Question that seems pretty easy, but I can't formalize it:
Let $L \subset C$ be a lattice, and $f(z)$ be an elliptic function for $L$, that is a meromorphic function so that $f(z+w) = f(z)$ for all $\omega \in L$. Assume that $f$ is analytic except for double poles at each point of the lattice $L$. Show that $f = a\wp + b$ for some constants $a,b$.
What I tried: $\displaystyle f(z) = \prod_{\omega \in L} {\frac{g(z)}{(z-\omega)^2}}$ , $g(z)$ is analytic and therefore constant in the fundamental domain. Now what is left to do, is to take the product apart to partial fractions, and then I get almost what needed, except it's not one constant $a$ and $b = \sum_{\omega \in L} -\frac1{\omega^2}$.
Am I right? How do I proceed?
Thanks in advance.