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

X ---> X/C

Browse Source
This commit is contained in:
jgarnek 2024-12-18 12:52:42 +01:00
parent 6778e16191
commit a1a831d315
4 changed files with 235 additions and 175 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

300
article_de_rham_cyclic.bbl
View File

@ -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}

2
article_de_rham_cyclic.out
View File

@ -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
\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000A\000n\000\040\000e\000x\000a\000m\000p\000l\000e\000\040\040\023\000\040\000a\000\040\000s\000u\000p\000e\000r\000e\000l\000l\000i\000p\000t\000i\000c\000\040\000c\000u\000r\000v\000e\000\040\000w\000i\000t\000h\000\040\000a\000\040\000m\000e\000t\000a\000c\000y\000c\000l\000i\000c\000\040\000a\000c\000t\000i\000o\000n}{}% 5
\BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 6

BIN
article_de_rham_cyclic.synctex.gz
View File

Binary file not shown.

108
article_de_rham_cyclic.tex
View File

@ -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*}
%
%