I am working on the following questions:
Let $X$ be a compact Riemann surface of genus $3$ with two points $p\neq q$.
Find a non-constant meromorphic function on $X$ with at least a double zero at $p$ and holomorphic everywhere except possibly at $q$.
What is the smallest possible pole under at $q$ we need to accept in order to guarantee the existence of such a function?
Construct an example of a Riemann surface of genus $3$ that has a holomorphic $1$-form with a zero of order $1$ at a point $p$, and zero of order $3$ at a different point $q$.
The second part of the first problem is easy. I think it is just some simple application of Riemann-Roch theorem.
But I am desperate to construct something on Riemann surface. Could anyone show me how to do this?