Take the 2-minute tour ×  Let $K$ be a finite extension of
$\mathbb{Q}_p$ and $L$ be a finite Galois extension of $K$. Let
$V$ be a $p$-adic representation of $G_L$ Let $D$ be the
$(\varphi, \Gamma)$-module associated to $V$ by Fontaine's
functor. Is there an easy way to describe the $(\varphi,
\Gamma)$-module associated to $Ind_{G_L}^{G_K} V$ in terms of
$D$ ?  share|improve this question add comment  2 Answers  up
vote 10 down vote accepted  If $L/K$ is unramified, then the
answer is exactly what Matt said. In the general case, you have
to take into account the fact that $\Gamma_K$ is larger than
$\Gamma_L$ and the construction is given in 2.2 of Ruochuan
Liu's "Cohomology and Duality for (phi,Gamma)-modules over the
Robba ring", see http://arxiv.org/abs/0711.4346  share|improve
this answer Thanks, it is exactly what I wanted to know. So
basically, on the $(\varphi, \Gamma)$-module side, the
construction is the "obvious" one. –  A M Dec 20 '10 at 18:12
add comment  I've always wondered this too. I think tha