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following notation for any $P \in X(\ol k)$ at $P$, \item $m_{X/Y, P} := \ord_p(e_{X/Y, P})$ is the maximal power of~$p$ dividing the ramification index, + \item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump, \item $m_{X/Y} := \max \{ m_{X/Y, P} : P \in X(k) \}$. \end{itemize} % @@ -219,7 +220,8 @@ For any $k[H]$-module $M$ denote: T^i M &= T^i_H M := M^{(i)}/M^{(i-1)} \quad \textrm{ for } i = 1, \ldots, p^n. \end{align*} % -Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????). +Recall that $\dim_k T^i M$ determines the structure of $M$ completely (see \cite[p. 108]{Valentini_Madan_Automorphisms} -- they give the argument for $M := H^0(X, \Omega_X)$, +but it works for an arbitrary module). Moreover, for $i > 0$: % \begin{equation} \label{eqn:dim_of_Ti_Jl} @@ -239,29 +241,7 @@ where we use the Iverson bracket notation: In the inductive step we use also the group $H' := \ZZ/p^{n-1}$. In this case we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$ -and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\ - -\noindent Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that: -% -\begin{align*} - u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\ - l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}. -\end{align*} -% -Assume now that $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$. Let $P' \in X'(k)$ be the image of $P \in X(k)$. Then, if -$e_{X/Y, P} = p^n$, we have: -% -\begin{align*} - i_{X/X', P}^{(t)} &= - \begin{cases} - i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(N)} \cdot p^N, & t = 0\\ - p^N \cdot i_{X/Y, P}^{(N+t)}, & t = 1, \ldots, n-N-1. - \end{cases}\\ - i_{X'/Y, P'}^{(t)} &= i_{X/Y, P}^{(t)} \qquad \textrm{ for } t < N. -\end{align*} -If $e_{X/Y, P} \le p^{n - N}$ then $i_{X/Y, P}^{(t)} = i_{X/X', P}^{(t)}$ -for all $t$. - +and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$. % \begin{Lemma} \label{lem:G_invariants_\'{e}tale} If the $G$-cover $X \to Y$ is \'{e}tale, then @@ -406,7 +386,28 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \end{equation} % Assume now that $Q \in B_{Y'/Y}$. Then there exists a unique point $Q' \in Y'(k)$ - in the preimage of $Q$ through $Y' \to Y$. Moreover, $m_{X/Y, Q} = m_{X/Y', Q'}$. By using ????above formulas???: + in the preimage of $Q$ through $Y' \to Y$. Moreover, $m_{X/Y, Q} = n$, $m_{X/Y', Q'} = n-1$. + Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that for every $t \ge 0$: + % + \begin{align*} + u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\ + l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}. + \end{align*} + % + Observe that: + % + \begin{align*} + i_{X/X', P}^{(0)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p,\\ + i_{X/X', P}^{(t)} &= p \cdot (i_{X/Y, P}^{(t + 1)} + \ldots + i_{X/Y, P}^{(n-1)}) \quad \textrm{ for } t > 0. + \end{align*} + % + This implies that + % + \begin{equation} \label{eqn:Q_in_B'} + p \cdot (u_{X/Y, Q} - 1) = (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1). + \end{equation} + + Indeed, by using the above formulas: % \begin{align*} p \cdot (u_{X/Y, Q} - 1) &= @@ -416,6 +417,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. &= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (u_{X/Y', Q'} - 1). \end{align*} % + The proof follows by summing~\eqref{eqn:Q_not_in_B'} and~\eqref{eqn:Q_in_B'} over all $Q \in B_{X/Y}$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] @@ -468,13 +470,9 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. 2(g_{Y'} - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ &= 2 p (g_Y - 1) + \sum_{Q' \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q'}^{(1)} + 1)\\ &+ 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ - &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right),\\ + &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right). \end{align*} % - where - % - \[ B := \{ Q \in Y(k) : e_Q > 1 \} = \{ Q \in Y(k) : e'_Q > 1 \}. \] - % In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$. Thus by Lemma~\ref{lem:lemma_mcT_and_T} for any $1 \le i \le p^n - p^{n-1}$: %