What are the $p$-adic representations of $\hat{\mathbb{Z}}$ ? A continuous representation $\hat{\mathbb{Z}} \rightarrow GL_n(\mathbb{Q}_p)$ is determined by the image of $1$. But the image of $1$ does not always defines such a representation (consider for example the representation which sends $1$ on $p$ from $\mathbb{Z}$ to $GL_1(\mathbb{Q}_p)$). So my question is : what are the conditions on the image of $1$ ?
For example if $n=1$, then I know that $1$ must be sent on an element of $\mathbb{Z}_p^\times$, but I don't know if the converse is true.
EDIT: Correction about the example.
 A: I am going to write a community wiki answer here which people can vote up.
(See this meta thread concerning the Mathoverflow user,
which bumps questions with no voted-up answer.)
Main result: A homomorphism $f: \mathbb Z \to GL_n(\mathbb Q_p)$ extends continuously to
$\hat{\mathbb Z}$ if and only if the image of $f$ can be conjugated into $GL_n(\mathbb Z_p)$.
Proof:
If $f:\hat{\mathbb Z} \to GL_n(\mathbb Q_p)$ is continuous, the image is compact, hence contained 
in a maximal compact subgroup, which can be conjugated into $GL_n(\mathbb Z_p)$.
Conversely, if $f:\mathbb Z \to GL_n(\mathbb Q_p)$ lands in a compact subgroup,
then the closure of the image is compact, hence profinite (any compact subgroup of $GL_n(\mathbb Q_p)$ is profinite), and hence $f$ extends to $\hat{\mathbb Z}$
(since $\hat{\mathbb Z}$ is precisely the profinite completion of $\mathbb Z$).
QED
As noted in the comments, to tell if a matrix (e.g. $f(1)$) can be conjugated into
$GL_n(\mathbb Z_p)$, one simply has to look at the characteristic polynomial,
and ask that all the coefficients lie in $\mathbb Z_p$, with the constant term being
a unit.  Thus to apply the theorem in practice, one simply computes the characteristic polynomial of $f(1)$ and see if its satisfies these conditions.
EDIT: Now actually made community wiki; sorry about that --- I thought I had already clicked the CW box,
but obviously not.  (The point is that the above argument is just a rephrasing of what is in the comments.)
