g_2 \in F^*$ such that for any point $P \in {{\mathscr{P}}}$, we have $m_P = 
(g_1, g_2)$ and $g_1$ defines $C$ at all $P \in {{\mathscr{P}}}\cap C$ and $g_2$ defines $E$ at all $P \in {{\mathscr{P}}}\cap E$. In particular, each $\alpha_i$ is unramified on $A_P$ except possibly at $(g_1)$ and $(g_2)$. Then, by (\[prop:ps2\]), $\alpha_i \otimes F_P(\sqrt[p^2]{h_1},
\cdots , \sqrt[p^2]{h_d}, \sqrt[p]{g_1},   \sqrt[p]{g_2}) = 0$ for all $i$.

Let $L = F(\sqrt[p]{f}, 
 \sqrt[p^2]{u_1}, \cdots , \sqrt[p^2]{u_{d}}, \sqrt[p]{u_{d+1}}, \sqrt[p^2]{h_1}, \cdots , 
 \sqrt[p^2]{h_d}, \sqrt[p]{g_1},  \sqrt[p]{g_2})$. We claim that $\alpha_i \otimes L = 0$ for all $i$. Let ${{{\mathscr{Y}}}}$ be a regular proper model of $L$ and $Y$ its special fiber. Let $y $ be a point of $Y$. If $y$ lies over a generic point of a component of the special fiber $X$ of ${{{\mathscr{X}}}}$, then by the choice of $f, u_1, \cdots , u_d$, $\alpha_i  \otimes L_y = 0$. Suppose the image of $y$ lies over a closed point $P$ of $X$. Suppose $P \in U_\eta$ for some $\eta$. Then $F_{U_\eta} \subset F_P$ and hence once again by the choice of $f, u_1, \cdots , u_d$, $\alpha_i \otimes L_y = 0$ for all $i$. Suppose $P \not\in U_\eta$ for all $\eta$. Then 