# Does there exist a complex analytic map between compact Riemann surfaces with certain conditions?

I am wondering if there exists a complex analytic map $\pi:X\rightarrow Y$ with $g(Y)=g(X)-1.$ By Riemann-Hurwitz formula, I think the simplest of such maps is a complex analytic map between compact Riemann surfaces of degree 1 with one branch point of index $3,$ or with two branch points of indices $2, 2.$ So I am wondering

Does there exist a complex analytic map of compact Riemann surfaces of degree $1$ with one branch point of index $3,$ or with two branch points of indices $2, 2?$

The closest example I found is the map $z\rightarrow z^2$ on the Riemann sphere, which has two branch points, both of index $2,$ but it is of degree $2,$ not $1.$

More generally, out of curiosity, does there exist a complex analytic map of compact Riemann surfaces of given degree with a given number of branch points, of given indices?
Thanks for any help in advance.

There certainly exist compact Riemann surfaces $$X$$ and $$Y$$ with $$g(Y)=g(X)-1$$ for which there exists a non-constant holomorphic mapping $$X\rightarrow Y$$. Here, $$g(-)$$ denotes the genus of a surface. As an example, consider any lattice $$\Gamma\subsetneq \mathbb{C}$$. Then let $$X\colon=\mathbb{C}/\Gamma$$ be the corresponding complex torus and let $$Y\colon=\mathbb{P}^1$$ be the Riemann sphere. Then, the Weierstrass $$\wp$$-function $$\wp(z,\Gamma)=\frac{1}{z^2}+\sum_{\omega\in\Gamma\setminus\{0\}}\Big(\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\Big)$$ defines a meromorphic doubly-periodic (with respect to the lattice $$\Gamma$$) function on $$\mathbb{C}$$. Thus, it extends to a non-constant holomorphic mapping $$X\rightarrow Y$$.
1. Any non-constant holomorphic mapping of degree $$1$$ (between compact Riemann surfaces) is a biholomorphism, thus in particular a homeomorphism. Homeomorphisms preserve the genus. Hence, there cannot exist a holomorphic mapping $$X\rightarrow Y$$ of degree $$1$$ if the genera of $$X$$ and $$Y$$ differ.
2. Additionally, there does not exist a holomorphic mapping $$F\colon X\rightarrow Y$$ of degree $$1$$ (between compact Riemann surfaces $$X$$ and $$Y$$) which has a ramification point (equivalently a branch point), say $$x\in X$$, since in this case we would have $$1=\operatorname{deg}(F)=\sum_{p\in F^{-1}(\{F(x)\})}\operatorname{mult}_p(F)\geq 2$$.