Lemma. Let \(C\) be a nice curve of genus \(g\ge 2\) over a number field \(k\). Then there is a finite extension \(k'/k\) and a finite set \(S'\) of places of \(k'\) satisfing the following. For every \(x\in C(k)\) there is a nice curve \(W_k\) over \(k'\) with good reduction outside \(S'\), and a morphism \(\varphi_x:W_x\to C_{k'}\) ramified exactly at \(x\) (with ramification index \(\le 2\)) s.t. \(\mathrm{deg}\,\varphi_x\le 2\cdot 4^g\)
Proof) Assume \(C(k)\ne \varnothing\), and let \(j:C\to J=\mathrm{Jac}\,C\) be the embedding induced by some fixed \(x\in C(k)\). The map "multiplication by two" is an etale self-covering of \(J\); we let \(\tilde{C}\) be its pullback:
$$ \begin{aligned} &\tilde{C}\longrightarrow J \\ & \downarrow \!{}^{\varphi} \,\,\,\,\,\,\, \downarrow \!{}^{2} \\ & C\overset{j}{\longrightarrow} J\end{aligned}$$
From the Chevalley-Weil theorem, we obtain a finite extension \(L/k\) s.t. \(\varphi^{-1}(x)\subseteq \tilde{C}(L)\) for all \(x\in C(k)\). For any \(x\in C(k)\), choose distinct \(x_1,x_2\in \tilde{C}(L)\) s.t. \(\varphi(x_i)=x\). There exists a divisor \(D\in \mathrm{Div}(\tilde{C})\) defined over some finite extension \(k'/k\) s.t. \(x_1-x_2+2D=(f)\) in \(\mathrm{Jac}\,\tilde{C}\) for some rational function \(f\). (which does not depend on \(x\)). Let \(\varphi_x:W_x\to \tilde{C}_x\) correspond to the inclusion \(k'(\tilde{C})\hookrightarrow k'(\tilde{C})[\sqrt{f}]\) of function fields. It's not to hard to show that \(W_x\) has the desired properties.
그냥 어떤 책 증명 글자 하나 안 바꾸고 배꼈어요 헤헿<< 아무래도 \(4^g\)가 튀어나오는 이유는 저 etale covering이라는 애의 degree때문에 그런 것 같고, \(x\)에서만 ramified인 이유는 etale때문에 그렇고 주저리주저리...