MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Either the following is a really stupid question or it is a really really stupid question, but here goes:

Does there exist a classification of $\ell$-adic 2-dimensional representations of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, where $\ell\neq p$?

I did a quick search of the internet that came up rather empty.

What about the subtler case of $\ell=p$?


share|cite|improve this question
False -- it is not a stupid question at all! I am not qualified to answer it, but the case $\ell = p$ involves the $p$-adic local Langlands correspondence, a topic of recent research interest. – Pete L. Clark Apr 27 '10 at 18:06
I have a nagging suspicion though that the question as stated is way too "wild"; probably, I should've restricted my attention to some subclass of representations (such as potentially semi-stable ones). But I'll let it stand as it is for now. – Daniel Larsson Apr 27 '10 at 18:20
Daniel: every ell-adic representation of the absolute Galois group of Q_p is potentially semi-stable, in the sense that a finite index subgropu of inertia will act as unipotent matrices. There is always a Weil-Deligne representation attached to the representation, for example. – Kevin Buzzard Apr 27 '10 at 20:57
Kevin: is this really true? Given a pst-representation we get a Weil--Deligne representation, but is it true the other way around also? I'm out on very thin ice here so there can certainly be something that I'm missing. – Daniel Larsson Apr 27 '10 at 21:15
Let me stress that I'm talking about ell not p! Take a look at Tate's article in Corvallis; it explains the construction, due to Deligne. Given the Galois representation you get a Weil-Deligne representation. Conversely, given a Weil-Deligne representation such that the eigenvalues of Frobenius are ell-adic units, you get a Galois representation. – Kevin Buzzard Apr 27 '10 at 21:26
up vote 10 down vote accepted

When $\ell \neq p,$ these are rather straightforward to classify (except when $p = 2$); see Tate's article in the second volume of Corvallis, for example.

The idea is that if $\rho$ is irred., then (unless $p = 2$), it must be induced from a character of a quadratic extension; thus the classification is given by local class field theory for quadratic extensions of $\mathbb Q_p$. (When $p = 2$, there are some exceptional irreps. that are not induced.)

If $\rho$ is reducible, it is an extension of characters. The characters of $\mathbb Q_p^{\times}$ are classified by local class field theory of $\mathbb Q_p$. There are lots of ways to compute the possible extensions; Tate local duality/local Euler char. formula gives one way.

When $\ell = p$, these are classified in terms of etale $(\phi,\Gamma)$-modules. To learn about this, you can e.g. read one of many expository articles on Laurent Berger's website. (In fact there are many recent papers by Berger, Breuil, and Colmez involving $(\phi,\Gamma)$-modules, all online, and most of them include an introductory page or two recalling the basics of the theory.)

Pete is correct that this $\ell = p$ case is also the starting point of $p$-adic Langlands, just as the case $\ell \neq p$ is related to classical local Langlands. However, as the above discussion shows, you don't need any Langlands theory to classify these reps.

Added: As JT points out in another answer, the (potentially) semi-stable representations also admit a nice classification, in terms of weakly admissible filtered $(\phi,N)$-modules.

Note that $(\phi,\Gamma)$-modules are themselves pretty nice objects. What is perhaps the most complicated part of the story is how, in the case of a potentially semi-stable representation, one compares its $(\phi,\Gamma)$-module description to its weakly admissible filtered $(\phi,N)$-module description. In the case of crystalline reps., this comparison is made via the theory of Wach modules. In general, it plays an important role in $p$-adic local Langlands, as well as in local Iwasawa theory. Laurent Berger has a number of papers discssing it (beginning with his thesis), and in the case of two-dimensional pst representations it is the subject of the most technical part (Chapter VI) of Colmez's recent long text on $p$-adic local Langlands.

share|cite|improve this answer
Thanks! Do you have another reference other than the above by Tate? One that I can easily find on the web perchance? – Daniel Larsson Apr 27 '10 at 19:37
The entire corvallis proceedings are available online for free at the AMS website. Volume 2 is here: – Rob Harron Apr 27 '10 at 20:23
Brilliant! Thanks. – Daniel Larsson Apr 27 '10 at 21:02

A small post script to Emerton's post (that would not fit in the comment box): as you suggest, there is a nicer (easier to understand) classification of potentially semi-stable representations. Basically the idea is that via B_st semi-stable representations are easy to understand, and a potentially semi-stable representation can be given in terms of a semi-stable representation of some field extension and a descent datum, to get you back to where you started.

A nice exposition of the potentially crystalline case (with a nice application) can be found in Volkov's paper, "A class of p-adic Galois representations arising from abelian varieties over Q_p".

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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