diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index ac78983..d6ef83b 100644 --- a/article_de_rham_cyclic.bbl +++ b/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 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} diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index f7839a2..554d79d 100644 Binary files a/article_de_rham_cyclic.synctex.gz and b/article_de_rham_cyclic.synctex.gz differ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 610e717..4aacecf 100644 --- a/article_de_rham_cyclic.tex +++ b/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_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}). @@ -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: % \[ - D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad - s \mapsto -1. +D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad +s \mapsto -1. \] % 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} diff --git a/bibliografia.bib b/bibliografia.bib index f73fcc5..1b086ca 100644 --- a/bibliografia.bib +++ b/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},