relation between jumps

article_de_rham_cyclic.tex

@@ -133,8 +133,15 @@ hyperref, bbm, mathtools, mathrsfs}
 %
 \section{Cyclic covers}
 %
-Let $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) denote the $t$th upper (resp. lower)
-ramification jump of $X \to Y$ at $P$.
+Let for any $\ZZ/p^n$-cover $X \to Y$
+%
+\begin{align*}
+	u_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_P^{(t)} \cong \ZZ/p^{n-t???} \},\\
+	l_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_{P, t} \cong \ZZ/p^{n-t???} \}.
+\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*}