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+\end{align*} +% +Note that if $G_P = \ZZ/p^n$, this coincides with the standard definition of +the $t$th upper (resp. lower) ramification jump of $X \to Y$ at $P$. % \begin{Theorem} Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k) \rangle = \ZZ/p^m = G_{P_0}$ for $P_0 \in X(k)$. Then, as $k[\ZZ/p^n]$-modules: @@ -158,6 +165,21 @@ In the inductive step we use also the group $\ZZ/p^{n-1}$. In this case we denote the irreducible $k[\ZZ/p^{n-1}]$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$ and $\mc T^i M := T^i_{\ZZ/p^{n-1}} M$ for any $k[\ZZ/p^{n-1}]$-module $M$. +Note also that for $j \ge 1$: +% +\[ + l_{X/Y, P}^{(j)} - l_{X/Y, P}^{(j-1)} = \frac{1}{p^{j-1}} (u_{X/Y, P}^{(j)} - u^{(j-1)}_{X/Y, P}) +\] +% +(in particular, $u_{X/Y, P}^{(1)} = l_{X/Y, P}^{(1)}$). Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ then: +% +\begin{itemize} + \item $u_{X'/Y, P}^{(t)} = u_{X'/Y, P}^{(t)}$ for $t \le N$, + + \item $l_{X/X', P}^{(t)} = l_{X/X', P}^{(t + N)}$ for $t \le n-N$. +\end{itemize} + + \begin{Lemma} If the $G$-cover $X \to Y$ is \'{e}tale, then the natural map % @@ -192,10 +214,20 @@ and $\mc T^i M := T^i_{\ZZ/p^{n-1}} M$ for any $k[\ZZ/p^{n-1}]$-module $M$. By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules: % \[ - M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n-m ??} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2 + M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2 \oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}} \] % + where + % + \[ + m' := + \begin{cases} + n-1, & \textrm{ if } m = n,\\ + n, & \textrm{ otherwise.} + \end{cases} + \] + Therefore, for $???$ % \begin{align*}