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pf of Thm 2.1 ctn - jumps 1

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jgarnek 2024-10-22 16:46:30 +02:00
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@ -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: 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:
% %
\[ \[
H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{P \neq P_0} J_{p^n - p^n/e_P}^2 H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\ P \neq P_0}} J_{p^n - p^n/e_P}^2
\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{P}^{(t+1)} - u_{P}^{(t)}}, \oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{P}^{(t+1)} - u_{P}^{(t)}},
\] \]
% %
where $e_P := e_{X/Y, P}$ and $u_P^{(t)} := u_{X/Y, P}^{(t)}$. 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: (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} \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} \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) \tr_{X/Y} : H^1(X, \mc O_X) \to H^1(Y, \mc O_Y)
\end{equation*} \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. the trace map on the de Rham cohomology must be surjective as well.
% %
\end{proof} \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 $x \in M^{(i)}$. 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 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. shows that $m_{\sigma - 1}$ is well-defined and injective.
\end{proof} \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: By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
% %
\[ \[
\mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2 \mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\P \neq P_0}} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2
\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}} \oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
\] \]
% %
where $e'_P := e_{X/Y', P}$ and where $e'_P := e_{X/Y', P}$. Note that
% %
\[ \[
m' := ??? p \cdot u^{(n)} = u^{(n-1)} + (p-1) \cdot l^{(1)} ???.
\begin{cases}
n-1, & \textrm{ if } m = n,\\
m, & \textrm{ otherwise.}
\end{cases}
\] \]
% %
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*} \begin{align*}
\dim_k \mc T^i \mc M = \dim_k \mc T^i \mc M &=
\begin{cases} 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)\\
\end{cases} &+ 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*} \end{align*}
% %
where
%
\[ R := \{ P \in X(k) : e_P > 1 \} = \{ P \in X(k) : e'_P > 1 \}. \]
%
In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$. 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} Thus by Lemma~\ref{lem:lemma_mcT_and_T}
% %
\[ \begin{align*}
\dim_k T^1 \mc M = \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M = ????. \dim_k T^1 \mc M &= \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M\\
\] &= 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 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: in $\FF_p[x]$ we have the identity: