%%This is a standard LaTeX2e article document template. personal version 12/5/200%% \documentclass[10pt,twoside]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%packages%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagestyle{empty} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amstext} \usepackage{multicol} \usepackage[all]{xy} \usepackage{amsxtra} \usepackage[pdftex, breaklinks, linktocpage=true, bookmarksopen=true,bookmarksopenlevel=0,bookmarksnumbered=true]{hyperref} \hypersetup{ pdftitle = {Math 726 Assignment 1}, pdfauthor = {\textcopyright\ Bryden Cais}, pdfcreator = {\LaTeX\ with package \flqq hyperref\frqq}, colorlinks = {true} } \DeclareFontFamily{OT1}{rsfs}{} \DeclareFontShape{OT1}{rsfs}{n}{it}{<-> rsfs10}{} \DeclareMathAlphabet{\mathscr}{OT1}{rsfs}{n}{it} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%formatting%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\topmargin}{-0.3in} %%% This sets all the spacing stuff to use the page more \setlength{\oddsidemargin}{-0.5in} %%% efficiently than the normal "article" setup would. \setlength{\evensidemargin}{-0.5in} %%% It's OK to play with these some. \setlength{\textheight}{9.5in} %%% \setlength{\textwidth}{7.5in} %%% \setlength{\headsep}{0in} %%% \setlength{\headheight}{0in} %%% %\setlength{\footskip}{0in} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\cont}{cont} \DeclareMathOperator{\dR}{dR} \DeclareMathOperator{\Fil}{Fil} \newcommand{\Q}{\mathbf{Q}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\V}{\mathbf{V}} \newcommand{\D}{\mathbf{D}} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\mathbf{F}} \newcommand{\Kbar}{\overline{K}} \renewcommand{\O}{\mathscr{O}} \newcommand{\E}{\mathscr{E}} \usepackage[all]{xy} \begin{document} \begin{center} {\bf \Large \underline{$p$-adic Hodge Theory, MATH 726 Fall 2008}}\\ {\large Assignment 3} \end{center} \begin{enumerate} \item Let $K$ be a $p$-adic field and set $B_{\dR}^{\mathrm{naive}}:=\C_K(\!(t)\!)$, equipped with the $\C_K$-semilinear $G_K$-action defined by $g.t^n:=\chi^n(g)t^n$ where $\chi:G_K\rightarrow \Z_p^{\times}$ is the $p$-adic cyclotomic character. Give $B_{\dR}^{\mathrm{naive}}$ the $t$-adic filtration, so it becomes a filtered $\C_K$-vector space with semilinear $G_K$-action. We define $$D_{\dR}^{\mathrm{naive}}: \Rep_{\Q_p}(G_K)\rightarrow \Fil_K$$ by $D_{\dR}^{\mathrm{naive}}(V):= (V\otimes_{\Q_p} B_{\dR}^{\mathrm{naive}})^{G_K}$ with filtration induced by the filtration on $B_{\dR}^{\mathrm{naive}}$, and we call $V\in \Rep_{\Q_p}(G_K)$ ``naively de Rham" if $\dim_K D_{\dR}^{\mathrm{naive}}(V) = \dim_{\Q_p}(V)$. Prove that $V$ is naively de Rham if and only if it is Hodge-Tate. \item Let $K$ be a $2$-adic field, and consider any choice of $\epsilon = (1,\zeta_2,\zeta_4,\zeta_8,\ldots) \in R_K$, with $\{\zeta_{2^i}\}$ a collection of compatible primitive $2^{i}$ th roots of 1 in $\O_{\C_K}$. Show that $[\epsilon]-1\in W(R)$ is a generator of the principal ideal $\ker \theta$. Bonus: Show that the corresponding statement is {\em false} for $p>2$. \item Do Exercise 4.4.8 in the notes. \item Suppose $V\in \Rep_{\Q_p}(G_K)$ is 1-dimensional. Show that $V$ is Hodge-Tate if and only if it is de Rham. \item Prove that the Frobenius automorphism of $W(R)[1/p]$ does not preserve $\ker \theta_K$, and so does not naturally extend to $B_{\dR}^+$. \item Prove $W(R)\cap (\ker\theta_K)^j = (\ker\theta)^j$ for all $j\ge 1$. % \item ????? Let $K$ be a $p$-adic field. An $\O_{K}$-{\em pro-infinitesimal thickening} of $\O_{\C_K}$ is a pair % $(D,\theta)$ consisting of an $\O_K$-algebra $D$ together with a surjective homomorphism of $\O_K$-algebras % \begin{equation*} % \xymatrix{ % {\theta: D} \ar@{->>}[r] & {\O_{\C_K}} % } % \end{equation*} % such that $D$ is separated and complete for the $\ker\theta$-adic topology. Such pairs naturally form a category, with morphisms % $(D,\theta)\rightarrow(D',\theta')$ given by $\O_K$-algebra maps $\alpha:D\rightarrow D'$ respecting the given surjections onto $\O_{\C_K}$; % i.e. $\theta'\alpha=\theta$ as maps $D\rightrightarrows \O_{\C_{K}}$. Show that the pair $(D,\theta)$ given by % $D:=\O_K\otimes_{W(k)} W(R)$ and $\theta$ the map induced by the natural surjection $W(R)\twoheadrightarrow \O_{\C_K}$ % is ``the" initial object of the category of $\O_K$-pro-infinitesimal thickenings of $\O_{\C_K}$. Hint: You will have to % use the universal mapping property of $W(R)$ in the category of $p$-rings. It may be helpful to look at Fontaine's % Ast\'erisque article ``Expos\'e II: Le corps des p\'eriodes $p$-adiques." \medskip \noindent{The next two problems are taken from Berger's article ``An introduction to the theory of $p$-adic representations".} \item Let $K$ be a $p$-adic field, fix $q\in K$ with $|q|<1$ and set $E_q:=\Kbar^{\times}/q^{\Z}$, considered as a $G_K$-module through the action on $\Kbar^{\times}$. We saw on Assignment 2, problem 4 that $V_p(E_q):=\Q_p\otimes_{\Z_p}\varprojlim_r E_q[p^r]$ is 2-dimensional $\Q_p$-representation of $G_K$, and that \begin{equation*} e:=(\epsilon^{(r)})_{r\ge 0}\quad\text{and}\quad f:=(q^{(r)})_{r\ge 0} \end{equation*} give a basis of $V_p(E_q)$ where $\epsilon^{0}=1$, $\epsilon^{(1)}\neq 1$, $q^{(0)}=q$ and for all $r\ge 1$, we have $(\epsilon^{(r+1)})^p = \epsilon^{(r)}$ and $(q^{(r+1)})^p = q^{(r)}$. Denote by $\underline{\epsilon}$ and $\underline{q}$ the elements of $R$ defined by the $p$-power compatible sequences $(\epsilon^{(r)})$ and $(q^{(r)})$. \begin{enumerate} \item Show that $g.e= \chi(g)e$ and $g.f=f+c(g)e$ for some $c(g)\in \Z_p$ depending on $g$. \item Show that the series $\sum_{n\ge 1}(-1)^{n+1}\frac{([\underline{q}]/q -1)^n}{n}$ for $\log(\frac{1}{q}[\underline{q}])$ makes sense and converges in $B_{\dR}^+$. We define $$u:=\log_p(q) + \log(\frac{1}{q}[\underline{q}]).$$ Morally, $u=\log([\underline{q}])$. \item Let $t=\log([\underline{\epsilon}])\in B_{\dR}$. Show that $g.t = \chi(g)t$ and $g.u = u+c(g)t$ for $c(g)$ as in (1). \item Prove that $V_p(E_q)$ is de Rham. Hint: all you have to show is that the $K$-vector space $(B_{\dR}\otimes_{\Q_p} V_p(E_q))^{G_K}$ has dimension 2. Do this by using $u$ and $t$ to appropriately modify the $B_{\dR}$-basis $1\otimes e$ and $1\otimes f$ of $B_{\dR}\otimes_{\Q_p} V_p(E_q)$ to be $G_K$-invariant. \end{enumerate} \item We can generalize exercise (7). Let $V$ be any extension of $\Q_p$ by $\Q_p(1)$ in $\Rep_{\Q_p}(G_K)$. Prove that $V$ is de Rham as follows: \begin{enumerate} \item Let $\widehat{K^{\times}}$ be the projective limit $\varprojlim_n (K^{\times}/(K^{\times})^{p^n})$ with transition maps the natural projection maps. Fix a choice $(\epsilon^{(n)})$ of a compatible system of $p$-power roots of unity in $\overline{K}$ and Consider the map $\delta: \widehat{K^{\times}}\rightarrow H^1_{\mathrm{cont}}(G_K,\Z_p(1))$ defined as follows: for $q=q^{(0)}$ in $\widehat{K^{\times}}$, choose a sequence $(q^{(n)})_{n\ge 0}$ in $\overline{K}$ with $(q^{(n+1)})^p=q^{(n)}$ for all $n$ and let $\delta(q)$ be the cocycle $c$ determined by $g(q^{(n)})= (q^{(n)})\cdot (\epsilon^{(n)})^{c(g)}$. Show that any two choices of $(q^{(n)})$ give cohomologous cycles, so $\delta$ is well-defined. \item Prove that $\delta$ induces an isomorphism $\Q_p\otimes_{\Z_p} \widehat{K^{\times}} \simeq H^1_{\mathrm{cont}}(G_K,\Q_p(1))$. \item Look over your work on Assignment 2, problem 3 and convince yourself that $H^1_{\mathrm{cont}}(G_K,\Q_p(1))$ classifies isomorphism classes of $G_K$-extensions of $\Q_p$ by $\Q_p(1)$. Conclude that we can choose a basis $\{e,f\}$ of $V$ such that $g.e = \chi(g)e$ and $g.f = f + c(g)e$ where $c(g)$ is the cocycle corresponding to $q\in \Q_p\otimes \widehat{K^{\times}}$ as above. \item Defining $u=``\log([\underline{q}])"$ as above, show that we can appropriately modify the basis $\{1\otimes e,1\otimes f\}$ of $B_{\dR}\otimes_{\Q_p} V$ so as to be $G_K$-invariant. Conclude that $V$ is de Rham. \end{enumerate} \end{enumerate} \end{document}