Let $\Lambda$ be a lattice in the complex plane. And Weierstrass Elliptic Function
$$\wp(z)=\frac{1}{z^2}+\sum_{\omega \in \Lambda - \{0\}}\frac{1}{(z-\omega)^2}-\frac{1}{\omega ^2}$$
How can I show that every $\Lambda $-periodic meromorphism function $f$ with poles of order 2 only at the lattice points can be written as
$$f = a \wp +b$$
where $a,b\in \mathbb C$ and $a$ is non-zero.
Let $$ \frac{a_{-2}}{z^2} + \frac {a_{-1}}z $$ be the principal part of $f$ at $z = 0$. For an elliptic function, the sum of all residues at the poles in a fundamental parallelogram is zero, therefore $$ \DeclareMathOperator*{\res}{Res} a_{-1} = \res (f; 0) = 0 \, . $$ It follows that $$ f(z) - a_{-2} \wp(z) $$ is an elliptic function without poles and therefore constant.