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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'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,

The * modular function* is defined by

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

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

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).

LetThe modular equation for usually has a simpler form in the associated variables and . In this form the 5th-order modular equation is given by In particular are related by an algebraic equation of degreezbe a primitivepth root of unity forpan odd prime. Consider thepth order modular equation for as defined by where and Then the function is a polynomial inxand (independentofq), which has integer coefficients and is of degreep+1in bothxand .

The miracle is not over. The **p*** th-order multiplier* (for ) is defined by

One is now in possession of a **p**th-order algorithm for , namely: Let . Then

The function is **1-1** on **F** and has a well-defined inverse,
, with branch points only at **0, 1** and . This can be used to
provide a one line proof of the ``big'' Picard theorem that a nonconstant entire
function misses at most one value (as does ). Indeed, suppose **g** is an entire
function and that it is never zero or one; then is a
bounded entire function and is hence constant.
Littlewood suggested that, at the right point in history, the above would have been a
strong candidate for a `one line doctoral thesis'.

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