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 $Ind_{G_L}^{G_K}$ Ind_{G_L}^{G_K} V$in terms of$D$? 1 # Induced representations and$(\varphi, \Gamma)-modules
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 $Ind_{G_L}^{G_K}$ in terms of $D$ ?