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# The Miracle of Theta Functions

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).

### Theorem 1

Let z be a primitive pth root of unity for p an odd prime. Consider the pth order modular equation for as defined by

where

and

Then the function is a polynomial in x and ( independent of q), which has integer coefficients and is of degree p+1 in both x and .
The 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 degree p+1.

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

and turns out to be a rational function of and .

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

This is an entirely algebraic algorithm. One needs to know the pth-order modular equation for to compute from and one needs to know the rational multiplier . The speed of convergence (, for some c < 1) is easily deduced from (5.13) and (5.9).

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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