Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G\_K$ on a p-adic field $E$, where $K$ is also a p-adic field. And a p-adic field means a finite extension of $Q\_p$.


(1) $\phi$ is hodge-Tate if and only if $\phi$ is de Rham if and only if $\phi\chi^n$ is potentially unramified for some integer $n$. 


(2)$\phi$ is crystalline if and only if $\phi$ is semi-stable if and only if $\phi\chi^n$ is unramified for some integer $n$.


Symbols: $\chi$ is the cyclotomic character of $G\_K$.



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> I want to know how to classify 2-dim p-adic Galois representations. When it is Hodge-Tate or de Rham or crystalline or semi-stable?
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Is there some reference on this? I prefer English reference to Frence reference because I am a beginner on French...


Thanks for any answers.



Classify 2-dim p-adic galois representations
The classification of 1-dimensional representations, i.e. characters $G\_K \to E^\times$, is a bit more complicated than you imply in your question. Any such character lands in $O\_E^\times$ by compactness; and since $O\_E^\times$ is profinite, and class field theory identifies the abelianisation of $G\_K$ with the profinite completion of $K^\times$, we conclude that there is a canonical bijection between continuous characters $G\_K \to O\_E^\times$ and continuous characters $K^\times \to O\_E^\times$.


In terms of this bijection, we can read off which characters are crystalline, semistable, de Rham or Hodge--Tate using the restriction of the character to $O\_K^\times$: the details are given in Appendix B of [B. Conrad, "Lifting global representations with local properties"](http://math.stanford.edu/~conrad/papers/locchar.pdf); see also [this question](https://mathoverflow.net/questions/98103) and [this question](https://mathoverflow.net/questions/61998/crystalline-characters).


As for $n = 2$, the best answer I can give is that for $K = \mathbf{Q}\_p$, there is a bijection between 2-dimensional representations of $G\_{\mathbf{Q}\_p}$ and certain $p$-adic Banach space representations of $GL\_2(\mathbf{Q}\_p)$: this is the $p$-adic local Langlands correspondence of Colmez. You can then classify which 2-dimensional reps are de Rham / crystalline in terms of their associated $GL\_2$ representation. However, this is a bijection between one kind of formidably complicated object and another kind of equally complicated object; there's no hope of getting a down-to-earth parametrisation of all the representations involved. Moreover, extending this correspondence to 2-dimensional reps of $G\_K$, for arbitrary $K$, or to 3-dimensional reps of $G\_{\mathbf{Q}\_p}$, is an open problem despite a decade or more of very intensive effort.


(Another viewpoint: if you just want to classify crystalline / semistable / de Rham reps, but you don't necessarily ask for a classification of *all* reps and how the crys/ss/dR ones sit inside that, then you can do this in some cases by classifying all filtered $\varphi$-modules, $(\varphi, N)$-modules, etc satisfying the relevant conditions. See e.g. [this paper of Dousmanis](https://arxiv.org/pdf/0711.2137.pdf). But this tells you nothing about what the non-de Rham guys look like.) 


