Regarding property of odd elliptic functions

Question

While self studying analytic number theory from Tom M Apostol modular functions and Dirichlet series in number theory I am unable to think about an argument which Apostol doesn't proves but uses it in Theorem 1.14 of chapter - elliptic functions.

This same problem was also asked in lecture notes on elliptic functions which I was studying.

Half periods of odd elliptic functions are either zeroes or poles .

I have doubt in only the part when $\omega$ /2 is proved to not to be zero. How to prove such half period must be pole .

Can someone please help on how to prove it.

Answer 1

Let $L \subseteq \mathbb{C}$ be a lattice and let $f$ be an odd elliptic function with respect to $L$. Recall that this means that $f(z+\omega) = f(z)$ for all $z \in \mathbb{C}, \omega \in L$ such that $f$ does not have a pole at $z$.

Given any $\omega \in L$, there are precisely two possibilities:

(i) $\omega/2$ is a pole of $f$.

(ii) $\omega/2$ is not a pole of $f$. In this case, we may conclude that $$f(\omega/2) = f(\omega/2 - \omega) = f(- \omega/2) = - f(\omega/2),$$ where we used at the first '$=$' that ($\omega \in L$ and hence) $-\omega \in L$, and at the third '$=$' that $f$ is odd. This now implies that $2 f(\omega/2) = 0$ and hence that $f(\omega/2) = 0$.