diff --git a/ramification.tex b/ramification.tex new file mode 100644 index 0000000..0d80184 --- /dev/null +++ b/ramification.tex @@ -0,0 +1,492 @@ +\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} \ No newline at end of file