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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 $\ell=p$?


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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
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2 Answers

up vote 7 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.

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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: ams.org/publications/online-books/pspum332-index –  Rob Harron Apr 27 '10 at 20:23
Brilliant! Thanks. –  Daniel Larsson Apr 27 '10 at 21:02
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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".

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