Definition: Let $f$ be an elliptic function on $\mathbb{C}$ and $\Omega \subset \mathbb{C}$ be the periodic lattice of $f$. The divisior of $f$ is defined as $$(f)=\sum_j n_jP_j$$ where $P_j$ are the zeros/poles of $f$ on $T=\mathbb{C}/\Omega$ with mutiplicities $n_j$.
The degree of the divisor is defined as $$deg(f)=\sum_j n_j$$
Theorem: (Abel) There exists some elliptic function $f$ on $\mathbb{C}/\Omega$ with divisor $$\sum_j n_jP_j$$ iff
$(1) \sum_j n_j=0$
$(2) \sum_j n_jP_j \in \Omega$
I can't quite understand the statement $(1)$ above because doesn't it imply that $n_j=0 \forall j$?