Applications of two representation theorems of elliptic functions and modular forms

Question

I'd like to know some elementary applications of these theorems, and also some examples of how one could work out the representations in question given a particular $f$.

If $f$ is an elliptic function it can always be written as $$f(z) = R_1 [\mathcal{P(z)}] + \mathcal{P}'(z) R_2[\mathcal{P}(z)]$$ for some rational function $R_1, \, R_2$ where $\mathcal{P}$ is the Weierstrass function.

As well as

If $f$ is a modular function it can always be written as a rational function of Klein's $j$-invariant.

References are also greatly appreciated. Thanks!

Answer 1

The theorems are a relatively advanced part of "The Elliptic Realm". They are of use if you already have an elliptic function or a modular function and you want to express it in a simple form relative to $\,\wp(z)\,$ or $\,j(\tau).\,$

To actually use the result is not hard. For example, the $\,j\,$-function has an inverse $\,g(z)\,$ which satisfies $\,z = j(g(z))\,$ and $\,\tau = g(j(\tau)).\,$ Thus, if you have another modular function $\,f(\tau),\,$ then find the power series expansion of $\,R(z) := f(g(z))\,$ where the function $\,R(z)\,$ is rational according to the theorem. Of course, there are other ways to find the rational function $\,R(z)\,$ using undetermined coefficients for instance. There is a recent MSE question "Regarding doubt in proof that every modular function can be represented as rational function in J" which refers to a book by Apostol. There is a pdf Lecture Notes which states

Theorem $20.8.$ Every modular function for $\Gamma(1)$ is a rational function of $\,j(\tau).\,$

For elliptic functions, the function $\,\wp(z)\,$ is even and $\,\wp'(z)\,$ is odd. Thus, given an elliptic function $\,f(z)\,$ we can split it into its even and odd part. For example, $\,f(z) = f_1(z) + \wp'(z) f_2(z)\,$ where both $\,f_1(z)\,$ and $\,f_2(z)\,$ are even functions of $\,z.\,$ The $\,\wp\,$ function (or rather $\,1/\sqrt{\wp(z)}\,$) has an inverse $\,g(z) = z + \frac{c_2}2 z^5 + \frac{c_3}2 z^7 + O(z^9)\,$ which satisfies $\,z = 1/\sqrt{\wp(g(z))} = g(1/\sqrt{\wp(z)}).\,$ Thus, $\,R_1(z) = f_1(g(1/\sqrt{z}))\,$ and $\,R_2(z) = f_2(g(1/\sqrt{z}))\,$ where the functions $\,R_1\,$ and $\,R_2\,$ are rational according to the theorem. This is referenced in the Wikipedia article Weierstrass's elliptic functions which states

The totality of meromorphic double periodic functions with given period defines an algebraic function field associated to that curve. It can be shown that this field is $$\mathbb{C}(\wp,\wp'),$$ so that all such functions are rational functions in the Weierstrass function and its derivative.

This theorem is also mentioned in Abramowitz and Stegun's Handbook of Mathematical Functions, section 18.11 page 651:

If $f(z)$ is any elliptic function and $\cal{P}(z)$ has the same periods, write $$ {\bf 18.11.1} \quad f(z) = \frac12 [f(z)+f(-z)] + \frac12[\{f(z)-f(-z)\} \{\cal{P}'(z)\}^{-1}]\cal{P}'(z). $$

Since both brackets represent even elliptic functions, we ask how to express an even elliptic function $\,g(z)\,$ (of order $2k$) in terms of $\cal{P}(z).\,$

There follows equation $18.11.2$ which expresses $\,g(z)\,$ as an explicitly factorized rational function in $\cal{P}(z)\,$ with linear factors.