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. 


Thanks!



$p$-adic Langlands correspondence
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](http://www.math.northwestern.edu/~emerton/preprints.html) . 


Since $GL\_2(F)$ is locally pro-$p$ this problem doesnot arrise if you are working over $\mathbb C$ or $\mathbb F\_l$, $l\neq p$.


