Let $\wp$ denote the Weierstrass elliptic function and $\wp^{'}$ its derivative.
Now, consider a two dimensional lattice $\Lambda\subset\mathbb{C}$ and let $\Lambda_{2}=\lbrace z\in\mathbb{C}:2z\in\Lambda\rbrace$. Let $f$ have triple poles on $\Lambda$ and simple zeros on $\Lambda_{2}$\ $\Lambda$.
How can I show that $f(z)=c\wp(z)^{'}$ for some $c\in\mathbb{C}$? (where $\wp$ is the Weierstrass function with respect to $\Lambda$)
Thank you!
First show that $\wp'(z)$ has triples poles on $\Lambda$ and simple zeroes on $\Lambda_2 \setminus \Lambda$. Now consider the function $f(z)/\wp'(z)$; observe that it is holomorphic and bounded (the latter because it is periodic) and so it is constant.