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\begin{document}

\begin{center}
{\bf \Large \underline{$p$-adic Hodge Theory, MATH 847 Spring 2011}}\\ 
	{\large Homework Problems}
\end{center}

\begin{enumerate}



	\item Let $I$ be a directed  set and $\{G_i\}_{i\in I}$ an inverse system of finite groups with projection maps $\phi_{ij}:G_i\rightarrow G_j$ 
	for all $i,j\in I$ satisfying $j\le i$.
	  Give each $G_i$ the discrete topology and denote by $\pi$ the product
	$\pi:=\prod_{i\in I} G_i$ endowed with the product topology.     Define 
	$$G:=\varprojlim_{i\in I} G_i:=\{(g_i)_{i\in I}\ |\ \phi_{ij}(g_i) = g_j\ \text{for all}\ j\le i\}\subseteq \pi$$
	\begin{enumerate}
	  \item Show that $G$ is a closed subset of $\pi$.
	  
	  \item Give $G$ the subspace topology. Show that $G$ is compact and totally disconnected for this topology.
	  
	  \item Prove that the natural projection maps $\phi_i: G\rightarrow G_i$ are continuous, and that the (open) subgroups
	  $K_i:=\ker \phi_i$ for a basis of open neighborhoods of the identity. 
	  
	  \item Show that a subgroup of $G$ is open if and only if it is closed and of finite index.
	  \end{enumerate}

%\item Prove that $(1)\implies (2)$ below:
%\begin{enumerate}
%	\item	Let $K$ be an algebraic extension of $\Q$ that is unramified outside a finite set of places $S$,
%	and put $G:=\Gal(K/\Q)$.
%		The union over all $\ell\not\in S$ of the conjugacy classes $[\Frob_{\ell}]$ in $G$ is dense in $G$.
%	
%	\item The absolute Galois group $G_{\Q}$ is topologically generated by the decomposition groups
%	at any set of primes of density one.
%\end{enumerate}

\item Let $I\subseteq \Gal(\overline{\Q}_p/\Q_p)$ be the inertia subgroup and $W\subseteq I$ the wild inertia subgroup.  Show that there is a 
non-canonical isomorphism of topological groups
$$I/W\simeq \prod_{\ell\neq p} \Z_{\ell}.$$
%What can be said if one replaces $\Q_p$ with a general $p$-adic field $K$?

\item Let $\rho: G_{\Q}\rightarrow \GL_n(\Q_p)$ be a continuous representation.  Show that for all $\ell\neq p$, the image under $\rho$ of
any wild inertia group $W_{\ell}$ at $\ell$ is finite.  Is the same necessarily true of the image of any $I_{\ell}$?

\item Let $F$ be a finite extension of $\Q_{\ell}$, and 
suppose $\rho:G_F\rightarrow \GL_n(\Q_p)$ is a continuous representation.  Show that $\overline{F}^{\ker(\rho)}$ 
is infinitely (wildly) ramified if and only if  the image of (wild) inertia under $\rho$ is infinite.  

\item Do Exercise 1.4.3 in the notes.

\item Let $K$ be a $p$-adic field.  Show that the image of the $p$-adic cyclotomic character $\chi:G_K\rightarrow \Z_p^{\times}$
is closed.

\item Show that the two definitions of {\em continuous representation} given in Definition 1.2.1 of the notes really are equivalent.

\item Let $\rho:G_{\Q}\rightarrow \GL_n(\C)$ be a continuous representation.  
\begin{enumerate}
	\item Prove that up to conjugation by an element of $\GL_n(\C)$, the representation $\rho$ factors through $\GL_n(K)$ for some field $K$ of finite degree over $\Q$. 
	(You may use the fact that any compact, totally disconnected subgroup of $\GL_n(\C)$ is finite).
 
	\item Prove that we may take $K$ above to be an abelian extension of $\Q$. 	
	
	\item For a prime $p$, is it the case that any continuous $\rho:G_{\Q}\rightarrow \GL_n(\C_p)$
	must factor through $\GL_n(K)$ for some $K/\Q_p$ of finite degree? 
\end{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 Do Exercise 2.5.1 in the notes.
	
	
	  \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 Do Exercise 3.4.1 in the notes.
	
	\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 $\V_{\E}(\E/\O_{\E}) = \Q_p/\Z_p$.

\item Do Exercise 3.4.3 in the notes.

\item Let $\O_{\E}$ and $\O_{\E'}$ be two complete discrete valuation rings, each with fraction field
of characteristic zero and uniformizer $p$.  Let $\varphi$ and $\varphi'$ be endomorphisms of $\E$ and
$\E'$ which lift the $p$-power Frobenius map modulo $p$, and suppose that $f:\O_{\E}\rightarrow \O_{\E'}$
is a local homomorphism which intertwines $\varphi$ and $\varphi'$.  If the induced map on residue fields
$E\rightarrow E'$ is finite and {\em purely inseparable}, show that the base change map
$\Phi M^{\et}_{\O_{\E}}\rightarrow \Phi M^{\et}_{\O_{\E'}}$ given by sending $(M,\varphi_{M})$  
to $(M\otimes_{\O_{\E}} \O_{\E'}, \varphi_{M}\otimes\varphi)$ is an equivalence of categories.
%In other words, the category of \'etale $\varphi$-modules is insensitive to inseparable extansion
%of the residue field.



	  \item Do Exercise 4.5.1 in the notes.

	\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 4.5.2 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 (cf. Example 6.3.9 of the notes).
	  
	  \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 Do Exercise 4.5.3. 
	  
%	  \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}



	\item Let $K$ be a $p$-adic field.  This exercise gives an alternative way of seeing that $D_{\dR}:\Rep^{\dR}_{\Q_p}(G_K)\rightarrow \Fil_K$ is not full.
	\begin{enumerate}	
		\item Let $V,V'\in \Rep^{\dR}_{\Q_p}(G_K)$.  Prove that $D_{\dR}(V)$ and $D_{\dR}(V')$ are isomorphic in $\Fil_K$ if and only if
		$V$ and $V'$ have the same Hodge-Tate numbers; i.e. if and only if they have the same Hodge-Tate weights and for each Hodge-Tate weight
		$i$, the multiplicities $\dim_K \gr^i(D_{\dR}(V))$ and $\dim_K \gr^i(D_{\dR}(V'))$ are equal.
		
		\item Show that there exists a non-split extension of $\Q_p$ by $\Q_p(1)$ in $\Rep^{\dR}_{\Q_p}(G_K)$.  Hint: Think back to previous assignments.
		
		\item Show that $D_{\dR}$ can not be full.
		
	\end{enumerate}

	\item Let $F$ be a field.  Do one (or more) of the following:
	\begin{enumerate}
		\item For objects $D,D'$ of $\Fil_F$, show that the canonical $F$-linear isomorphism $D\otimes_F {D'}^*\simeq \Hom_F(D',D)$
		is an isomorphism in $\Fil_F$, where the tensor product is given its usual tensor-product filtration and $\Hom_F(D',D)$ is given the filtration
		$\Fil^i\Hom_F(D',D) := \Hom_{\Fil_F}(D',D[i])$.		
		\item Show that the canonical $F$-linear isomorphisms
		$$\det(D^*)\simeq \det(D)^*\qquad\text{and}\qquad \det(D\otimes D') \simeq \det(D)^{\dim_F D'} \otimes \det(D')^{\dim_F D}$$
		are isomorphisms in $\Fil_F$.
		
		\item Prove that for a short exact sequence in $\Fil_F$
		\begin{equation*}
			\xymatrix{
				0\ar[r] & D'\ar[r] & D\ar[r] & D''\ar[r] & 0
			}
		\end{equation*}
		the canonical $F$-linear isomorphism $\det(D')\otimes\det(D'')\simeq \det(D)$ is an isomorphism in $\Fil_F$.
	\end{enumerate}

	\item Let $n,m$ be positive integers and $K$ a $p$-adic field.  Show that if $V$ is any extension 
	\begin{equation*}
		\xymatrix{
			0\ar[r] & {\Q_p(n)} \ar[r] & V \ar[r] & {\Q_p(m) } \ar[r] & 0
		}
	\end{equation*}
	in $\Rep_{\Q_p}(G_K)$, then $V$ is always Hodge-Tate, and is de Rham if and only if $n>m$.  
	Hint: Use Corollary 6.3.4 and
	Examples 6.3.5--6.3.6 in the notes.  Make sure you understand Example 6.3.6!




\item Do Exercise 7.4.1 in the notes.

\item Do Exercise 7.4.2 in the notes.

\item Do Exercise 7.4.3 in the notes.

\item Do Exercise 7.4.5 in the notes.

\item Do Exercise 7.4.7 in the notes.

\item Do Exercise 7.4.9 in the notes.


	\item Let $D$ be a $K_0$-vector space with a $\sigma$-semilinear endomorphism $\phi:D\rightarrow D$.  If
	$D$ has finite $K_0$ dimension, show that $\phi$ is injective if and only if it is bijective.  Give a counterexample
	to this with $D$ of infinite dimension.
	
	\item Let $D$ be an isocrystal over $K_0$.  Prove that $t_N(D)=t_N(\det D)$.  Hint: first show that if $D(\alpha)$ and $D(\beta)$ are isoclinic
	of slopes $\alpha$ and $\beta$ respectively, then $D(\alpha)\otimes_{K_0} D(\beta)$ is isoclinic of slope $\alpha+\beta$.  Then
	work with a basis for $D$ adapted to the isoclinic decomposition of $D$ as guaranteed by Lemma 7.2.7.
	
	\item Let $D$ be a filtered $(\varphi,N)$-module over $K$.  Prove that $D$ is weakly admissible if and only if $D^*$ is.
	
	\item Let $h:M'\rightarrow M$ be a bijective morphism in $\Fil_K$.  Show that $h$ is an isomorphism in $\Fil_K$ if and only if
	$\det(h):\det(M')\rightarrow \det(M)$ is an isomorphism.
	
	
\item Do Exercise 8.4.1 of the notes.

\item Do Exercise 9.4.1 of the notes.

\item Do Exercise 9.4.2 of the notes.
	

\end{enumerate}  


	
	  
  








\end{document}
