a finite number of holomorphic maps onto a Riemann surface with genus>1?

Question

I have a question on Riemann surface after reading a survey paper 'The proof of the Mordell conjecture' by Spencer Bloch.

Question 1: Is it true that there is only a finite number of holomorphic maps onto a Riemann surface with genus$>1$ ?

The holomorphic maps the article mentioned above concerned with is in fact more special, Fix two Riemann surfaces $M$ and $N$, both with genus$\geq 2$. then consider the holomorphic map $f: M \rightarrow N$ branched covering over only a single (but not fixed) point on $N$, and the degree of the map also fixed.

If I understand correctly, Parshin's argument of the Shafarevich conjecture (on curves defined over number fields) implying the Mordell conjecture relies on this fact.

I am OK to see the special holomorphic maps mentioned in the above are of finite number, if we fix the branch point on $N$. It is also reasonable to see if we replace $N$ by $\mathbb{P}^1$ then they are infinitely many such maps (one can use Auto($P^1$) to move such a map.). But in general, I am stuck. Thanks for help!

Answer 1

There are two theorems along these lines. One is due to de Franchis:

Given any pair of compact Riemann surfaces $X, Y$ where genus of $Y$ is $\ge 2$, there are only finitely many nonconstant holomorphic maps $X\to Y$.

The second, harder, theorem, is due to Severi:

Given a compact Riemann surface $X$, there are only finitely many (up to an isomorphism) pairs $(f, Y)$, where $Y$ is a Riemann surface of genus $\ge 2$ and $f: X\to Y$ is a nonconstant holomorphic map.

See my answer here for a proof of the de Franchis theorem and references.