Let $f$ be a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$. Let $P$ be a fundamental parallelogram whose boundary contains no zeros or poles of $f$. Show that the number of zeros of $f$ in $P$ is the same as the number of poles of $f$ in $P$, both counted with multiplicities.

Suppose additionally that $f$ is even. Show that there exists a rational function $Q(z)$ such that $f=Q(\wp)$, where $\wp$ is the Weierstrass $\wp$-function.

Suppose $f$ is a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$, and $F$ is a meromorphic antiderivative of $f$, so that $F^{\prime}=f$. Is it necessarily true that $F$ is an elliptic function? Justify your answer.

[You may use standard properties of the Weierstrass $\wp$-function throughout.]