pf of Lemma p*(u-1) - two cases

article_de_rham_cyclic.tex

@@ -145,9 +145,10 @@ of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{?
 \section{Cyclic covers}
 %
 \red{For any $\ZZ/p^n$-cover $\pi : X \to Y$ and $P \in X(k)$ write $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) for the $t$th ramification jump at $P$.}
-We use also the convention $u^{(0)}_{X/Y, P} = 1$ and $u^{(t)}_{X/Y, P} := u^{(m)}_{X/Y, P}$, if $p^t \ge |G_P| = p^m$.
+We use also the convention $u^{(0)}_{X/Y, P} = 1$.
 By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $n_{X/Y, P}$
 by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$.
+We abbreviate the last ramification jump to $u_{X/Y, P}$.
 \red{For any $Q \in Y(k)$ we denote also $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
 $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.}
 %
@@ -156,7 +157,7 @@ $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.}
 	%
 	\[
 	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\red{\substack{Q \in Y(k)\\ Q \neq Q_0}}} J_{p^n - p^n/e_{\red{Q}}}^2
-	\oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t \ge 0} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}},
+	\oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t = 0}^{n_{X/Y, P}} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}},
 	\]
 	%
 	where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$.
@@ -182,7 +183,8 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\
 	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*}
 %
-Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ and $P'$ is the image of $P \in X(k)$ on $X'$ then:
+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)} &=
@@ -192,6 +194,8 @@ Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ and
 	\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 $t$.
 }
 %
 \begin{Lemma} \label{lem:G_invariants_\'{e}tale}
@@ -318,16 +322,30 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 \end{proof}
 %
 \begin{Lemma} \label{lem:u_equals_ul}
+	Assume that $Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
 	For any $Q \in Y(k)$:
 	%
 	\[
-	p \cdot (u^{(n_Q)}_{X/Y, Q} - 1) = \sum_{Q'} \left( (u^{(n_{Q'})}_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right),
+	p \cdot (u_{X/Y, Q} - 1) = \sum_{Q'} \left( (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right),
 	\]
 	%
-	where we sum over points $Q' \in Y'(k)$ lying above $Q$ and $n_Q := n_{X/Y, Q}$, $n_{Q'} := n_{X/Y', Q'}$.
+	where we sum over points $Q' \in Y'(k)$ lying above $Q$.
 \end{Lemma}
 \begin{proof}
-	????
+	%
+	Consider the following two cases. If $e_{Y'/Y, Q} = 1$ then
+	$l^{(1)}_{Y'/Y, Q} = - 1$ and $u_{X/Y, Q} = u_{X/Y', Q'}$ for all $p$ points $Q' \in Y'(k)$ in the preimage of $Q$. This easily implies the desired equality.\\
+	If $e_{Y'/Y, Q} = 1$, then there exists a unique point $Q' \in Y'(k)$
+	in the preimage of $Q$ through $Y' \to Y$. Moreover, $n_{X/Y, Q} = n_{X/Y', Q'}$. By using ????above formulas???:
+	%
+	\begin{align*}
+		p \cdot (u_{X/Y, Q} - 1) &=
+		p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(n_Q - 1)}_{X/Y, Q} - 1)\\
+		&= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + \left( (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) + p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \right)\\
+		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\
+		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + u_{X/Y', Q'} - 1.
+	\end{align*}
+	%
 \end{proof}
 
 \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}]
@@ -377,10 +395,10 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	%
 	\begin{align*}
 		\dim_k \mc T^i \mc M &=
-		2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'}^{(n_{Q'})} - 1)\\
+		2(g_{Y'} - 1) + 2 + 2(\# R - 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(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'}^{(n_{Q'})} - 1)\\
+		 &+ 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
-		&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'}^{(n_Q)} - 1) \right)
+		&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right)
 	\end{align*}
 	%
 	where
@@ -392,7 +410,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	%
 	\begin{align*}
 		\dim_k T^1 \mc M &= \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M\\
-		&= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P}^{(n_Q)} - 1).
+		&= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1).
 	\end{align*}
 	%
 	By Lemma~\ref{lem:trace_surjective} since $X \to X''$ is not \'{e}tale, the map $\tr_{X/X''} : H^1_{dR}(X) \to H^1_{dR}(X'')$ is surjective. Recall that
@@ -415,7 +433,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 		\ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})}
 	\]
 	%
-	and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus:
+	and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus, if $i \in [p^{n-1} - p^k, p^{n-1} - p^{k-1}]$:
 	%
 	\[
 		\dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = ....