From a1a831d3151b3a8693a063895200875d43d869dc Mon Sep 17 00:00:00 2001 From: jgarnek Date: Wed, 18 Dec 2024 12:52:42 +0100 Subject: [PATCH] X ---> X/C --- article_de_rham_cyclic.bbl | 300 +++++++++++++++--------------- article_de_rham_cyclic.out | 2 +- article_de_rham_cyclic.synctex.gz | Bin 144444 -> 151853 bytes article_de_rham_cyclic.tex | 108 ++++++++--- 4 files changed, 235 insertions(+), 175 deletions(-) diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index d6ef83b..d4cad17 100644 --- a/article_de_rham_cyclic.bbl +++ b/article_de_rham_cyclic.bbl @@ -1,150 +1,150 @@ - \def\cprime{$'$} -\begin{thebibliography}{10} - -\bibitem{MR2035696} -Alejandro Adem and R.~James Milgram. -\newblock {\em Cohomology of finite groups}, volume 309 of {\em Grundlehren der - mathematischen Wissenschaften [Fundamental Principles of Mathematical - Sciences]}. -\newblock Springer-Verlag, Berlin, second edition, 2004. - -\bibitem{Alperin_local_rep} -J.~L. Alperin. -\newblock {\em Local representation theory}, volume~11 of {\em Cambridge - Studies in Advanced Mathematics}. -\newblock Cambridge University Press, Cambridge, 1986. -\newblock Modular representations as an introduction to the local - representation theory of finite groups. - -\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{Bleher_Wood_polydiffs_structure} -F.~M. Bleher and A.~Wood. -\newblock The {G}alois module structure of holomorphic poly-differentials and - {R}iemann-{R}och spaces. -\newblock {\em J. Algebra}, 631:756--803, 2023. - -\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{Chevalley_Weil_Uber_verhalten} -C.~Chevalley, A.~Weil, and E.~Hecke. -\newblock \"{U}ber das verhalten der integrale 1. gattung bei automorphismen - des funktionenk\"{o}rpers. -\newblock {\em Abh. Math. Sem. Univ. Hamburg}, 10(1):358--361, 1934. - -\bibitem{Curtis_Reiner_Methods_II} -C.~W. Curtis and I.~Reiner. -\newblock {\em Methods of representation theory. {V}ol. {II}}. -\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc., - New York, 1987. -\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{Ellingsrud_Lonsted_Equivariant_Lefschetz} -G.~Ellingsrud and K.~L{\o}nsted. -\newblock An equivariant {L}efschetz formula for finite reductive groups. -\newblock {\em Math. Ann.}, 251(3):253--261, 1980. - -\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 - Mathematics}}. -\newblock Springer-Verlag, New York-Berlin, 1979. -\newblock Translated from the French by Marvin Jay Greenberg. - -\bibitem{Steinberg_Representation_book} -B.~Steinberg. -\newblock {\em Representation theory of finite groups}. -\newblock Universitext. Springer, New York, 2012. -\newblock An introductory approach. - -\bibitem{Valentini_Madan_Automorphisms} -R.~C. Valentini and M.~L. Madan. -\newblock Automorphisms and holomorphic differentials in characteristic~{$p$}. -\newblock {\em J. Number Theory}, 13(1):106--115, 1981. - -\bibitem{Weibel} -Ch.~A. Weibel. -\newblock {\em An introduction to homological algebra}, volume~38 of {\em - Cambridge Studies in Advanced Mathematics}. -\newblock Cambridge University Press, Cambridge, 1994. - -\end{thebibliography} + \def\cprime{$'$} +\begin{thebibliography}{10} + +\bibitem{MR2035696} +Alejandro Adem and R.~James Milgram. +\newblock {\em Cohomology of finite groups}, volume 309 of {\em Grundlehren der + mathematischen Wissenschaften [Fundamental Principles of Mathematical + Sciences]}. +\newblock Springer-Verlag, Berlin, second edition, 2004. + +\bibitem{Alperin_local_rep} +J.~L. Alperin. +\newblock {\em Local representation theory}, volume~11 of {\em Cambridge + Studies in Advanced Mathematics}. +\newblock Cambridge University Press, Cambridge, 1986. +\newblock Modular representations as an introduction to the local + representation theory of finite groups. + +\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{Bleher_Wood_polydiffs_structure} +F.~M. Bleher and A.~Wood. +\newblock The {G}alois module structure of holomorphic poly-differentials and + {R}iemann-{R}och spaces. +\newblock {\em J. Algebra}, 631:756--803, 2023. + +\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{Chevalley_Weil_Uber_verhalten} +C.~Chevalley, A.~Weil, and E.~Hecke. +\newblock \"{U}ber das verhalten der integrale 1. gattung bei automorphismen + des funktionenk\"{o}rpers. +\newblock {\em Abh. Math. Sem. Univ. Hamburg}, 10(1):358--361, 1934. + +\bibitem{Curtis_Reiner_Methods_II} +C.~W. Curtis and I.~Reiner. +\newblock {\em Methods of representation theory. {V}ol. {II}}. +\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc., + New York, 1987. +\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{Ellingsrud_Lonsted_Equivariant_Lefschetz} +G.~Ellingsrud and K.~L{\o}nsted. +\newblock An equivariant {L}efschetz formula for finite reductive groups. +\newblock {\em Math. Ann.}, 251(3):253--261, 1980. + +\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 + Mathematics}}. +\newblock Springer-Verlag, New York-Berlin, 1979. +\newblock Translated from the French by Marvin Jay Greenberg. + +\bibitem{Steinberg_Representation_book} +B.~Steinberg. +\newblock {\em Representation theory of finite groups}. +\newblock Universitext. Springer, New York, 2012. +\newblock An introductory approach. + +\bibitem{Valentini_Madan_Automorphisms} +R.~C. Valentini and M.~L. Madan. +\newblock Automorphisms and holomorphic differentials in characteristic~{$p$}. +\newblock {\em J. Number Theory}, 13(1):106--115, 1981. + +\bibitem{Weibel} +Ch.~A. Weibel. +\newblock {\em An introduction to homological algebra}, volume~38 of {\em + Cambridge Studies in Advanced Mathematics}. +\newblock Cambridge University Press, Cambridge, 1994. + +\end{thebibliography} diff --git a/article_de_rham_cyclic.out b/article_de_rham_cyclic.out index a6b4d0f..75870ec 100644 --- a/article_de_rham_cyclic.out +++ b/article_de_rham_cyclic.out @@ -2,5 +2,5 @@ \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n\000\040\000a\000n\000d\000\040\000p\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i\000e\000s}{}% 2 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 3 \BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000P\000r\000o\000o\000f\000\040\000o\000f\000\040\000M\000a\000i\000n\000\040\000T\000h\000e\000o\000r\000e\000m}{}% 4 -\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000E\000x\000a\000m\000p\000l\000e\000s}{}% 5 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zk@%^?V6c_xRtbhP@(yrEIK-O7DOwO{(0F>URUtH_!3kzXR8Oky+=l3+#yKskP%OXd z=_QnZPAY-^Qo__pB;%9Yby6#?j3p^!6`Iajr2kHuSTX}T`VQCcSP%;U=?OEU^LK*y z;LZguarkOcFK}8veZj2euzmUm3U$jTQpVz&ztbl2??3_~@V~GF& diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index d2e9f7d..28295c5 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -834,25 +834,24 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo is determined by higher ramification data as well. \end{proof} % -\section{Examples} -% -Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order~$p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$ +The method of proof of Main Theorem allows to obtain explicit formulas in the style of the result of Chevalley--Weil for +particular group. Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order~$p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$ for a homomorphism $\chi : C \to \FF_p^{\times}$. % \begin{Proposition} -Keep the above notation. {\color{red} Assume that $k$ is algebraically closed.} If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then: -% -\[ + Keep the above notation. {\color{red} Assume that $k$ is algebraically closed.} If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then: + % + \[ H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(C)} \mc V(W, p)^{\oplus a^{dR}_{Y, C}(W)} \oplus \mc V(W, p-1)^{\oplus b_W}, -\] -% -where for any $W \in \Indec(k[C])$ the number $a_W$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $X$, -$a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ and -% -\begin{align*} - b_W &:= a^{dR}_{X, C}(W) - \frac 1p \sum_{i = 0}^{p-2} a^{dR}_{X, C}(W \otimes \chi^i) - a^{dR}_{Y, C}(W \otimes \chi). -\end{align*} -% + \] + % + where for any $W \in \Indec(k[C])$ the number $a_W$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $X$, + $a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ and + % + \begin{align*} + b_W &:= a^{dR}_{X, C}(W) - \frac 1p \sum_{i = 0}^{p-2} a^{dR}_{X, C}(W \otimes \chi^i) - a^{dR}_{Y, C}(W \otimes \chi). + \end{align*} + % \end{Proposition} \begin{proof} Theorem~\ref{thm:cyclic_de_rham} easily implies that @@ -864,9 +863,14 @@ $a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ for some $A_W, B_W \in \ZZ$. ?? \end{proof} % -Let $p > 2$ be a prime and $p \nmid m$ an natural number. Fix a primitive root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$. + +\section{An example -- a superelliptic curve with a metacyclic action} +% + +Let $p > 2$ be a prime and $p \nmid m$ an natural number. Let $k$ be an algebraically closed field of characteristic~ $p$. +Fix a primitive root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$. Note that $\zeta^m \in \FF_p$. -We compute now the equivariant structure of the de Rham cohomology for the superelliptic curve $X$ with the affine part given by: +In this section we compute the equivariant structure of the de Rham cohomology for the superelliptic curve $X$ with the affine part given by: % \begin{equation*} y^m = x^{p^n} - x. @@ -887,24 +891,80 @@ This action is given by: \rho(x, y) &= (\zeta^m \cdot x, \zeta \cdot y). \end{align*} % +\begin{Proposition} + \[ + H^1_{dR}(X) \cong ????. + \] +\end{Proposition} +% Note that $X/G \cong \PP^1$ and the quotient map is given by $(x, y) \mapsto (x^p - x)^{p-1}$. Indeed, ????. -We claim that the set of branch points is given by $B := \{ 0, \infty \} \cup B'$, where +We claim that the set of branch points is given by $B := \{ Q_{\infty}, Q_0, Q_1, \ldots, Q_N \}$, where +$N := \frac{p^{n-1} - 1}{p - 1}$, $Q_0 = 0$, $Q_{\infty} = \infty$ and $Q_1, \ldots, Q_N$ are +the elements of the set % \[ - B' := \{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}. + \{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}. \] % -The set $B'$ has $\frac{p^{n-1} - 1}{p - 1}$ elements. We claim that: +Write $C' := \langle \rho^{p-1} \rangle \cong \ZZ/m$ and note that $C'$ is in the center of $G$. We claim that: % \begin{itemize} - \item $G_{Q_0} = C$, + \item $G_{Q_0}$ is the conjugacy class of the subgroup $C$, - \item $G_Q = \langle \rho^{p-1} \rangle \cong \ZZ/m$ for $Q \in B'$, + \item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$, - \item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$. + + \item $G_{Q_i} = C'$ for $i = 1, \ldots, N$. \end{itemize} % -Indeed, ????. +Indeed, ????. The ramification points of $\pi : X \to X/G$ are as follows: +% +\begin{itemize} + \item points $P_0^{(1)}, \ldots, P_0^{(p)}$ above $Q_0$ + \item[] (their stabilizers are subgroups $C_1 = C$, $\ldots$, $C_p$ + conjugated to $C$), + + \item point $P_{\infty}$ above $Q_{\infty}$ (its stabilizer is $G$), + + \item points $P_i^{(1)}, \ldots, P_i^{(p \cdot (p-1))}$ above $Q_i$ for $i = 1, \ldots, N$ + \item[] (their stabilizers equal $C'$). +\end{itemize} +% +The same points are in the ramification locus of the morphism $X \to X/C$ with the following +ramification groups: +% +\begin{align*} + C_{P_0^{(1)}} &= C\\ + C_{P_0^{(i)}} &= C' \qquad \textrm{ for } i > 1,\\ + C_{P_{\infty}} &= C\\ + C_{P_i^{(j)}} &= C' \qquad \textrm{ for } i = 1, \ldots, N, \, j = 1, \ldots, p \cdot (p-1). +\end{align*} + + + + + +Note that $Y := X/H$ is given by the equation: +% +\[ + y^m = z^{p^{n-1}} + \ldots + z^p + z. +\] +% +Let $\psi : C \to k^{\times}$ be a primitive character. We claim that: +% +\begin{align*} + a^{dR}_{X, C}(\psi^i) &= + \begin{cases} + p \cdot N, & \textrm{ if } m \nmid i,\\ + 0, & \textrm{ otherwise. } + \end{cases}\\ + a^{dR}_{Y, C}(\psi^i) &= + \begin{cases} + \frac{p^{n-1} - 1}{p - 1}, & \textrm{ if } m \nmid i,\\ + 0, & \textrm{ otherwise. } + \end{cases} +\end{align*} +% %

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