Definition (Ramification index). Suppose $P\in C_1$, $Q\in C_2$, $\varphi :P\mapsto Q$. Let $t\in K(C_2)$ be a uniformiser at $Q$.

The ramification index of $\varphi$ at $P$ is

$$e_{\varphi }(P)={\mathrm{ord}}_P({\varphi }^{\ast }t)$$

(always $\ge 1$, independent of choice of $t$).