Given two nonzero complex numbers $\omega_1, \omega_2$, with nonreal ratio, we define the period module $$M= \omega_1 \mathbb Z+ \omega_2 \mathbb Z= \{n_1 \omega_1+ n_2 \omega_2:n_1,n_2 \in \mathbb Z \} $$ and the Weierstrass $\wp$ function $$\wp(z; \omega_1, \omega_2) \equiv \wp(z;M)= \frac{1}{z^2}+ \sum_{\omega \in M \setminus \{0 \}} \frac{1}{(z- \omega)^2}-\frac{1}{\omega^2} .$$
I want to solve the following exercise from Ahlfors' complex analysis text (page 274):
Show that any even elliptic function with periods $\omega_1$, $\omega_2$ can be expressed in the form $$C \prod_{k=1}^n \frac{\wp(z)-\wp(a_k)}{\wp(z)-\wp(b_k)} \text{ ($C$=const.)} $$ provided that $0$ is neither a zero nor a pole. What is the corresponding form if the function either vanishes or becomes infinite at the origin?
My attempt:
Let $f$ be an even elliptic function with periods $\omega_1,\omega_2$, and suppose for the moment that $f$ has neither a zero nor a pole at the origin. If $f$ is constant, we have an empty product representation $$f(z)= C \prod_{k=1}^0 \left( \dots \right). $$ Suppose now that $f$ isn't constant. As an elliptic function $f$ has equal number of (congruent) zeros and poles. Denote its zeros by $a_1, \dots,a_n$ and its poles by $b_1, \dots ,b_n$ (multiple points being repeated), and define $$g(z)=f(z) \bigg/ \prod_{k=1}^n \frac{\wp(z)-\wp(a_k)}{\wp(z)-\wp(b_k)} $$ What I want to say is that any numerator $$\wp(z)-\wp(a_k) $$ has a simple zero at $a_k$, and any denominator $$\frac{1}{\wp(z)-\wp(b_k)}$$ has a simple pole at $b_k$. If that's true then $g$ is a holomorphic elliptic function, which reduces to a constant $C$.
If $f$ has a zero of order $2m$ at the origin we repeat the proof for $\tilde{f}= \wp^m f$ and we obtain the representation $$f(z)=C \wp(z)^{-m} \prod_{k=1}^n \frac{\wp(z)-\wp(a_k)}{\wp(z)-\wp(b_k)} \text{ ($C$=const.)} $$
If $f$ has a pole of order $2m$ at the origin we repeat the proof for $\tilde{f}= \wp^{-m} f$ and we obtain the representation $$f(z)=C \wp(z)^{+m} \prod_{k=1}^n \frac{\wp(z)-\wp(a_k)}{\wp(z)-\wp(b_k)} \text{ ($C$=const.)} .$$
My question: Why are all values of $\wp$ (except $\infty$) taken "simply" (that is with non-vanishing derivative at the point)?
I tried considering the "fundamental parallelogram" with vertices at $a,a+\omega_1,a+\omega_2,a+\omega_1+\omega_2 $ where $a=-\frac{1}{2} \omega_1-\frac{1}{2} \omega_2$, and WLOG the uppermost and rightmost edges are included. It is known that in this parallelogram all complex values are taken twice. Since $\wp$ is even, if $a$ is an interior point, the value $\wp(a)$ is taken at least twice, at the points $\pm a$.
However, if $a$ lies on the part of the boundary of the parallelogram which is included, I can't use the evenness argument, and as far as I can tell $a$ might be a double value of $\wp$ (?)
Is my solution correct so far? and can you please help me with the question in boldface?
Thanks!