6
$\begingroup$

Let $E/\mathbb{Q}_p$ be a finite extension. Let $\rho$ be a continuous irreducible representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ on a two dimensional $E$-vector space. To $\rho$, via the p-adic Langlands correspondence, there is an associated $E$-Banach vector space representation of $GL_2(\mathbb{Q}_p)$, which we denote by $\Pi(\rho)$.

Is it known if this construction behaves well in $p$-adic families? More precisely, if $S$ is a reduced affinoid algebra and $\rho_S$ is a continuous $S$-linear representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ on a free rank two $S$-module is there an associated $S$-Banach module, $\Pi(\rho_S)$, equipped with a contiuous $S$-linear action of $GL_2(\mathbb{Q}_p)$, such that for all $x \in Sp(S)$, $S/x \otimes\Pi(\rho_S) \cong \Pi(\rho_{S,x})$?

$\endgroup$
2
  • 2
    $\begingroup$ I will remark that the classical local Langlands correspondence does not behave well in families. For if you consider a reducible representation $\chi_1\oplus\chi_2$ and then let $\chi_1$ and $\chi_2$ vary in a complex family, then occasionally $\chi_1/\chi_2$ can be $|.|^{\pm1}$ and in these cases the associated $\pi$ is 1-dimensional, so even the dimension of $\pi$ jumps. $\endgroup$ Apr 3, 2011 at 22:56
  • 3
    $\begingroup$ Dear Kevin, When considering the behaviour of the (classical) local Langlands correspondence in families, I think it is better to replace the correspondence as usually considered by the Breuil--Schneider correspondence, in which non-generic representations are replaced by the parabolic inductions of which they are the Langlands quotient. (This is discussed in my recent preprint with David Helm, for example.) I should add that the behaviour of this modified correspondence in families is still quite subtle (and this is the subject of my preprint with David). Best wishes, Matt $\endgroup$
    – Emerton
    Apr 4, 2011 at 2:46

1 Answer 1

8
$\begingroup$

There is a discussion of the behaviour of $\Pi(\rho_S)$ in Colmez's Asterisque paper which says that the answer is essentially yes. (Probably it deals with the case of a formal family rather than a rigid analytic one, though.) See also the discussion in section 3 of my paper on local-global compatibility, which presents the deformation-theoretic formulation of $p$-adic local Langlands as originally suggested by Kisin. (Again, this will handle formal families rather than rigid ones.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.