Classification of l-adic representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:26:24Z http://mathoverflow.net/feeds/question/22753 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22753/classification-of-l-adic-representations Classification of l-adic representations Daniel Larsson 2010-04-27T17:45:33Z 2010-04-28T05:07:07Z <p>Either the following is a really stupid question or it is a really really stupid question, but here goes:</p> <p>Does there exist a classification of $\ell$-adic 2-dimensional representations of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, where $\ell\neq p$? </p> <p>I did a quick search of the internet that came up rather empty. </p> <p>What about the subtler case of $\ell=p$?</p> <p>References?</p> http://mathoverflow.net/questions/22753/classification-of-l-adic-representations/22765#22765 Answer by Emerton for Classification of l-adic representations Emerton 2010-04-27T19:24:01Z 2010-04-28T05:07:07Z <p>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.</p> <p>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.)</p> <p>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.</p> <p>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.)</p> <p>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.</p> <p>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. </p> <p>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 <a href="http://www.math.jussieu.fr/~colmez/kirilov.pdf" rel="nofollow">long text</a> on $p$-adic local Langlands.</p> http://mathoverflow.net/questions/22753/classification-of-l-adic-representations/22772#22772 Answer by JT for Classification of l-adic representations JT 2010-04-27T19:59:50Z 2010-04-27T19:59:50Z <p>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.</p> <p>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".</p>