by (5.9).
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