intro - references

article_de_rham_cyclic.bbl

@@ -1,10 +1,21 @@
-\begin{thebibliography}{1}
+\begin{thebibliography}{10}
+
+\bibitem{Bleher_Camacho_Holomorphic_differentials}
+F.~M. Bleher and N.~Camacho.
+\newblock Holomorphic differentials of {K}lein four covers.
+\newblock {\em J. Pure Appl. Algebra}, 227(10):Paper No. 107384, 27, 2023.
 
 \bibitem{Bleher_Chinburg_Kontogeorgis_Galois_structure}
 F.~M. Bleher, T.~Chinburg, and A.~Kontogeorgis.
 \newblock Galois structure of the holomorphic differentials of curves.
 \newblock {\em J. Number Theory}, 216:1--68, 2020.
 
+\bibitem{Borevic_Faddeev}
+Z.~I. {Borevi\v{c}} and D.~K. Faddeev.
+\newblock Theory of homology in groups. {II}. {P}rojective resolutions of
+  finite groups.
+\newblock {\em Vestnik Leningrad. Univ.}, 14(7):72--87, 1959.
+
 \bibitem{Curtis_Reiner_Methods_II}
 C.~W. Curtis and I.~Reiner.
 \newblock {\em Methods of representation theory. {V}ol. {II}}.
@@ -13,22 +24,72 @@ C.~W. Curtis and I.~Reiner.
 \newblock With applications to finite groups and orders, A Wiley-Interscience
   Publication.
 
+\bibitem{Dummigan_99}
+N.~Dummigan.
+\newblock Complete {$p$}-descent for {J}acobians of {H}ermitian curves.
+\newblock {\em Compositio Math.}, 119(2):111--132, 1999.
+
 \bibitem{Garnek_equivariant}
 J.~Garnek.
 \newblock Equivariant splitting of the {H}odge-de {R}ham exact sequence.
 \newblock {\em Math. Z.}, 300(2):1917--1938, 2022.
 
+\bibitem{Garnek_p_gp_covers}
+J.~Garnek.
+\newblock {$p$}-group {G}alois covers of curves in characteristic {$p$}.
+\newblock {\em Trans. Amer. Math. Soc.}, 376(8):5857--5897, 2023.
+
+\bibitem{Garnek_p_gp_covers_ii}
+J.~Garnek.
+\newblock $p$-group {G}alois covers of curves in characteristic $p$ {II}, 2023.
+
 \bibitem{garnek_indecomposables}
 J.~Garnek.
 \newblock Indecomposable direct summands of cohomologies of curves, 2024.
 \newblock arXiv 2410.03319.
 
+\bibitem{Gross_Rigid_local_systems_Gm}
+B.~H. Gross.
+\newblock Rigid local systems on {$\Bbb G_m$} with finite monodromy.
+\newblock {\em Adv. Math.}, 224(6):2531--2543, 2010.
+
 \bibitem{Hartshorne1977}
 R.~Hartshorne.
 \newblock {\em {Algebraic geometry}}.
 \newblock Springer-Verlag, New York-Heidelberg, 1977.
 \newblock Graduate Texts in Mathematics, No. 52.
 
+\bibitem{Heller_Reiner_Reps_in_integers_I}
+A.~Heller and I.~Reiner.
+\newblock Representations of cyclic groups in rings of integers. {I}.
+\newblock {\em Ann. of Math. (2)}, 76:73--92, 1962.
+
+\bibitem{Higman}
+D.~G. Higman.
+\newblock Indecomposable representations at characteristic {$p$}.
+\newblock {\em Duke Math. J.}, 21:377--381, 1954.
+
+\bibitem{laurent_kock_drinfeld}
+L.~Laurent and B.~K{\"{o}}ck.
+\newblock The canonical representation of the drinfeld curve.
+\newblock {\em Mathematische Nachrichten}, online first, 2024.
+
+\bibitem{Lusztig_Coxeter_orbits}
+G.~Lusztig.
+\newblock Coxeter orbits and eigenspaces of {F}robenius.
+\newblock {\em Invent. Math.}, 38(2):101--159, 1976/77.
+
+\bibitem{WardMarques_HoloDiffs}
+S.~Marques and K.~Ward.
+\newblock Holomorphic differentials of certain solvable covers of the
+  projective line over a perfect field.
+\newblock {\em Math. Nachr.}, 291(13):2057--2083, 2018.
+
+\bibitem{Prest}
+M.~Prest.
+\newblock Wild representation type and undecidability.
+\newblock {\em Comm. Algebra}, 19(3):919--929, 1991.
+
 \bibitem{Serre1979}
 J.-P. Serre.
 \newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in

article_de_rham_cyclic.tex

@@ -136,25 +136,23 @@ The classical Chevalley--Weil formula (cf. ????) gives an explicit description
 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 $X \to X/G$. ????
 
 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
-
+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. \cite{Prest}). There are many results concerning equivariant structure of cohomologies  for particular groups
+(see e.g.~\cite{Valentini_Madan_Automorphisms} for the case of cyclic groups, \cite{WardMarques_HoloDiffs} for abelian groups, \cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} for groups with a cyclic Sylow subgroup, or \cite{Bleher_Camacho_Holomorphic_differentials} for the Klein group) or curves (cf. \cite{Lusztig_Coxeter_orbits}, \cite{Dummigan_99}, \cite{Gross_Rigid_local_systems_Gm}, \cite{laurent_kock_drinfeld}). Also, one may expect that that (at least in the case of $p$-groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. \cite{Garnek_p_gp_covers}, \cite{Garnek_p_gp_covers_ii}). 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. \cite{Higman}, \cite{Borevic_Faddeev}, \cite{Heller_Reiner_Reps_in_integers_I}). While their representation theory
+seems too complicated to derive a formula for the cohomologies, 
+the article~\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} proved that 
+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 main result of this article is a similar statement for the de Rham cohomology.
+%
 \begin{mainthm}
 	Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
 	Let $X$ be a curve with an action of~$G$ over a field $k$ of characteristic $p$.
-	The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the lower ramification groups and the fundamental characters of closed
-	points $x$ of $X$ that are ramified in the cover $X \to X/G$.
+	The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the lower ramification groups and the fundamental characters of
+	points $P \in X(k)$ that are ramified in the cover $X \to X/G$.
 \end{mainthm}
 %
 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. 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
+of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data. Indeed, see \cite{garnek_indecomposables} for a construction of $G$-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}