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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?

and

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 $q$.


My attempt:

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?

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Take the Riemann surface $X$ as the Klein's surface, i.e. the Riemann surface associated to the algebraic function $$w^7=z(z-1)^2$$ Applying Riemann-Hurwitz formula, we have the genus of $X$ is $3$.

To construct the holomorphic $1$-form, you can see the section 7.2 of

http://www.indiana.edu/~minimal/research/klein.pdf

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