TITLE: Classify 2-dim p-adic galois representations
QUESTION [7 upvotes]: 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 $G_K$.

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?

Is there some reference on this? I prefer English reference to Frence reference because I am a beginner on French...
Thanks for any answers.

REPLY [3 votes]: As Loeffler points out, the classification of $2$-dimensional $p$-adic crystalline Galois representations is complicated. There has been considerable interest in classifying (in a way that is explicit and thus what you are probably seeking) the residual representation $\bar{\rho}_{\restriction D_p}^{ss}$ as $\rho=\rho_f$ ranges over the crystalline Galois representations associated to a cuspidal Hecke-eigenform $f$. Let me quote Buzzard and Gee from "Explicit Reduction Mod $p$ of Certain $2$-dimensional Crystalline Representations"
"Question: If $f=\sum_n a_n q^n$ be a normalized cuspidal level $N$ eigenform and $p$ is a prime, and if $\bar{\rho}_f$ is the associated semisimple representation, then can one explicitly read off $\bar{\rho_f}_{\restriction D_p}^{ss}$ from the weight character and $q$-expansion of $f$?"
This question is interesting when $f$ ranges over eigenforms which are supersingular at $p$ and has proved to be a whole lot more tractable and yielded interesting results though is not yet settled. Let $\lambda\in \mathbb{Q}_{>0}$ be the slope of $a_p$ (with respect to the normalization with respect to which the slope of $p$ is 1 and a chosen embedding of the field of fourier coefficients of the eigenform). As far as I''m aware the classification is for $\lambda<3$ and the number of possibilities increases considerably as $\lambda$ exceeds $3$. 
This is a culmination of the results of Berger-Li-Zhu, Buzzard-Gee, Ganguli-Ghate and Pande. Thus in response to what you probably have in mind, the most natural step to pursue would be to try and extend this classification to some slopes $\lambda>3$.