I'm trying to prove the theorem:
Theorem. Let $\{z_j\}$ and $\{\zeta_k\}$ be finite sets of distinct points in $M := \mathbb{C}/\langle z \mapsto z + \omega_1, z \mapsto z + \omega_2\rangle$, $n_j$ and $\nu_k$ be positive integers, and $\Lambda$ be the group relative to which we are factoring. Suppose $$ \sum_j n_j = \sum_k \nu_k $$ and $$ \sum_j n_j z_j - \sum_k \nu_k \zeta_k = 0 \qquad \mod{\Lambda}. $$ Then there exists an elliptic function which has zeros and poles at $\{z_j\}$ and $\{\zeta_k\}$ with the given orders.
Attempt. Listing the points $z_j$ and $\zeta_k$ with their respective multiplicities, we obtain sequences $z_j'$ and $\zeta_k'$ of the same length, say $n$. Shifting by a lattice element if needed, we have $$ \sum_{j = 1}^n z_j' = \sum_{k = 1}^n \zeta_k'. $$ Define, for $\sigma(z) := z \prod_{\omega \in \Lambda^*} E_2 (z/\omega)$, $$ f(z) = \prod_{j = 1}^n \frac{\sigma(z - z_j')}{\sigma(z - \zeta_j')}. $$ This has the desired zeros and poles, but I don't know how to show periodicity.