the exercise 2.10(a) in silverman AEC says $(\phi_*f)(D)=f(\phi^*D)$.(any $f\in \bar K \ (C_1),D\in Div(C_2) ) ( \phi:C_1 \rightarrow C_2 is \ a \ nonconstant \ map \ of\ smooth\ curve $ )
(Here we need the support of divisor D and the support of div(f) disjoint)
I want to prove the simple case. When $D$ is $Q(Q \;\text{is a point in} \ C_2$).
Then we need prove
$$N_{K(C_1)/\phi^* K(C_2))}(f)(\phi^{-1}(Q))=\prod_{P\in\phi^{-1}(Q)}f(P)^{e_{\phi}(P)}$$
For $$N_{K(C_1)/\phi^* K(C_2))}(f)=\prod_{\sigma\in Gal({K(C_1)/\phi^* K(C_2))}}\sigma(f)$$,
then i want to prove $\sigma(f)(P)=f(P_1) (some\ P_1\ is \ in \ \phi^{-1}(Q))$
and it is true for all Q in $C_2$.
Then i guess $Gal({K(C_1)/\phi^* K(C_2)})$ may have action on $\phi^{-1}(Q)$,
$Gal({K(C_1)/\phi^* K(C_2)}) \times\phi^{-1}(Q)\rightarrow\phi^{-1} Q)$,$(\sigma,P)\rightarrow \sigma (P)$
which keeps $\sigma(f)(P)=f(\sigma (P)) $.
Then we can prove $N_{K(C_1)/\phi^* K(C_2))}(f)(\phi^{-1}(Q))=\prod_{P\in\phi^{-1}(Q)}f(P)^{e_{\phi}(P)}$.
But i don't know whether it is true. Thank you very much for any help.