From dcba4a07a2a9644e61dc6609d89948612209a621 Mon Sep 17 00:00:00 2001 From: jgarnek Date: Fri, 22 Nov 2024 19:06:06 +0100 Subject: [PATCH] intro - references --- article_de_rham_cyclic.bbl | 63 +++++++++++++++++++++++++++++- article_de_rham_cyclic.synctex.gz | Bin 97143 -> 102416 bytes article_de_rham_cyclic.tex | 22 +++++------ 3 files changed, 72 insertions(+), 13 deletions(-) diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index 332bc41..db7563a 100644 --- a/article_de_rham_cyclic.bbl +++ b/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. 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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 diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 952065ed00a0ee10c8787b1e7588d20815191d9e..2eb163a35405b9d0a6fa275a3a64b0b6ece16c79 100644 GIT binary patch literal 102416 zcmV)|KzzR+iwFP!000003begxk6yWwCicC4MK_QiQh@4$`=W1y85ppQ!5Fs37^CW< zp;Bqd*4k3)j%ph7-~Bv1hs(<#Ghda;_d=DKCl+qOV2}(({9k{3c=3O~{BQsC|GfPC zXHub)1@eE#|V?!~*8zn(w;<@cBW^7+m4_y6+Zj~DOmUi|I; z<+qc)*|Pr+pFaJk=l9Ql{XaiGe}DJk^{@Z=?te%9eg)~T@1FC&AD+MFzrTIC|Knt4 zf0)_N@67%4-@g9s#jDpRPxs?IKYqS{^YQnO-v+M!&vSkL{>zJZU-$!^`#1c0vJ7PT ze|-M<`t$Sm|Mb7yy~9APXx~4-e^Y-BT^GNP8h_po)am=EU%&h?5a;hB{_V>j1F=@$ zNB!~BShUvHQNQ~5;r`37U(w}%d-Z#zHQz&9e;Et^0dakQ_u})%`#(Ot?7xki-4#^> z?9-dO7hisT`|^)}JbyQm`zy#_-hce~j?=&XGBOB%Y0G~8^!qD-{`ULJSG9on*LGc> zm^%FcXUG`z-`(pE!>?nvTqEhfJ^%9Z*SlBuZ!YGDQGZRfe*f*oo8!Mjo6TOw|5@fk z&3{0{UohwN8AQPR$m|*Uet9)>{sEPL!TJ7v=6ub>`gY96`g82chgCeZ>dVh}F9d^O 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z_08lxFqs%w&xvwua8*THk+6MzrhLaQ_2Sctn~2 diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 79ec5a4..39c0936 100644 --- a/article_de_rham_cyclic.tex +++ b/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}