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\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}
