I'm thinking about properties of "limits" of p-adic representations, in the following sense.


Notations: $p$ denotes a prime. For a field $F$, let $G\_F$ be the absolute Galois group of $F$. Representations are always continuous.


Definition: Let $\rho:\ G\_{\Bbb{Q}\_p}\rightarrow GL\_d(\Bbb{Q}\_p)$ be a representation, 
 $(\rho\_n:\ G\_{\Bbb{Q}\_p}\rightarrow GL\_d(\Bbb{Q}\_p))\_{n\geq 1}$ a sequence of representations. We say that the sequence converges uniformly to $\rho$ if it converges uniformly as a sequence of continuous functions. More explicitly, for every $\epsilon>0$,
for all sufficiently large $n$, and for every $\sigma$ in $G\_{\Bbb{Q}\_p}$, the (sup)norm of the matrix $\rho\_n(\sigma)-\rho(\sigma)$ is less than $\epsilon$. From now on, simly say $(\rho\_n)$ converges to $\rho$ (ie: drop the word "uniform"). 


Remark+ Questions:


(0) Obviously, we can define convergence for representations from topological groups over topological rings. But just stick with the case above. 


(1) Do we have an existing definition of limits of p-adic representations that are different (better?) than the naive one above?


(2) Assume the sequence $(\rho\_n)$ converges to $\rho$, and all (actually infinitely many) of them are de Rham/ potentially semistable / semistable / crystalline, can we say that $\rho$ has the same property?


(3) Conversely, if $\rho$ is DR/ pst/ st/ cryst, can we say that almost all of the $\rho\_n$ are too?


More remarks:


(1) A person gave me the following counterexample of (2) in the de Rham case: look at a nonclassical point $x$ on the eigencurve, it is the limit of points $x\_n$ that have big integer weights hence classical (Coleman's result). The restriction to $G\_{\Bbb{Q}\_p}$ of the representation induced by $x$ is not de Rham (because it is not classical, and by the Fontaine-Mazur conjecture (?)). But the restrictions of representations given by $x\_n$ are!


(2) I realize there is a paper of Laurent Berger "Limites De Representations Cristallines". Thm IV.2.1 in page 25, he proved that under extra conditions on the Hodge-Tate weights of the $\rho\_n$, then (2) holds for the crystalline case. It seems to me his word "limites" have the same meaning as in the definition above.



Limits of p-adic Representations
Both $2$ and $3$ are immediately false by considering limits of ($p$-adic) powers of the cyclotomic character.


