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

intro - pt 2

Browse Source
This commit is contained in:
jgarnek 2024-11-22 18:49:45 +01:00
parent 1b32a59bea
commit 76d0f18196
2 changed files with 14 additions and 4 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

@ -133,9 +133,17 @@ hyperref, bbm, mathtools, mathrsfs}
\section{Introduction} \section{Introduction}
% %
The classical Chevalley--Weil formula (cf. ????) gives an explicit description The classical Chevalley--Weil formula (cf. ????) gives an explicit description
of the equivariant structure of the cohomology of a curve with a group action over a field of characteristic~$0$. It is hard to expect such a formula over fields of characteristic~$p$. of the equivariant structure of the cohomology of a curve $X$ with a group action over a field of characteristic~$0$. Their formula depends on the so-called \emph{fundamental characters} of points $x \in X$ that are ramified in the cover $X \to X/G$. ????
Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set of indecomposable $k[G]$-modules is infinite. If, moreover, $p > 2$ then the indecomposable $k[G]$-modules are considered impossible to classify (cf. ???). This brings attention to groups
with ??? It is hard to expect such a formula over fields of characteristic~$p$.
Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set of indecomposable $k[G]$-modules is infinite. If, moreover, $p > 2$ then the indecomposable $k[G]$-modules are considered impossible to classify (cf. ???). There are many results concerning equivariant structure of cohomologies for particular curves (cf. ????), groups (cf. ????). Also, one may expect that that (at least in the case of $p$-groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. ???). However, there is no hope of obtaining a result similar to the one of Chevalley and Weil.\\
This brings attention to groups with cyclic $p$-Sylow subgroup. For those, the set of indecomposable modules is finite (cf. ???). Even though for this ?????
The article~\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} shows that in this case,
the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the lower ramification groups and the fundamental characters of the ramification locus. The representation theory of those groups is
\begin{mainthm} \begin{mainthm}
Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup. Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
@ -145,7 +153,9 @@ with ???
\end{mainthm} \end{mainthm}
% %
Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure
of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{garnek_indecomposables}. of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data. Indeed, see \cite{garnek_indecomposables} for an example of a family of $\ZZ/p \times \ZZ/p$-covers
with the same ramification data, but different $k[G]$-module structure of $H^0(X, \Omega_X)$ and
$H^1_{dR}(X)$.
\section{Cyclic covers} \section{Cyclic covers}
% %