7
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ 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 $n = 2$. $\endgroup$ – David Loeffler Jun 30 '19 at 9:34
  • $\begingroup$ @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. $\endgroup$ – Sssss Jun 30 '19 at 9:44
  • $\begingroup$ Have you looked at any of the recent papers/preprints by Aubert, Herzig, et al.? $\endgroup$ – Jim Humphreys Jun 30 '19 at 12:00
  • $\begingroup$ @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. $\endgroup$ – Sssss Jun 30 '19 at 13:13
18
$\begingroup$

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.)

$\endgroup$
4
$\begingroup$

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 $\lambda>3$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.