new example

article_de_rham_cyclic.bbl

@@ -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 Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc.,
-\newblock With applications to finite groups and orders, A Wiley-Interscience Publication.
+  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}

article_de_rham_cyclic.tex

@@ -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(x, y) &= (x+1, y),\\
-	\sigma_1(x, y) &= (x, y+1),\\
+	\rho(x, y) &= (\zeta^m \cdot x, \zeta \cdot y).
-	s(x, y) &= (y, x),\\
-	\theta(x, y) &= (\zeta \cdot x, \zeta^{-1} \cdot y) \quad \textrm{ for } \FF_p^{\times} = \langle \zeta \rangle.
 \end{align*}
 %
 Recall representation theory of $D_{2(p - 1)}$ (cf. \cite[Example~8.2.3]{Steinberg_Representation_book}).
@@ -875,7 +885,7 @@ For $1 \le j \le p-2$ let $\chi_j$ be the character of the representation of $D_
 induced from
 %
 \[
-	\ZZ/(p-1) = \langle \theta \rangle \to \FF_p^{\times}, \quad \theta \mapsto \zeta^j.
+\ZZ/(p-1) = \langle \theta \rangle \to \FF_p^{\times}, \quad \theta \mapsto \zeta^j.
 \]
 %
 One easily checks that $\chi_j$ is given by the matrices:
@@ -902,25 +912,16 @@ Moreover, $\chi_j$ is irreducible and isomorphic to $\chi_{p - 1 - j}$.
 Let also $\chi_0$ be the representation:
 %
 \[
-	D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad
+D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad
-	s \mapsto -1.
+s \mapsto -1.
 \]
 %
 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}

bibliografia.bib

@@ -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},