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A $p$-adic representation of a group $G$ is continuous group homomorphism $$\rho: G \to GL_n(\mathbb{Q}_p)$$

How are those representations related to $p$-adic analytic groups (= $p$-adic Lie groups)? (i.e. groups that have the additional structure of a $p$-adic manifold s.t. the group operation and the inverse operation are analytic functions)

I think that $GL_n(\mathbb{Q}_p)$ is itself a $p$-adic analytic group with the manifold structure derived from $\mathbb{Q}_p^{n^2}$. As preimage of the sub-manifold $U = im(\rho) \subseteq GL_n(\mathbb{Q}_p)$ under a continuous group homomorphism, should $G$ carry a manifold structure, too? And is continuity enough for transfering the manifold structure? Is $G$ therefore a $p$-adic analytic group?

Background: I'm trying to understand a consequence of the Fontaine-Mazur conjecture, that asserts - for a special group $G$ - the above $p$-adic representation to factor through a finite quotient.

Thank you in advance :-)

BIS HD
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    Sure, GLn(Qp) is a p-adic analytic group but that doesn't imply that G is p-adic analytic. E.g. consider the obvious map $\rho$ from the direct product GLn(Qp) x H to GLn(Qp), for any topological group H you like -- this will be continuous, but that doesn't imply anything about the topology of H. – David Loeffler Nov 19 '13 at 19:48
  • @DavidLoeffler Thank you for this hint! Ok, I see that one cannot "pull back" the manifold structure to $G$. If $G = G_S(k)$, ie. the absolute Galois group of a number field $k$, unramified outside a finite set $S$ of primes of $k$ that contains no primes dividing $p$, how can we relate $p$-adic analytic quotients of $G$ with the representation above? – BIS HD Nov 19 '13 at 19:52

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If $G$ is compact, then its image in $GL_n(\mathbb Q_p)$ is closed, and hence an analytic subgroup (via the $p$-adic version of a theorem of Cartan). This is most likely the statement being used in what you are reading about the Fontaine--Mazur conjecture.

user160609
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