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\begin{document}

\begin{center}
{\bf \Large \underline{$p$-adic Hodge Theory, MATH 726 Fall 2008}}\\ 
	{\large Assignment 1}
\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 1.2.5 in the notes.

\item Do Exercise 1.3.2 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}  


\end{enumerate}
\end{document}
