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 $\operatorname{Ind}\_{G\_L}^{G\_K} V$ in terms of $D$?



Induced representations and $(\varphi, \Gamma)$-modules
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> 


