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\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}  


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