Let $f$ be a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$. Let $P \subset \mathbb{C}$ be a fundamental parallelogram and let the degree of $f$ be $n$. Let $a_{1}, \ldots, a_{n}$ denote the zeros of $f$ in $P$, and let $b_{1}, \ldots, b_{n}$ denote the poles (both with possible repeats). By considering the integral (if required, also slightly perturbing $P$ )

$$\frac{1}{2 \pi i} \int_{\partial P} z \frac{f^{\prime}(z)}{f(z)} d z$$

show that

$$\sum_{j=1}^{n} a_{j}-\sum_{j=1}^{n} b_{j} \in \Lambda$$

Let $\wp(z)$ denote the Weierstrass $\wp$-function with respect to $\Lambda$. For $v, w \notin \Lambda$ with $\wp(v) \neq \wp(w)$ we set

$$f(z)=\operatorname{det}\left(\begin{array}{ccc} 1 & 1 & 1 \\ \wp(z) & \wp(v) & \wp(w) \\ \wp^{\prime}(z) & \wp^{\prime}(v) & \wp^{\prime}(w) \end{array}\right)$$

an elliptic function with periods $\Lambda$. Suppose $z \notin \Lambda, z-v \notin \Lambda$ and $z-w \notin \Lambda$. Prove that $f(z)=0$ if and only if $z+v+w \in \Lambda$. [You may use standard properties of the Weierstrass $\wp$-function provided they are clearly stated.]