change of the residue of meromorphic differential by a covering map.

Question

Let $C_1$ and $C_2$ denote complex algebraic curves, and let $f : C_1 \rightarrow C_2$ be a non-constant map. Let $x$ be a point on $C_1$ and its ramification degree is $e_x$.

I want to compute the residue of the meromorphic differential $f^*(\omega)$ at $x$, where $\omega$ is a meromorphic differential on $C_2$ and $f^*$ is the pullback. I think the following might be true, but I cannot find a reference or a proof. $$ Res_x f^*(\omega) =^? e_x Res_{f(x)} \omega. $$ Does anyone know whether this is true or not? If it is true, then is there any reference or a proof?

Answer 1

Another way to see this: if $\gamma$ is a small loop in $C_1$ winding once around $x$, then $f(\gamma)$ is a small loop in $C_2$ winding $e_x$ times around $f(x)$, so $$ \mathrm{Res}_x f^*\omega=\frac{1}{2\pi i}\int_\gamma f^*\omega=\frac{1}{2\pi i}\int_{f(\gamma)}\omega=\frac{1}{2\pi i} \left(e_x\cdot 2\pi i\cdot\mathrm{Res}_{f(x)}\omega\right)=e_x \mathrm{Res}_{f(x)}\omega. $$

Answer 2

Do the calculations locally. Let $z$ be the co-ordinate at $x$ and $w$ at $f(x)$. Then $\omega=(\cdots+ a w^{-1}+\mathrm{hol})dw$ and the map looks like $w\mapsto z^{e_x}$ (by changing units etc.) and the residue is just $a$. Then $f^*(\omega)= (\cdots+az^{-e_x}+\mathrm{hol}) d(z^{e_x})=(\cdots+ae_xz^{-1}+\mathrm{hol})dz$.