# Classify 2-dim p-adic galois representations

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...

• Your supposed classification in the 1-dimensional case is incorrect (e.g. the Groessencharacter attached to a CM elliptic curve, restricted to the decomposition group of a prime of good supersingular reduction, is a 1-dimensional crystalline representation $\phi$ but there is no $n$ such that $\phi \chi^n$ is unramified). Your classification is valid if $K = \mathbf{Q}_p$, but not for general finite extensions., and even for $K = \mathbf{Q}_p$ there is simply no hope of getting a simple classification like this for $n = 2$. – David Loeffler Jun 30 '19 at 9:34
• @DavidLoeffler Thank you for your correction. And I want to know what the classification is for 1-dim case for a general finite extension of $Q_p$...And is there some progress on 2-dim case? Thanks again. – Sssss Jun 30 '19 at 9:44
• Have you looked at any of the recent papers/preprints by Aubert, Herzig, et al.? – Jim Humphreys Jun 30 '19 at 12:00
• @JimHumphreys Sorry, I just finished Fontaine's book on p-adic Galois representations and have seen some notes on p-adic Hodge...So I'm not familiar with recent progress...But I'm interested in these things. – Sssss Jun 30 '19 at 13:13

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.)

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 $$\lambda>3$$.