Show that the sum of zeros and poles of elliptic function is lattice point

Question

I try to solve the following problem:

Let $f$ be an elliptic function with respect to a lattice $\Lambda$. Let $z_1,\cdot,s z_2$ be the zeroes and poles of $f$ inside a fundamental parallelogram $\Pi$, of degree $d_1,\cdots,d_n$. Show that $$\sum_{k=1}^{n}d_kz_k\in \Lambda$$

(Hint: considering the integral $$\int\limits_{\partial ~\Pi} z\cdot\frac{f'(z)}{f(z)}~\mathrm{d}z$$

I have no clue to prove this with the hint. How to use the hint?

Also, the sum seems like the divisor of $f$. So is this result related to the theory of Riemann surface and even algebraic geometry? Thanks a lot!

Answer 1

HINT: Recall that $f$ is doubly periodic, so $f(z+\lambda) = f(z)$ for any $\lambda\in\Lambda$. Think about $\Pi$ and how you might parametrize $\partial\Pi$.