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Basic question: Is it correct that the $p$-adic Langlands correspondence is known for $GL_2$ only over $Q_p$ but not other $p$-adic fields? If so, I would like to request some light to be shed on this restriction, i.e., why only $Q_p$ but not its extensions. Any reference for this (and prospects) would be much appreciated.


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Start with Pierre Berger's recent Bourbaki exposé 1017 arxiv.org/abs/1002.4111. –  Chandan Singh Dalawat Apr 15 '11 at 2:28
If you read the proofs of the construction of the $p$-adic Langlands correspondence, you will see that the restriction $F=\mathbb Q_{p}$ is ubiquitous, starting from (but not restricted to) the fact that you want the residual field to be cyclic. @Chandan Pierre Berger is an entrepreneur and benefactor unlikely ever to contribute to Bourbaki, it is of course Laurent you want. –  Olivier Apr 15 '11 at 3:46
Olivier : quelle bêtise de ma part ! C'est inexcusable, d'autant plus que je connais Laurent Berger, et j'ai beaucoup entendu parler de l'ami Pierre Bergé du feu Yves Saint-Laurent... Je demande pardon à Laurent. –  Chandan Singh Dalawat Apr 15 '11 at 13:44
@Chandan : tu es tout excusé! –  Laurent Berger Apr 15 '11 at 13:52
Breuil has written an article for the ICM ("The emerging p-adic Langlands programme" see his webpage math.u-psud.fr/~breuil/publications.html), where he explains the difficulties in going from $Q_p$ to an extension. –  Laurent Berger Apr 15 '11 at 13:54
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2 Answers

up vote 18 down vote accepted

Yes, this is correct.

The problem is that when you replace $Q_p$ by an extension, the dimension of $GL_2(F)$ as a $p$-adic analytic group increases. This also means that the cohomological dimension of its open subgroups increases. This leads to representation theory of $GL_2(F)$ of being much more complicated than $GL_2(Q_p)$. For example, smooth irreducible $\overline{\mathbb F}_p$-representations have not been classified if $F\neq Q_p$.

Prototypical examle: Let $\mathbb G=\mathbb G_a$, and let $K=\mathbb{G}(\mathcal O_F)$. So that $K$ is $(\mathcal O_F, +)$. Then the completed group agebra $\mathcal O[[K]]$ is isomorphic to $\mathcal O[[x_1, ..., x_d]]$, where $d=[F:Q_p]$, where $\mathcal O$ is a ring of inegers in a finite extension of $Q_p$. The theory of modules of $\mathcal O[[K]]$ is much easier, when $d=1$. If you want to see this in action have a look at Emerton's "On a class of coherent rings, with applications to the smooth representation theory of GL_2(Q_p) in characteristic p", available on his website .

Since $GL_2(F)$ is locally pro-$p$ this problem doesnot arrise if you are working over $\mathbb C$ or $\mathbb F_l$, $l\neq p$.

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Many thanks for your very clear answer! –  SGP Apr 15 '11 at 21:09
Thanks a lot for a very clear answer! I wish I could accept more than one answer! –  SGP Apr 15 '11 at 21:14
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Regarding prospects for extending the correspondence to $GL_2(F)$ for other $F$, one could look at Paškūnas's papers "Coefficient systems and supersingular representations", "Towards a modulo $p$ Langlands correspondence for $GL_2(F)$" (joint with C. Breuil), and "Admissible unitary completions of locally $\mathbb Q_p$-rational representations of $GL_2(F)$", available on his website and/or the arXiv.

There is also Breuil's ICM talk from last summer, "The emerging $p$-adic Langlands program", available at his website. This gives a very nice survey of the whole state of the theory (which has remained relatively stable since then).

Some commentary on Paškūnas's papers: In the $GL_2(\mathbb Q_p)$ case, Breuil found that the numbers of irred. supersingular reps. of $GL_2(\mathbb Q_p)$ mod $p$ matches with the numbers of $2$-dim'l irred. mod $p$ reps. of $G_{\mathbb Q_p}$, and that there is even a natural way to match them (which is e.g. compatible with Serre's conjecture on weights of modular forms giving rise to mod $p$ global Galois reps.).

It was then natural to conjecture that the same was true for $GL_2(F)$. The first of these papers has the goal of verifying this conjecture. Indeed, it succeeds in constructing the right number of supersingular reps. mod $p$ of $GL_2(F)$. However, it was later realized that there was no way to match these with irred. Galois reps. in any way that is compatible with the Buzzard--Diamond--Jarvis (BDJ) conjecture (the generalization of Serre's conjecture to Hilbert modular forms).

The second paper extends the techniques of the first, and shows in fact that when $F \neq \mathbb Q_p$ there are many, many more supersingulars than there are $2$-dim'l. irreps of $G_F$. It attempts to find order among this chaos by identifying certain classes of supersingulars which seem to have something to do with the Galois side (in the sense that they match with the predictions of the BDJ conjecture).

The third paper shows how to lift mod $p$ representations to $p$-adic Banach spaces representations in interesting ways, and so can be thought of as (i) giving some evidence that there will be a $p$-adic local Langlands for $GL_2(F)$, but also (ii) showing that understanding it will be at least as difficult as understanding the mod $p$ situation.

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A very minor comment: I assume 'Breuil' instead of 'Breiul' is meant. –  quid Apr 15 '11 at 13:02
I have taken the liberty of correcting the spelling. –  Chandan Singh Dalawat Apr 15 '11 at 13:46
Thank you both! –  Emerton Apr 15 '11 at 16:37
Thank you for a very clear and detailed answer! Thanks also for providing clear answers to many many questions on MO! –  SGP Apr 15 '11 at 21:11
I wish I could select both yours and Vytas answers. Thanks again –  SGP Apr 15 '11 at 21:13
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