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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.

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  • $\begingroup$ I guess that you want the $n$ in $\mathrm{GL}_n(\mathbb{Q}_p)$ to be something else, since you're using $n$ for the index variable of the sequence? $\endgroup$ – Keenan Kidwell Jun 29 '11 at 2:01
  • $\begingroup$ Re (3): No. Run the eigencurve example in the other direction, approximating a classical point by a sequence of nonclassical points. $\endgroup$ – David Hansen Jul 1 '11 at 2:30
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Both $2$ and $3$ are immediately false by considering limits of ($p$-adic) powers of the cyclotomic character.

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  • $\begingroup$ thanks. I didn't know integral powers of the cyclotomic characters satisfy all those properties (except Hodge-Tate !!!). Now I can understand why Berger required conditions on the Hodge-Tate weights of the $\rho_n$. $\endgroup$ – ndk Jun 29 '11 at 6:46
  • $\begingroup$ @ndk: Integral powers of cyclotomic characters, and direct sums of unramified twists thereof, are basic examples of crystalline representations. :) $\endgroup$ – David Hansen Jul 1 '11 at 2:39
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Regarding $(1)$, the paper Converging Sequences of p-adic Galois Representations and Density Theorems by Bellaiche-Chenevier-Khare-Larsen studies four different notions of convergence: (pointwise) trace convergence, (pointwise) physical convergence, uniform trace convergence, and uniform physical convergence. The definition you give is that of uniform physical convergence. The results in the first section of the paper are of the form "trace convergence implies physical convergence."

http://arxiv.org/abs/math/0406102

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