# Construct a meromorphic function on Riemann surface of genus $3$

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?

a) Consider the divisor $$D=-2\cdot p+N\cdot q$$ on $$X$$. We have $$L(D)=\Gamma (X,\mathcal O(D))\neq 0$$ as soon as $$N\geq 5$$ (by Riemann-Roch). Choose such an $$N$$ and then choose a non-zero $$f\in L(D)$$.
That meromorphic function $$f$$ satisfies $$\operatorname {div}f-2\cdot p +N\cdot q\geq 0$$ and thus has a pole of order at most $$N$$ at $$q$$ and is holomorphic everywhere else with a a zero of order $$\geq2$$ at $$p$$, just as required.

b) Take any smooth plane projective quartic $$X\subset \mathbb P^2$$ (for example $$x^4+y^4+z^4=0$$): it has genus $$3$$.
The curve $$X$$ always has an inflexion point $$q\in X$$ and the tangent line $$T_q(X)\subset \mathbb P^2$$ cuts $$X$$ in another point $$p\in X$$.
Since our tangent line $$T_q(X)$$ cuts $$C$$ in a canonical divisor (like any line in the plane!), we see that $$3\cdot q+1\cdot p$$ is a canonical divisor and that divisor is the divisor of a holomorphic differential form $$\omega$$ vanishing at $$p$$ and having a zero of order $$3$$ at $$q$$, just as required.

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

https://minimal.sitehost.iu.edu/research/klein.pdf#page=27