I try to do problem from chapter 2 number 2 from [Silverman] "The Arithmetic of elliptic curves" .
(http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf exercise 2.2)
Let $\phi : C_1 \rightarrow C_2$ be a nonconstant map of smooth curves, let $f \in \overline{K}(C_2)^*$, and $P \in C_1$. Prove that $$\mbox{ord}_P(\phi^*f) = e_\phi(P)\ \mbox{ord}_{\phi(P)}(f)$$ where $e_\phi(P) := \ \mbox{ord}_P(\phi^*t_{\phi(P)}), t_{\phi(P)}$ is a uniformizer at $\phi(P).$
Attempt : It perhaps about unwrap the definitions, but I cannot see how the extra $e_\phi(P)$ comes from.
Let $f \in \overline{K}(C_2)^* = \overline{K}(C_2)\setminus\{0\}.$ Let $m = \ \mbox{ord}_P(\phi^*f) = \ \mbox{ord}_P(f \circ \phi)$ and $n = \ \mbox{ord}_{\phi(P)}(f)$.
Then by definition, $$f \circ \phi \in M^m_P \rightarrow f\circ \phi = \sum_{i=1}^k h_{i1}h_{i2}...h_{im}$$ and $$f \in M^n_{\phi(P)} \rightarrow f = \sum_{i=1}^l g_{i1}g_{i2}...g_{in}$$ where $h_{ij}(P) = 0 = g_{st}(\phi(P))$ for all $i, j, s, t.$
It does not clear where from definition, $e_\phi(P)$ should factor out.
Guess I might instead try to look at $g := f^{e_\phi(P)}$. So I now want $\mbox{ord}_P(\phi^*f) = \ \mbox{ord}_{\phi(P)}(g)$ (compute two function under ord).
Still why they are equal ? Perhaps more tools on ord will help, but [Silverman] only provide definitions, so it still not so clear.
You can write $f=gt_{\phi(P)}^{\alpha}$ for some $g \in \overline{K}(C_2)$ defined at $\phi(P)$ and equal to a nonzero scalar $\lambda$ that point and $\alpha$ is an integer.
Then $\phi^*g$ is a function on $C_1$ defined at $P$ and its value at $P$ is $\lambda \neq 0$, so it has vanishing order zero at $P$. Now, the vanishing order of $\phi^*t_{\phi(P)}^{\alpha}$ is by definition $e_{\phi}(P)\alpha$, and thus the vanishing order of $\phi^*f=\phi^*g \phi^*t_{\phi(P)}^{\alpha}$ at $P$ is $e_{\phi}(P)\alpha$, and $\alpha$ is the vanishing order of $f$ at $\phi(P)$, QED.