# Number of holomorphic map form genus 2 to genus 1 of given a ramification of fixed type

I want to count all the holomorphic maps(with some property) form connected genus 2 surfaces to Torus. Let $\sum_2$ denote genus 2 surface and $T$ denote the torus. Let fix a point say $0$ on the torus. So I want all the $f: \sum_2\rightarrow T$ such that the divisor $f^{-1}(0)=3*p_1+p_2$ that is $f^{-1}$ have two points $p_1$ and $p_2$ with ramification data $3,1$ respectievly and no other ramification.

I read that I could count using monodromy that is counting $\rho:Hom(\pi_{1}(T- \{0\}),S_4)$ such that $\rho(\gamma_0)=[\alpha_1,\alpha_2]$ where $\gamma_0$ is a cycle of type (3,1) and $\alpha_1$ and $\alpha_2$ are elements of $S_4$. I was wondering $\alpha_i$ have to be of particular type ? Or else I don't know how to count. Please help.

Monodromy anyway imply that the number of above maps is equal to number of 3 cycles in $A_4$?

• Look up Riemann-Hurwitz formula. That will identify the possibilities of degree and branching. Once you have that you puncture, – Charlie Frohman Mar 6 '17 at 13:20
• And specify holonomy, and then look at subgroups of the fundamental group corresponding to those choices. – Charlie Frohman Mar 6 '17 at 13:21