\documentclass[10pt]{amsart} \usepackage{calrsfs} \usepackage{amsmath,tikz} \usetikzlibrary{tikzmark,fit} \newcommand\bigzero{\makebox(0,0){\text{\huge0}}} \usepackage[greek,english]{babel} % \usepackage[iso-8859-7]{inputenc} \usepackage{graphicx} %\usepackage{hyperref} \usepackage{amsthm} \usepackage{amssymb} \usepackage{multirow} \usepackage{tikz-cd} \usepackage{amsmath} \usepackage{todonotes} \usepackage{amsbsy} \usepackage[all]{xy} \usepackage{qtree} % \usepackage{comment} \usepackage{enumitem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{xfrac} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{color, colortbl} \definecolor{LightCyan}{rgb}{0.88,1,1} \definecolor{Gray}{gray}{0.9} \usepackage{faktor} \usepackage{changes} \definechangesauthor[color=orange,name={Aristidesd Kontogeorgis}]{AK} \definechangesauthor[color=blue,name={Alex Terezakis}]{AT} \usepackage{longtable,booktabs} \usepackage{float} %gia to figure[H] na mpainei to figure ekei pou prepei % \usepackage[linenumbers]{MagmaTeX} %\magmanames(depthFirstSearch,nodeVisitor,strongComponents,acyclicQuotient,transitiveQuotient) %\setlength\textwidth{14cm} % \setlength\textwidth{15cm} \setlength\topmargin{0pt} % \addtolength\topmargin{-\headheight} % \addtolength\topmargin{-\headsep} % \setlength\textheight{8.9in} % \setlength\oddsidemargin{0pt} \setlength\evensidemargin{0pt} % \setlength\marginparwidth{0.5in} %\setlength\textwidth{5.5in} %d % \newtheorem{proposition}{Proposition} % % \numberwithin{proposition}{subsection} % \numberwithin{proposition} % \newtheorem{lemma}{Lemma} % \numberwithin{lemma}{subsection} % \newtheorem{example}{Example} % \numberwithin{example}{subsection} % \newtheorem{definition}{Definition} % \numberwithin{definition}{subsection} % \newtheorem{exercise}{Exercise} % \numberwithin{exercise}{subsection} % \newtheorem{corollary}{Corollary} % \numberwithin{corollary}{subsection} % \newtheorem{comments}{Comments} % \numberwithin{comments}{subsection} % \newtheorem{examples}{Examples} % \numberwithin{examples}{subsection} % \newtheorem{theorem}{Theorem} % \numberwithin{theorem}{subsection} % \newtheorem{problem}{Problem} % \numberwithin{problem}{subsection} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{convention}[theorem]{Convention} \newtheorem{problem}{Problem} \newcommand{\Ker}{\mathrm Ker} \newcommand{\adj}{\mathrm adj} \newcommand{\nullspace}{\mathrm Null} \newcommand{\la}{\latintext} \newcommand{\slg}{\selectlanguage{greek}} \renewcommand{\Im}{\mathrm Im} \newcommand{\Char}{\mathrm char} \newcommand{\Diff}{\mathrm Diff} \newcommand{\Spe}{\mathrm Spec } \newcommand{\mdeg}{\mathrm mdeg} \newcommand{\lc}{\left\lceil} \newcommand{\rc}{\right\rceil} \newcommand{\lf}{\left\lfloor} \newcommand{\rf}{\right\rfloor} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} % \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Heis}{\mathrm Heis} \newcommand{\Fer}{\mathrm Fer} \newcommand{\Dgl}{{D_{\mathrm gl}}} \newcommand{\codim}{{\mathrm codim}} \newcommand{\init}{{\mathrm in_\prec}} \newcommand{\HomC}{\mathcal{H}\!\mathit{om}} \DeclareMathOperator{\Ima}{Im} \newcommand{\cL}{{\mathcal{L}}} \newcommand{\Id}{{\mathrm Id}} % \newcommand{\Hom}{{\mathrm Hom}} \newcommand{\tg}{{\mathrm tg}} \newcommand{\cO}{ {\mathcal{O} } } \newcommand{\TO}{{\mathcal{T}_\mathcal{O} }} \newcommand{\T}{{\mathcal{T}}} % \newcommand{\Aut}{{ \mathrm Aut }} \newcommand{\Fp}{{\mathbb{F}_p}} \newcommand{\F}{{\mathbb{F}}} \newcommand{\im}{{ \mathrm Im }} \renewcommand{\O}{{\mathcal{O}}} % \newcommand{\asp}{ \begin{array}{c} \; \\ \; \\mathrm{end}{array}} % \newcommand{\Gal}{\mathrm{Gal}} \renewcommand{\mod}{{\;\mathrm{mod}}} \newcommand{\G}{{\mathcal{G}}} \newcommand{\rad}{{\mathrm rad}} \newcommand{\Rgl}{R} \newcommand{\m}{\mathfrak{m}} \newcommand{\bb}{\textbf} \newcommand{\uu}{\underline} \newcommand{\ol}{\overline} \newcommand{\mc}{\mathcal} \newcommand{\wh}{\widehat} \newcommand{\wt}{\widetilde} \newcommand{\mf}{\mathfrak} \newcommand{\ms}{\mathscr} \renewcommand{\AA}{\mathbb{A}} \newcommand{\II}{\mathbb{I}} \newcommand{\HH}{\mathbb{H}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\CC}{\mathbb{C}} \newcommand{\RR}{\mathbb{R}} \newcommand{\PP}{\mathbb{P}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\LL}{\mathbb{L}} \newcommand{\NN}{\mathbb{N}} \newcommand{\FF}{\mathbb{F}} \newcommand{\VV}{\mathbb{V}} \newcommand{\ddeg}{\textbf{deg}\,} \DeclareMathOperator{\SSh}{-Sh} \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Sh}{Sh} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Divv}{Div} \DeclareMathOperator{\Coind}{Coind} \DeclareMathOperator{\coker}{coker} % \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Tot}{Tot} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Gl}{Gl} \DeclareMathOperator{\Sl}{Sl} \DeclareMathOperator{\GCD}{GCD} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Proj}{Proj} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\cha}{char} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\Jac}{Jac} \DeclareMathOperator{\Lie}{Lie} \DeclareMathOperator{\GSp}{GSp} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\qlog}{qlog} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\divv}{div} \DeclareMathOperator{\mmod}{-mod} \DeclareMathOperator{\ev}{ev} \DeclareMathOperator{\Indec}{Indec} \DeclareMathOperator{\pole}{pole} % \DeclareMathOperator{\Ima}{Im} % \newcommand{\defeq}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt % \hbox{\scriptsize.}\hbox{\scriptsize.}}}% \newcommand\overmat[2]{% \makebox[0pt][l]{$\smash{\color{white}\overbrace{\phantom{% \begin{matrix}#2\end{matrix}}}^{\text{\color{black}#1}}}$}#2} \newcommand\bovermat[2]{% \makebox[0pt][l]{$\smash{\overbrace{\phantom{% \begin{matrix}#2\end{matrix}}}^{\text{#1}}}$}#2} \date{\today} \title{Galois Action on Homology of the Heisenberg Curve.} % \author[A. Kontogeorgis]{Aristides Kontogeorgis} % \address{Department of Mathematics, National and Kapodistrian University of Athens % Pane\-pist\-imioupolis, 15784 Athens, Greece} % \email{kontogar@math.uoa.gr} % \author[D. Noulas]{Dimitrios Noulas} % \address{Department of Mathematics, National and Kapodistrian University of Athens\\ % Panepistimioupolis, 15784 Athens, Greece} % \email{dnoulas@math.uoa.gr} % \author[I. Tsouknidas]{Ioannis Tsouknidas } % \address{Department of Mathematics, National and Kapodistrian University of Athens\\ % Panepistimioupolis, 15784 Athens, Greece} % \email{iotsouknidas@math.uoa.gr} \date \today \makeatletter \newcommand{\aprod}{\mathop{\operator@font \hbox{\Large$\ast$}}} \makeatother %\renewcommand{\thefootnote}{\fnsymbol{footnote}} \begin{document} \section{Cyclic Ramification} Serre Local fields p. 77 Hasse- Arf for cyclic groups. For a cyclic group $G= \mathbb{Z}/p^n$ and $G(i)=\mathbb{Z}/p^{n-i}$, there are integers $i_0,i_1, \ldots , i_{n-1}>0$ such that \begin{align} \label{eq:serreI} G(0) & =G_0= \cdots = G_{i_0} &&=G^0 = \cdots = G^{i_0} \\ G(1) & = G_{i_{0}+1 }= \cdots = G_{i_{0}+ p i_1 } &&= G^{i_0+1} = \cdots = G^{i_0+i_1} \nonumber \\ G(2) &= G_{i_0+p i_1 +1} = \cdots = G_{i_0 +p i_1 + p^2 i_2} & &= G^{i_0+i_1+1} = \cdots = G^{i_0+ i_1 +i_2} \nonumber \end{align} We also set $i_{-1}=-1$. This means that the lower jumps occur at the integers \[ i_0, i_0+i_1 p, i_0+ i_1 p + i_2 p^2, \ldots , i_0 + i_1 p + i_2 p^2 + \cdots i_{n-1} p^{n-1} \] while the upper jumps occur at \[ i_0, i_0+i_1, i_0+ i_1 + i_2, \ldots , i_0 + i_1 + i_2 + \cdots i_{n-1} \] % \begin{definition} Let for any $\ZZ/p^n$-cover $X \to Y$ % {\color{blue} \begin{align*} u_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_P^{(t)} \cong \ZZ/p^{n-t} \},\\ l_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_{P, t} \cong \ZZ/p^{n-t} \}. \end{align*} } {\color{red} \begin{align*} u_{X/Y, P}^{(t)} &:= \min \{ \nu \ge 0 : G_P^{(\nu)} \cong \ZZ/p^{n-t} \},\\ l_{X/Y, P}^{(t)} &:= \min \{ \nu \ge 0 : G_{P, \nu} \cong \ZZ/p^{n-t} \}. \end{align*} but maybe you mean \begin{align} u_{X/Y, P}^{(t)} &:= \min \{ \nu \ge 0 : G_P^{(\nu)} \cong \ZZ/p^{n-t} \}-1,\\ l_{X/Y, P}^{(t)} &:= \min \{ \nu \ge 0 : G_{P, \nu} \cong \ZZ/p^{n-t} \}-1. \end{align} } \end{definition} % {\color{blue} Note that if $G_P = \ZZ/p^n$, this coincides with the standard definition of the $t$th upper (resp. lower) ramification jump of $X \to Y$ at $P$. If $G_P = \ZZ/p^m$, then (??relation with usual jumps??). By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. % } Observe that if $G_P= \ZZ/ p^{n}$ with corresponding integers $i_0=i_0(P), \ldots , i_{n-1}=i_{n-1}(P)$ at $P$ then eq. (\ref{eq:serreI}) gives us \begin{align*} l^{(t)}_{X/Y,P} &= \begin{cases} 0 , &\text{ if } t=0 \\ i_0 + i_{1} p+ \cdots+ i_{t-1} p^{t-1} % {\color{blue} +1}, &\text{ if } t>0 \\ \end{cases} \\ u^{(t)}_{X/Y,P} &= \begin{cases} 0 , &\text{ if } t=0 \\ i_0 + i_{1} + \cdots+ i_{t-1} % {\color{blue} +1}, &\text{ if } t>0 \\ \end{cases} \end{align*} % that is not the upper jump but the next number. We then have: \[ i_{j-1} =u_{X/Y,P}^{(j)}-u_{X /Y,P}^{(j-1)}= \frac{1}{p^{j-1}} (l_{X/ Y,P}^{j} - l_{X/ Y,P}^{j-1}) = \frac{1}{p^{j-1}} p^{j-1} i_{j-1} \] {\color{red} you have written it in the other way out, do you agree?} % Now the ramification jumps for a subgroup I thing are a little bit different from what you write. The lower ramification jumps for the subgroup $ \mathbb{Z}/p^{n-N} =G(N)$ are given by \begin{align*} I_0(N) &=i_0 + i_1 p + \cdots + i_{N} p^{t},\\ I_0(N) + p^{N+1} i_{i+1} &=I_0(N)+ p I_1(N),\\ I_0(N) + p I_1(N) + p^{N+2} i_{N+2} &= I_0(N) + p I_1(N) + p^2 I_2(N),\\ \ldots \end{align*} that is \begin{align*} I_0(N)&= i_0 + i_1 p + \cdots + i_{N} p^{N},\\ I_1(N)&= p^N i_{N+1}, \\ I_2(N)& = p^N i_{N+2},\\ \ldots \end{align*} This proves that if $\Gal(X/ X^{\prime} )= G(N)$ then \begin{align*} l_{X/X^{\prime},P}^{(t)} &=I_0 + I_1 p + \cdots + I_{t-1}p^{t-1}+1\\ &= i_0 + i_1 p + \cdots + i_{t+N-1} p^{t+N-1}+1\\ &= l_{X/Y,P}^{(t+N)} \end{align*} and \begin{align*} u_{X/X^{\prime},P}^{(t)} &=I_0 + I_1 + \cdots + I_{t-1}+1\\ &= (i_0 + i_1 p + \cdots + i_{N} p^{N})+ i_{N+1} p^N + \cdots + i_{N+t} p^N + 1\\ &= \end{align*} \vskip 2cm Assume now that $X \to Y$ is not \'{e}tale. Therefore $X \to X''$ is also not \'{e}tale. $\Gal(X / X'')= \ZZ/p$ and $\Gal(X /Y^{\prime} ) = \ZZ/p^{n-1}$. \[ \xymatrix{ & X \ar[dl]_{ \ZZ / p\cong \langle \sigma^{p^{n-1} } \rangle } \ar[dr]^{H^{\prime} =\langle \sigma^p \rangle \cong \ZZ / p^{n-1} =G(1) } \\ X '' \ar[dr]& & Y^{\prime} \ar[dl] \\ & Y } \] Note that for any $P \in X(k)$: {\color{blue} % \begin{equation} \label{eq:pul} p \cdot u^{(n)}_{X/Y, P} = u^{(n-1)}_{X/Y', P} + (p-1) \cdot l^{(1)}_{Y'/Y, Q}, \end{equation} % } Indeed, {\color{blue} blue color means that it is going to be erased} {\color{red} \begin{align} u^{(n)}_{X/ Y, P } &= (i_0 + i_1 + \cdots + i_{n-1} {\color{blue} +1 } ) \\ u^{(n-1)}_{X/Y', P} &= (i_0 + i_1 p) + i_2 p + \cdots + i_{1+n-1} p {\color{blue} +1} \\ \label{eq:l1} l^{(1)}_{Y'/Y, Q} &= u^{(1)}_{Y^{\prime} /Y,Q}= u^{(1)}_{(X/ Y,Q)} = l^{(1)}_{(X/ Y,Q)} = i_0 {\color{blue} + 1}. \end{align} } where $Q$ denotes the image of~$P$ in~$Y'$. Riemann Hurwitz formula for the cover $Y^{\prime} / Y$, together with eq. (\ref{eq:l1}), implies that \begin{equation} \label{eq:RH} 2(g_{Y^{\prime} }-1) = 2p(g_Y- 1) + \sum_{P\in Y^{\prime} (k)} (p-1)(l^{(1)}_{Y^{\prime} / Y}+1) \end{equation} By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules: % \[ \mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\ P \neq P_0}} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2 \oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}} \] % where $e'_P := e_{X/Y', P}$. % \begin{align*} \dim_k \mc T^i \mc M &= 2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 1)\\ & \stackrel{(\ref{eq:RH})}{=} 2 p (g_Y - 1) + \sum_{Q \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q}^{(1)} + 1)\\ &+ 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 1)\\ & \stackrel{(\ref{eq:pul})}{=} p \cdot \left( 2(g_Y - 1) + 2\# R {\color{red}/p} {\color{red}+ \frac{1}{p}\sum_{Q \in Y'(k)} (p-1) } + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1 {\color{red}/p} ) \right) \\ &= {\color{red} p \cdot \left( 2(g_Y - 1) + \# R + \sum_{P \in R} u_{X/Y, P}^{(n)} \right) } \end{align*} {\color{red} I guess that we want to combine $2\# R/ p + \frac{1}{p}\sum_{Q \in Y'(k)} (p-1)$ together. This depends on the ramification of all ramified points in $H^{\prime}$... } % where % \[ R := \{ P \in X(k) : e_P > 1 \} = \{ P \in X(k) : e'_P > 1 \}. \] % In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$. Thus by Lemma~\ref{lem:lemma_mcT_and_T} % \begin{align*} \dim_k T^1 \mc M &= \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M\\ &= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1). \end{align*} % By Lemma~\ref{lem:trace_surjective} since $X \to X''$ is not \'{e}tale, the map $\tr_{X/X''} : H^1_{dR}(X) \to H^1_{dR}(X'')$ is surjective. Recall that in $\FF_p[x]$ we have the identity: % \[ 1 + x + \ldots + x^{p-1} = (x - 1)^{p-1}. \] % Therefore in the group ring $k[H]$ we have: % \[ \tr_{X/X''} = \sum_{j = 0}^{p-1} (\sigma^{p^{n-1}})^j = (\sigma^{p^{n-1}} - 1)^{p-1} = (\sigma - 1)^{p^n - p^{n-1}}. \] % This implies that: % \[ \ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})} \] % and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus: % \[ \dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = .... \] % This ends the proof. \end{document}