Sorry for the delay. This answer is aimed at proving an explanation for the $(c)$ case only. When thinking about the question I realised that should one know about the basic theory of elliptic functions, there is almost nothing to say, i.e. the result is immediate. Thus I will frame my answer as more of an exposé into elliptic functions, and sketch out the path one would take when proving the required result. There are already many good books on the introductory analysis that one could read to gain a good entry into this field. I would recommend A Course of Modern Analysis by Whittaker and Watson. All of the result involving elliptic functions that I give can be found therein. Unfortunately I do not have a copy to hand thus cannot give precise reference to results.
Lets start by defining what doubly periodic / elliptic functions are:
Doubly Periodic & Elliptic Functions
Suppose that $f: U\to\mathbb C$ is a holomorphic function from an open
subset $U \subseteq \mathbb C$. For any $\omega \in \mathbb C$ we say that $f$ is $\omega$-periodic if for all $u \in U$ both $u + \omega$ and $u - \omega$ lie in $U$ with $(0)$ holding. $$f(u - \omega) = f(u) = f(u +\omega) \tag{0}$$
Furthermore, we say that $f$ is doubly periodic if there are two $\omega,\omega^\prime \in \mathbb C$ that are linearly independent in $\mathbb C_\mathbb R$ such that $f$ is both $\omega$ and $\omega^\prime$-periodic. Lastly we say that $f$ is elliptic if $U = \mathbb C \setminus \mathfrak P_f$, where $\mathfrak P_f$ is the set of poles of $f$ (i.e. $f$ is a meromorphic function on $\mathbb C$).
Note that the definitions in the literature of a doubly periodic / elliptic function invariably vary with most authors not being as particular with the detail and many not differentiating between a doubly periodic function and an elliptic function. However, the above definition is a suitable working definition which we will adopt here.
What makes elliptic functions nice to work with is much of their study can be reduced down to the study of their zeros and poles and their periods, insofar as knowing an elliptic function's poles and zero's (including their order) and periods is enough to define an elliptic function up-to multiplicative constant.
Note that the set $\Lambda_f$ of periods of an elliptic function is an additive subgroup of $\mathbb C$. If $f$ isn't constant then this group is discrete and thus it follows that $\Lambda_f$ is generated by a pair of elements $(\omega, \omega)$. We call any such pair fundamental periods for $f$. Note that multiple choices of fundamental periods exists (compare and contrast with the singly periodic case where the fundamental period is unique upto sign). Denote the $(\mathbb C,+)$ subgroup generated by a set $S$ of elements we will use the notation $\Lambda(S)$. In particular $\Lambda(\omega) = \omega\mathbb Z$ and $\Lambda(\omega,\omega^\prime)= \omega\mathbb Z + \omega^\prime\mathbb Z$.
Irreducible Zeros & Poles
Suppose that one is given independent $(\omega,\omega^\prime)$ and
elliptic function $f$ with set $\mathfrak Z$ of zeros of $f$. We say
that $\mathfrak Z$ is an irreducible set of zero's (w.r.t.
$(\omega,\omega^\prime)$) if for every zero $z_0$ of $f$ there exists
a $z \in \mathfrak Z$ such that $z - z_0 \in \Lambda(\omega,\omega^\prime)$ and further for no two distinct
$z,z^\prime \in \mathfrak Z$ do $z - z^\prime \in
\Lambda(\omega,\omega^\prime)$. Such a set gives all of the periods of
$f$ up-to period. Usually we look at the irreducible zeros w.r.t.
fundamental periods but sometimes it is useful to look at w.r.t.
other periods. Likewise we can define the notion of irreducible poles
of $f$.
The above construction is here unmotivated but the notion is extremely useful for proving facts about elliptic functions in practise.
(Fundamental) Parallelogram & The order of an Elliptic Function
Suppose that $\omega,\omega^\prime$ are $\mathbb C_\mathbb R$
independent. Call the convex hull $H$ of the points $\{z,z +
\omega,z+\omega^\prime,z+\omega + \omega^\prime\}$ (given explicitly
by $H := \left\{x \in \mathbb C \ \big | \ \exists s,t \in [0,1]: x =
z + s \omega + t\omega^\prime\right\}$) an
$(\omega,\omega^\prime)$-parallelogram. Suppose then that $f:U \to \mathbb C$ is an elliptic function with periods $(\omega,\omega^\prime)$. If $f \neq 0$ then as the poles and zeros of $f$ are discreet we may always find some $z_0 \in \mathbb C$ such that $H_{z_0}$ contains no pole or zero on its boundary. By contour integrating around said boundary and using $(0)$ to perform cancelation we deduce that said integral and vanishes. Now by the argument principle integrating the elliptic function $\frac f{f^\prime}$ around a period parallelogram is just a nonzero constant times the difference between the difference of the zeros and poles in the parallelogram accounting for order. This means that the sum of the orders of the zeros of $f$ in $H$ is equal to that of the poles. This value can be proved to be invariant of choice of $z_0$ and is called the order of an elliptic function (with respect to $(\omega,\omega^\prime)$). Furthermore a short argument shows that the order of $f$ is equal to the sum of the orders of any irreducible set of zeros or poles respectively.
When considering the order of an elliptic function, we usually mean with respects to its fundamental periods (if noncostant), but sometimes (as you will see below) it is useful to consider its order with respect to other periods. The order of an elliptic function is a very useful property to know, and whilst the results I give hence do not heavily use the concept, their proofs invariably involve reference to it at some stage in their argument. It is for this reason I introduce the concept.
Elliptic Functions Defined By their Zeros & Poles
Suppose that $\omega,\omega^\prime$ are $\mathbb C_\mathbb R$-linearly independent and $f,g$ are two elliptic functions with said periods. Suppose further that $f$ and $g$ have the same zeros and poles and of identical order. Then there is some constant $C \in \mathbb C^*$ such that $f = C \cdot g$
Whilst proofs aren't the point of this answer I will provide a proof of the above as it is quick, satisfying and illustrative of how to work with elliptic functions in practise.
Suppose that $f$,$g$ and $\omega$, $\omega^\prime$ are as given. If $g$ is identically zero then so is $f$ and there is nothing to prove. Otherwise the meromrohpic function given by $h := \frac fg$ is well defined and has periods $\omega$, $\omega^\prime$. Note that it has by construction no poles and is thus entire. Let $H$ be any $(\omega,\omega^\prime)$-parallelogram. Note that $H$ is compact. Furthermore, by periodicity argument we see that for any $x \in \mathbb C$ there is a $y \in H$ such that $h(x) = h(y)$. Thus $h[\mathbb C] = h[H]$. Now, by compactness $h$ is bounded on $H$. however, as $h[\mathbb C] = h[H]$ so $h$ is a bounded entre function thus by the louisville theorem is identically some constant $C$. pick some $x \in \mathbb C$ that is neither a zero or pole of $f$ and $g$ then $C = \frac {f(x)}{g(x)} \neq 0$ Remark we have shown that $f = C \cdot g$ identically as required.
Now we have the background stuffs out of the way, lets get to showing $R$ exists. We'll start with the $\wp$ function.
The Weierstrass $\wp$ Function
Suppose that $(\omega,\omega^\prime)$ are linearly independent in $\mathbb C_\mathbb R$. Then $\wp_{\omega,\omega^\prime}$ is an elliptic function that may be defined via an explicit summation (which is beyond the scope of this discussion but see Wikipedia). It satisfies the following:
- $\wp_{\omega,\omega^\prime}$is an order two elliptic function.
- $\wp_{\omega,\omega^\prime}$ has fundamental periods $\omega$, $\omega^\prime$.
- The set of poles of $\wp_{\omega,\omega^\prime}$ is $\Lambda(\omega,\omega^\prime)$ and are all of order $2$.
- $\wp_{\omega,\omega^\prime}$ is even.
It is common knowledge that $\wp$ is uniquely defined by satisfying a certain differential equation, but this is not relevant here. The following result is what allows us to produce the $R$ that we speak.
$\wp$-Decomposition of Even Elliptic Functions
Suppose that $f: \mathbb C \to \mathbb C$ is an even elliptic function of degree $2n$ for some $n \in \mathbb N_1$ with linearly independent periods $\omega$, $\omega^\prime$. Then, there exists two finite sets $Z$, $P$ and functions $\mathfrak z: Z \to \mathbb C \setminus \Lambda(\omega,\omega^\prime)$ and $\mathfrak p: P \to \mathbb C \setminus \Lambda(\omega,\omega^\prime)$ such that $(1)$ holds for some rational function $R(Z) \in \mathbb C(Z)$ defined via $(2)$ for $C \in \mathbb C^*$. Furthermore $R$ has degree $n$.
$$\begin{align}
f(z) &= R(\wp_{\omega,\omega^\prime}(z)) \tag{1}\\
R(Z) &= C \cdot
\frac{\prod_{i \in Z} Z - \wp_{\omega,\omega^\prime}(\mathfrak z_i)}{\prod_{i \in P} Z - \wp_{\omega,\omega^\prime}(\mathfrak p_i)} \tag{2}
\end{align}
$$
The proof of the above result is not hard, but does require a little bit of faff to dot the i's and cross the t's (which Whatson and Whittaker handwaves for the most part admittedly). The methodology reduces to choosing $\mathfrak z$ and $\mathfrak p$ such that $R(\wp_{\omega,\omega^\prime}(z))$ has the same zero's and poles as $f(z)$.This in turn reduces down to finding the zeros and poles of $f$ (upto period and sign).Throwing all the zeros and poles together to produce $R$ does however require a few technical considerations and special working to be done when $f$ has either a zero or pole at a period or half period. It goes without saying that the proof of the above relies on knowing the basic results about $\wp_{\omega.\omega^\prime}$ (namely those I have given in the basic facts about $\wp$ section).
Given then the preliminary results outlined, we can prove the existence of the required $R$. Suppose then that $M = \left[\begin{matrix}a &b\\c &d\end{matrix}\right] \in \text{Gl}(2,\mathbb Z)$ has $\det M := N \neq 0$ and further suppose that $(\omega,\omega^\prime)$ and $(\Omega,\Omega^\prime)$ are two pairs of $\mathbb C_ \mathbb R$-linearly independent complex numbers such that $(3)$ holds.
$$
M\left[\begin{matrix}\omega
\\\omega^\prime\end{matrix}\right] = \left[\begin{matrix}\Omega
\\\Omega^\prime\end{matrix}\right]. \tag{3}
$$
Without loss of generality assume that $a \neq 0$. Otherwise left-multiply $(3)$ by $\left[\begin{matrix}0 & -1 \\ 1 & 0 \end{matrix}\right]$ and try again. It follows from a matrix-manipulation argument that $\Lambda(\Omega,\Omega^\prime) = \Lambda(n\omega +k \omega^\prime,n^\prime\omega^\prime)$ for some $n,n^\prime, k \in \mathbb N$ such that $n \cdot n^\prime = N$. Thus $\Lambda(\Omega,\Omega^\prime)$ is generated by $n\omega + k \omega^\prime$ and $n^\prime\omega^\prime$. As $\wp_{\omega,\omega^\prime}$ has periods $n\omega + k\omega^\prime$ and $n^\prime\omega^\prime$ it is immediate that $\wp_{\omega,\omega^\prime}$ is a $(\Omega,\Omega^\prime)$
-periodic function. Furthermore we may also deduce that it is of $(\Omega,\Omega^\prime)$-order $N$ as the set $\left\{x \in \mathbb C \ \big |\ \exists m \in \{0,\ldots, n- 1\} \ \exists m^\prime \in \{0, \ldots, n^\prime - 1\}: x = m \omega + (m^\prime - mk)\omega^\prime\right\}$ is an $(\Omega,\Omega)$-irreducible set of poles for $\wp_{\omega,\omega^\prime}$.
By $(1)$ we deduce that there is some rational function $R \in \mathbb C(Z)$ such that $R(\wp_{\omega,\omega^\prime}(z)) = \wp_{\Omega,\Omega^\prime}(z)$, which was what was to be proved.
Now, depending on how particular you are being about an explicit definition for $R$, you still may not be satisfied as one needs to explicitly find the values $\mathfrak z_i$ and $p_i$. As alluded to prior this amounts to finding the zeros of $\wp_{\Omega,\Omega^\prime}$. However in the general case there is no nice form for such zeros, so using the methodology I have outlines one will not be able to provide a better formula, (although I did once see a paper that gave a very complicated integral for computing them). Additionally, I'm not quite sure about how to express the Inverse of $R$ in terms of radicles (which the paper assets is possible). That is something I myself am interested in hearing an answer to.
One pitfall I should make you aware of is that some authors (myself included) use $\omega,\omega^\prime$ as the fundamental periods for $\wp_{\omega,\omega^\prime}$, whereas others (your paper and my source) use them as half periods. To convert from my notation to theirs one will need to either divide by two or multiply by two in certain places. For example I would define $e_i := \wp_{\omega_1,\omega_3}(\frac12 \omega_i)$ for $i \in \{1,2,3\}$ whereas your source defines it as $\wp_{\omega_1,\omega_3}(\omega_i)$. Aside from this terminology between authors is pretty much the same and pretty much uses that set out in Whittaker and Watson.
It was always the intention to provide a brief outline to a subject in my answer, and I am aware that much of what I have said lacks sufficient justification. I hope I have written enough detail that the ideas are laid out with some coherence.
Whilst I know the basics of elliptic functions, I should add that my study was particularly focused on proving the basic results with a lot of rigour, thus there is every possibility that there is an easier method or idea that will give the required $R$, but it is not known to myself. This answer is made community because it is only a partial answer and there is much room for more commentary to be had, I hope that making this post a wiki-thread will encourage more experienced readers to add their knowledge.
