%%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} \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 2} \end{center} \begin{enumerate} \item Let $\Gamma$ be a profinite group and $R$ a complete discrete valuation ring with fraction field $K$ that is a $p$-adic field. We suppose that $\Gamma$ acts on $R$ via continuous automorphisms (and hence also on $K$). Recall that if $V$ is a finite dimensional vector space over $K$, an $R$-lattice in $V$ is a finite free $R$-submodule $\Lambda$ of $V$ with the property that $\Lambda\otimes_{R} K \simeq V$. Show that any $V$ with semilinear $\Gamma$ action (i.e. $g(\alpha v) = g(\alpha)g(v)$ for all $\alpha\in K$ and $v\in V$) admits a $\Gamma$-stable $R$-lattice $\Lambda$ as follows: \begin{enumerate} \item Choose any $R$-lattice $\Lambda_0\subseteq V$. By choosing bases, show that $\Aut_{R}(\Lambda_0)$ is an open subgroup of $\Aut_{K}(V)$. \item Conclude that the preimage $\Gamma_0$ of $\Aut_{R}(\Lambda_0)$ in $\Gamma$ under the representation $\rho: \Gamma\rightarrow \Aut_{K}(V)$ is of finite index in $\Gamma$. \item Letting $\{\gamma_i\}$ be any finite set of coset representatives for $\Gamma/\Gamma_0$, show that the sum (taken inside $V$) $$\Sigma_{i} \rho(\gamma_i)\Lambda_0$$ is a $\Gamma$-stable $R$-lattice in $V$. \end{enumerate} \item Let $K$ be a $p$-adic field and $W\in \Rep_{\C_K}(G_K)$. Define the dual of $W$ by $W^*:=\Hom_{\C_K-\text{lin}}(W,\C_K)$ with $G_K$-action given by $g.\varphi(w):=g\varphi(g^{-1}w)$ (i.e. $W^*$ as a $\C_K$-vector space is the usual $\C_K$-linear dual of $W$). Verify that indeed $W^*\in \Rep_{\C_K}(G_K)$ and that $W^{**}\simeq W$ in $\Rep_{\C_K}(G_K)$. Show that $W^*$ is Hodge-Tate if and only if $W$ is. Hint: you may want to use the ``concrete" characterization of Hodge-Tate representations given in class. \item It may be helpful to know a little Galois cohomology for this exercise. I recommend looking at Tate's article \url{http://modular.math.washington.edu/Tables/Notes/tate-pcmi.html} or Serre's book. Let $\eta:G_K\rightarrow \Z_p^{\times}$ be any continuous character. Fix an extension \begin{equation} \xymatrix{ 0\ar[r] & {\C_K(\eta)} \ar[r] & W\ar[r] & {\C_K}\ar[r] & 0 }\label{ext} \end{equation} in $\Rep_{\C_K}(G_K)$. \begin{enumerate} \item By choosing a $\C_K$-linear vector space splitting of this exact sequence, show that we may identify $W$ with $\C_K(\eta)\oplus \C_K$ with $g\in G_K$-acting via $$g(v,\alpha)=(g.v+g\alpha\cdot\tau(g), g\alpha)$$ where $\tau:G_K\rightarrow \C_K(\eta)$ is a function satisfying $\tau(hg)=\eta(g)\tau(h)+\tau(g)$, i.e. $\tau$ is a 1-cocycle. \item Prove that $\tau$ is continuous, and that making a different choice of splitting alters $\tau$ by a coboundary. %Show also %that if $W'$ is any other extension equipped with a map $W'\rightarrow W$ in $\Rep_{\C_K}(G_K)$ that induced the identity %on $\C_K(\eta)$ and on $\C_K$, then the $\tau$ corresponding to $W'$ and the $\tau$ corresponding to $W$ \item Show that the association $W\rightsquigarrow \tau$ induces a bijection between isomorphism classes of extensions of $\C_K$ by $\C_K(\eta)$ and the set $H^1_{\cont}(G_K,\C_K(\eta))$. If you feel energetic, show that this is even an isomorphism of abelian groups, where we add two extensions by taking their Baer sum. \item Deduce from the Ax-Sen-Tate theorem that if $\eta(I_K)$ is infinite, then (\ref{ext}) splits (as an extension in $\Rep_{\C_K}(G_K)$!) and that this splitting is {\em unique}. \end{enumerate} % \item Stuff about $\C_K$: alg closed, Galois action etc \item Let $K$ be a $p$-adic field and fix $q\in K$ with $|q|<1$. Then $q^{\Z}:=\{q^n\ |\ n\in \Z\}$ is a discrete subgroup (lattice) of $\Kbar^{\times}$. Consider the quotient $E_q:=\Kbar^{\times}/q^{\Z}$; this abelian group admits a natural structure of $G_K$-module through the action on $\Kbar^{\times}$. For each $r\ge 0$, let $E_q[p^r]$ be the subgroup of $E_q$ consisting of $p^r$-torsion elements. \begin{enumerate} \item Let $\zeta$ be a primitive $p^r$-th root of unity and choose a $p^r$-th root $\xi$ of $q$ in $\Kbar^{\times}$. Show that the natural map $i_{\zeta,q}: (\Z/p^r\Z)^{2}\rightarrow E_q[p^r]$ induced by $$(m,n)\mapsto \xi^{n}\zeta^m\in \Kbar^{\times}$$ is an isomorphism of abelian groups. What happens to $\iota_{\zeta,\xi}$ if we change our choices of $\zeta$ and $\xi$? \item Define $T_p(E_q):=\varprojlim_r E_q[p^r]$ by using the natural multiplication by $p$ maps $E_q[p^{r+1}]\rightarrow E_qp^r]$. Show that $T_p(E_q)$ is a free $\Z_p$-module of rank 2 and gives a continuous 2-dimensional representation $\rho_{E_q}: G_K\rightarrow \GL_2(\Z_p)$. \item Set $V_p(E_q):=T_p(E_q)\otimes_{\Z_p} \Q_p$. Using (a), show that the natural maps $\Z/p^r\Z\rightarrow (\Z/p^r\Z)^2$ and $ (\Z/p^r\Z)^2\rightarrow \Z/p^r\Z$ given by $m\mapsto (m,0)$ and $(m,n)\mapsto n$ realize $V_p(E_q)$ as an extension of $\Q_p$ by $\Q_p(1)$, i.e. that we have a canonical exact sequence of continuous $G_K$-modules \begin{equation} \xymatrix{ 0\ar[r] & {\Q_p(1)}\ar[r] & {V_p(E_q)} \ar[r] & {\Q_p} \ar[r] & 0 }.\label{tate} \end{equation} \item Prove that $V_p(E_q)$ is Hodge-Tate. Hint: Use Problem (3). \item Prove that (\ref{tate}) is non-split as an extension of representations of $G_K$, even if we extend scalars to $\Kbar$. \end{enumerate} \item Let $K$ be a $p$-adic field with finite residue field $\F_q$. Pick $\alpha\in \GL_n(\C_K)$ and consider the uramified Galois representation defined by \begin{equation*} \xymatrix{ G_K \ar@{->>}[r] & G_{\F_q}\simeq \widehat{\Z} \ar[r] & \GL_n(\C_K) } \end{equation*} defined by sending $1\in \widehat{\Z}$ to $\alpha$. Show that this is a continuous representation if and only if all eigenvalues of the matrix $\alpha$ have absolute value 1. Use this to give an example of a continuous, n-dimensional $G_K$-representation with $\C_K$ coefficients that does not factor through $\GL_n(L)$ for any algebraic extension $L/K$. \item Let $K$ be a $p$-adic field containing $\mu_p$ and let $\chi: G_K\rightarrow \Z_p^{\times}$ be the cyclotomoc character. \begin{enumerate} \item Show that $\chi$ has image in $1+p\Z_p$. \item For any $s\in \Z_p$, show that the character $\chi^s$ of $G_K$ defined by the composition of $\chi$ with the map $1+p\Z_p\rightarrow 1+p\Z_p$ given by $x\mapsto x^s$ makes sense and is continuous. %(where we define $x^s$ as $\exp(s\log x)$ using the usual power series %for $\exp$ and $\log$ on $\Z_p$ %logarithm map $1+p\Z_p\rightarrow \Z_p$ defined by $\log(x):=\sum_{n\ge 1} \frac{(1-x)^n}{n}$ and the exponential map %defined by $\exp(t):=\sum_{n\ge 0} \frac{t^n}{n!}$. \item Prove that $\chi^s$ is Hodge-Tate if and only if $s\in \Z$. \end{enumerate} %\item Prove tensor and duality statements in Thm 2.3.9. \item Fix a $p$-adic field and let $\eta$ be a nontrivial finite order continuous character $\eta:G_K\rightarrow \Q_p^{\times}$. \begin{enumerate} \item Show that $\eta$ factors through the natural inclusion $\Z_p^{\times}\hookrightarrow \Q_p^{\times}$. \item Prove that there are no nonzero $G_K$-homomorphisms $K\rightarrow K(\eta)$. \item Suppose that $L/K$ is finite Galois and the restriction of $\eta$ to $G_L$ is trivial. Show that there exists a nonzero homomorphism $L\rightarrow L(\eta)$ of $L$-modules with semilinear $G_K$-action, and hence that these two $G_K$-modules are isomorphic. \end{enumerate} \item Fix a field $E$ of characteristic $p$ and let $(M,\varphi_M)$ be an \'etale $\varphi$-module over $E$. Define $M^{\vee}$ to be the $E$-linear dual of $M$ and let $\varphi_{M^{\vee}}$ be the map \begin{equation} \xymatrix{ {M^{\vee}} \ar[r] & {(\varphi_E^*(M))^{\vee}} \ar[r] & {M^{\vee}} }\label{duality} \end{equation} where the first map takes a linear functional $\ell$ on $M$ to the linear functional on $\varphi^*_E(M):=M\otimes_{E,\varphi_E} E$ given by $m\otimes e\mapsto \varphi_E(\ell(m)) e$, and the second map is the $E$-linear dual of the inverse of the $E$-linear isomorphism $\varphi^*_E(M)\rightarrow M$ given by the linearization of $\varphi_M$. Prove that $\varphi_{M^{\vee}}$ is semilinear over $\varphi_E$, and that its linearization is an isomorphism. Hint: show that the linearization of first map in (\ref{duality}) is the canonical isomorphism \begin{equation*} \varphi_E^*(M^{\vee})=\Hom_E(M,E)\otimes_{E,\varphi} E \simeq \Hom_E(M,E_{\varphi}) \simeq \Hom_{\varphi-{\mathrm{sl}}}(M,E) \simeq \Hom_E(\varphi^*_E(M),E)= \varphi^*_E(M)^{\vee} \end{equation*} where $E_{\varphi}$ denotes $E$ as an $E$-module via $\varphi_E$, and $\Hom_{\varphi-{\mathrm{sl}}}$ is the $E$-module of $\varphi_E$-semilinear $E$-module homomorphisms. \item Let $M$ be any \'etale $\varphi$-module over $\O_{\E}$. Show that $\V_{\E}(M)$ is continuous as a $G_E$-representation. % \item Let Show that the $G_E$ action on $E_s\otimes V$ is cts: i.e. has open stabilizers. % \item Galois descent: infinite case from finite case % \item Decription of $\O_{\E}^{un}$ when $E=k((u))$? Will need alg. closeure of $k((u))$. \item Let $E=\F_p$, so $G_E\simeq \widehat{\Z}$. Let $\rho:G_E\rightarrow \Aut_{\F_p}(V)$ be a continuous representation on a $d$-dimensional $\F_p$-vector space $V$, and let $(M,\varphi_M)=\D_E(V)$ be the associated \'etale $\varphi$-module over $E$. Since $E=\F_p$, we canonically have $\varphi^*(M)=M$ so that the linearlization $\varphi_M^{\mathrm{lin}}$ of $\varphi_M$ is an $\F_p$-linear endomorphism of the $d$-dimensional $\F_p$-vector space $M$. Identifying $G_E$ with $\widehat{Z}$ show that $\det(\rho(1))$ is the inverse of $\det(\varphi_M^{\mathrm{lin}})$. \item Fix a pair $(\O_{\E},\varphi)$ as in the notes and let $(M,\varphi_M)$ be a $\varphi$-module over $\O_{\E}$; i.e. a finitely generated $\O_{\E}$-module with a $\varphi$-semilinear endomorphism $\varphi_M:M\rightarrow M$. Show that $\varphi_M$ is \'etale if and only if $\varphi_M \bmod p$ is \'etale. Hint: first show that $M$ and $\varphi^*(M)$ are abstractly isomorphic as $\O_E$-modules---i.e. that they have the same rank and invariant factors. Conclude that $\varphi_M$ is an isomorphism if and only if it is surjective, and show that surjectivity may be checked modulo $p$. \item Let $M$ be a finitely generated module over a complete discrete valuation ring $R$ of characteristic zero with uniformizer $p$. Suppose that $G$ is a monoid acting on $R$ by ring endomorphisms and on $M$ by semilinear module endomorphisms. Show that for each $n$, $G$ acts on $M/p^nM$ and that $\varprojlim_n (M/p^n)^G = M^G$. \item Prove that $\V_{\E}(\E/\O_{\E}) = \Q_p/\Z_p$. \end{enumerate} \end{document}