Meromorphic function on projective plane and complex torus

Question

I'm attempting to do the following questions from Miranda's Algebraic Curves and Riemann Surfaces.

Question II.2.A - Consider the projective line $\mathbb{P}^1$. Fix a point $p \in \mathbb{P}^1$, and a finite set $S \subset \mathbb{P}^1$ with $p \not \in S$. Show that there exists a meromorphic function $f$ on $\mathbb{P}^1$ with a simple at $p$ and no zeroes or poles at any of the points of $S$.

Question II.2.H - Consider the complex torus $X = \mathbb{C}/L$. Fix a point $p \in X$, and a finite set $S \subset X$ with $p \not \in S$. Show that there exists a meromorphic function $f$ on $X$ with a simple zero at $p$ and no zeroes or poles at any of the points of $S$.

I've been able to do the other questions, but am unsure of how to prove that there actually exists meromorphic functions with these properties.

Answer 1

Let $p=(p_{0}:p_{1})$. We consider $a=(a_0:a_1)\in\mathbb{P}^1-(S\cup\{p\})$. Then, $$ f=\frac{p_1 X_0-p_0 X_1}{a_1 X_0-a_0 X_1} $$ is a rational function on $\mathbb{P}^1$ with a simple zero at $p$ and no zeros or poles at any of the points of $S$.

I guess that the same idea can be applied for the torus with theta functions.