Classify 2-dim p-adic galois representations

Question

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.

Answer 1

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"; see also this question and this question.

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. But this tells you nothing about what the non-de Rham guys look like.)

Answer 2

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 $\lambda>3$.