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article_de_rham_cyclic.bbl
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@ -3,14 +3,18 @@
\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 {\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 {\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.
\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.
@ -22,21 +26,31 @@ 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 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 \"{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.
\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.
@ -100,7 +114,8 @@ G.~Lusztig.
\bibitem{WardMarques_HoloDiffs}
S.~Marques and K.~Ward.
\newblock Holomorphic differentials of certain solvable covers of the projective line over a perfect field.
\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}
@ -110,7 +125,8 @@ M.~Prest.
\bibitem{Serre1979}
J.-P. Serre.
\newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in Mathematics}}.
\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.
@ -127,7 +143,8 @@ R.~C. Valentini and M.~L. Madan.
\bibitem{Weibel}
Ch.~A. Weibel.
\newblock {\em An introduction to homological algebra}, volume~38 of {\em Cambridge Studies in Advanced Mathematics}.
\newblock {\em An introduction to homological algebra}, volume~38 of {\em
Cambridge Studies in Advanced Mathematics}.
\newblock Cambridge University Press, Cambridge, 1994.
\end{thebibliography}

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article_de_rham_cyclic.tex
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@ -234,13 +234,13 @@ Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, de
Keep the above notation and assume that $p \nmid \# G$. Then:
%
\begin{equation} \label{eqn:cw}
H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a(X, G, W)},
H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a(W)},
\end{equation}
%
where:
%
\begin{align*}
a(X, G, W) := (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{Q, i}(W) + \llbracket W \cong k \rrbracket,
a(W) := (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{Q, i}(W) + \llbracket W \cong k \rrbracket,
\end{align*}
%
and $N_{Q, i}(W)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $W \otimes_{k[G_Q]} \theta_{X/Y, Q}$.
@ -249,13 +249,13 @@ Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, de
\begin{Corollary}[Chevalley--Weil formula for the de Rham cohomology]
Keep the notation of Proposition~\ref{prop:chevalley_weil}. Then:
\begin{equation} \label{eqn:cw_dR}
H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a^{dR}(X, G, W)}.
H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a^{dR}(W)}.
\end{equation}
%
where:
%
\begin{align*}
a^{dR}(X, G, W) := 2 (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \dim_k W/W^{G_Q} + 2 \cdot \llbracket W \cong k \rrbracket.
a^{dR}(W) := 2 (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \dim_k W/W^{G_Q} + 2 \cdot \llbracket W \cong k \rrbracket.
\end{align*}
%
\end{Corollary}
@ -265,17 +265,17 @@ Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, de
\begin{align*}
H^1_{dR}(X) &\cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X)\\
&\cong H^0(X, \Omega_X) \oplus H^0(X, \Omega_X)^{\vee}\\
&\cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus (a(X, G, W) + a(X, G, W^{\vee}))}.
&\cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus (a(W) + a(W^{\vee}))}.
\end{align*}
%
Note moreover that $N_{Q, i}(W^{\vee}) = N_{Q, e_Q - i}(W)$
(since $\chi_{e_Q}^{e_Q - i}$ is the dual representation to $\chi_{e_Q}^i$) and:
(since $\chi_{e_Q}^{e_Q - i}$ is the dual representation to $\chi_{e_Q}^i$), $N_{Q, 0}(W) = \dim_k W^{G_Q}$ and:
%
\[
\sum_{i = 0}^{e_Q - 1} N_{Q, i}(W) = \dim_k W.
\]
%
Therefore $a(X, G, W) + a(X, G, W^{\vee})$ equals:
Therefore $a(W) + a(W^{\vee})$ equals:
%
\begin{align*}
2 (g_Y - 1) \cdot \dim_k W
@ -296,6 +296,8 @@ Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, de
This ends the proof.
%
\end{proof}
%
When considering an action of a group~$G$ on a curve $X$ we will write $a_{X, G}^{dR}(W)$ instead of $a^{dR}(W)$ for clarity.
}
\section{Cyclic covers}
@ -834,14 +836,14 @@ for a homomorphism $\chi : C \to \FF_p^{\times}$.
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:
%
\[
H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(C)} \mc V(W, p)^{\oplus a_W'} \oplus \mc V(W, p-1)^{\oplus b_W},
H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(C)} \mc V(W, p)^{\oplus a^{dR}_{Y, C}(W)} \oplus \mc V(W, p-1)^{\oplus b_W},
\]
%
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 &:= \frac 1p \left( p \cdot a^{dR}_{X, C}(W) - \sum_{i = 0}^{p-2} a^{dR}_{X, G}(W \otimes \chi^i) \right) - a^{dR}_{Y, G}(W \otimes \chi).
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}
@ -854,20 +856,28 @@ $a_W'$ 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}
\noindent Let $p > 2$. Consider the Mumford curve
%
Let $p > 2$ be a prime and $p \nmid m$ an natural number. Fix a 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:
%
\begin{equation*}
y^m = x^{p^m} - x.
\end{equation*}
%
Note that for $m = 2$ this curve was considered e.g. in \cite[Section~4]{Bleher_Wood_polydiffs_structure}.
It is a curve of genus $\frac 12 (p^2 - 1) (m-1)$ with an action of the group $G := H \rtimes_{\chi} C$,
where $H := \langle \sigma \rangle \cong \ZZ/p$, $C := \langle \rho \rangle \cong \ZZ/(m \cdot p - m)$ and
%
\[
X : (x^p - x) \cdot (y^p - y) = 1.
\chi : C \to H, \quad \rho \mapsto \sigma^{\zeta^m}.
\]
%
It is a curve of genus $(p-1)^2$ and an action of the group $(\ZZ/p \times \ZZ/p) \rtimes D_{2(p-1)}$ given by:
This action is given by:
%
\begin{align*}
\sigma_0(x, y) &= (x+1, y),\\
\sigma_1(x, y) &= (x, y+1),\\
s(x, y) &= (y, x),\\
\theta(x, y) &= (\zeta \cdot x, \zeta^{-1} \cdot y) \quad \textrm{ for } \FF_p^{\times} = \langle \zeta \rangle.
\sigma(x, y) &= (x+1, y),\\
\rho(x, y) &= (\zeta^m \cdot x, \zeta \cdot y).
\end{align*}
%
Recall representation theory of $D_{2(p - 1)}$ (cf. \cite[Example~8.2.3]{Steinberg_Representation_book}).
@ -909,18 +919,9 @@ Let also $\chi_0$ be the representation:
We claim that as $k[C]$-modules: ??k or $\FF_p$??
%
\begin{equation}
H^1_{dR}(X) \cong V_0^{\oplus (p-1)} \oplus \bigoplus_{j = 1}^{\frac{p-1}{2}} V_j^{\oplus 2(p-1)}.
H^1_{dR}(X) \cong ????.
\end{equation}
{\color{gray}
Basis of holomorphic differentials:
%
\[
\omega_{a, b} = \frac{x^a \cdot y^b \, dx}{(x^p - x)} \qquad 0 \le a, b \le p-2.
\]
}
\bibliography{bibliografia,AKGeneral}

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bibliografia.bib
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@ -1,3 +1,20 @@
@article {Bleher_Wood_polydiffs_structure,
AUTHOR = {Bleher, F. M. and Wood, A.},
TITLE = {The {G}alois module structure of holomorphic
poly-differentials and {R}iemann-{R}och spaces},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {631},
YEAR = {2023},
PAGES = {756--803},
ISSN = {0021-8693,1090-266X},
MRCLASS = {11G20 (14G17 14H05 20C20)},
MRNUMBER = {4595907},
MRREVIEWER = {Martha\ Rzedowski-Calder\'on},
DOI = {10.1016/j.jalgebra.2023.05.010},
URL = {https://doi.org/10.1016/j.jalgebra.2023.05.010},
}
@book {Alperin_local_rep,
AUTHOR = {Alperin, J. L.},
TITLE = {Local representation theory},