pf of Thm 2.1 ctn - jumps 1

article_de_rham_cyclic.tex

@@ -154,8 +154,8 @@ the $t$th upper (resp. lower) ramification jump of $X \to Y$ at $P$.
 	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:
 	%
 	\[
-	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{P \neq P_0} J_{p^n - p^n/e_P}^2
-	\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{P}^{(t+1)} - u_{P}^{(t)}},
+	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\ P \neq P_0}} J_{p^n - p^n/e_P}^2
+	\oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{P}^{(t+1)} - u_{P}^{(t)}},
 	\]
 	%
 	where $e_P := e_{X/Y, P}$ and $u_P^{(t)} := u_{X/Y, P}^{(t)}$.
@@ -182,9 +182,9 @@ Note also that for $j \ge 1$:
 (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 $u_{X'/Y, P}^{(t)} = u_{X/Y, P}^{(t)}$ for $t \le N$,
 	
-	\item $l_{X/X', P}^{(t)} = l_{X/X', P}^{(t + N)}$ for $t \le n-N$. 
+	\item $l_{X/X', P}^{(t)} = l_{X/Y, P}^{(t + N)}$ for $t \le n-N$. 
 \end{itemize}
 
 
@@ -240,7 +240,7 @@ and one shows similarly that the trace map
 	\tr_{X/Y} : H^1(X, \mc O_X) \to H^1(Y, \mc O_Y)
 \end{equation*}
 %
-is surjective. Therefore, since the outer trace maps in the diagram are surjective,
+is surjective. Therefore, since the outer vertical maps in the diagram are surjective,
 the trace map on the de Rham cohomology must be surjective as well.
 %
 \end{proof}
@@ -261,7 +261,7 @@ We define $m_{\sigma - 1}$ as follows:
 \]
 %
 where for $\ol x \in T^i M$ we picked any representative $x \in M^{(i)}$.
-If $x \in M^{(i+1)} := \ker((\sigma - 1)^{i+1})$ then clearly $(\sigma - 1) x \in M^{(i)}$.
+Indeed, if $x \in M^{(i+1)} := \ker((\sigma - 1)^{i+1})$ then clearly $(\sigma - 1) x \in M^{(i)}$.
 Moreover $(\sigma - 1) \cdot x \in M^{(i-1)}$ holds if and only if $x \in M^{(i)}$. This
 shows that $m_{\sigma - 1}$ is well-defined and injective.
 \end{proof}
@@ -325,35 +325,37 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
 	%
 	\[
-		\mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2
-		\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
+		\mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\P \neq P_0}} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2
+		\oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
 	\]
 	%
-	where $e'_P := e_{X/Y', P}$ and
+	where $e'_P := e_{X/Y', P}$. Note that
 	%
 	\[
-	m' :=
-		\begin{cases}
-			n-1, & \textrm{ if } m = n,\\
-			m, & \textrm{ otherwise.}
-		\end{cases}
+		??? p \cdot u^{(n)} = u^{(n-1)} + (p-1) \cdot l^{(1)} ???.
 	\]
 	%
-	Therefore, for $i \le p^n - p^{n-1}$
+	Therefore, for $i \le p^{n-1} - p^{n-2}$, using the Riemann--Hurwitz formula (cf. ????):
 	%
 	\begin{align*}
-		\dim_k \mc T^i \mc M =
-		\begin{cases}
-			???, 
-		\end{cases}
+		\dim_k \mc T^i \mc M &=
+		2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 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_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 1)\\
+		&= ?? p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1) \right)
 	\end{align*}
 	%
+	where
+	%
+	\[ R := \{ P \in X(k) : e_P > 1 \} = \{ P \in X(k) : e'_P > 1 \}. \]
+	%
 	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}
 	%
-	\[
-		\dim_k T^1 \mc M = \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M = ????.
-	\]
+	\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_{P \in X(k)} (u_{X/Y, P}^{(n)} - 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
 	in $\FF_p[x]$ we have the identity: