jgarnek/de_rham_cyclic
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

cw formula begin 3

Browse Source
This commit is contained in:
jgarnek 2024-12-06 17:00:04 +01:00
parent 9ab8e2e490
commit 23e0664e4a
2 changed files with 16 additions and 2 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
article_de_rham_cyclic.synctex.gz
View File

Binary file not shown.

18
article_de_rham_cyclic.tex
View File

@ -10,6 +10,7 @@
\usepackage[T1]{fontenc} \usepackage[T1]{fontenc}
\usepackage{tikz, tikz-cd, stmaryrd, amsmath, amsthm, amssymb, \usepackage{tikz, tikz-cd, stmaryrd, amsmath, amsthm, amssymb,
hyperref, bbm, mathtools, mathrsfs} hyperref, bbm, mathtools, mathrsfs}
\usepackage[all]{xy}
%\usepackage{upgreek} %\usepackage{upgreek}
\newcommand{\upomega}{\boldsymbol{\omega}} \newcommand{\upomega}{\boldsymbol{\omega}}
\newcommand{\upeta}{\boldsymbol{\eta}} \newcommand{\upeta}{\boldsymbol{\eta}}
@ -189,7 +190,11 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$
\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump. \item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump.
\end{itemize} \end{itemize}
% %
By Hasse--Arf theorem (cf. ???), if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers. By Hasse--Arf theorem (cf.
{\color{red}
\cite[p. 76]{Serre1979}),
}
if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers.
For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$, For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$. $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
Let Let
@ -422,7 +427,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
\end{proof} \end{proof}
% %
\begin{Lemma} \label{lem:u_equals_ul} \begin{Lemma} \label{lem:u_equals_ul}
Assume that $Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$. Assume that $ {\color{red} \phi:} Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
Then: Then:
% %
\[ \[
@ -483,6 +488,15 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
\begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}]
We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$. $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$.
{\color{red}
\[
\xymatrix{
& X \ar[rd] \ar[ld] \ar[dd]^{\pi}& \\
Y' \ar[rd]^{\phi} & & X'' \ar[ld]\\
& Y &
}
\]
}
Let also $\mc M := H^1_{dR}(X)$ and write $\mc M_0$ for the module~\eqref{eqn:HdR_formula}. Let also $\mc M := H^1_{dR}(X)$ and write $\mc M_0$ for the module~\eqref{eqn:HdR_formula}.
We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$:
% %