Classification of l-adic representations

2010-04-27T17:45:33Z

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 $\ell=p$?

References?

Answer by Emerton for Classification of l-adic representations

2010-04-27T19:24:01Z

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.

Answer by JT for Classification of l-adic representations

2010-04-27T19:59:50Z

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".

Classification of l-adic representations - MathOverflow

Classification of l-adic representations

Answer by Emerton for Classification of l-adic representations

Answer by JT for Classification of l-adic representations