Let $F$ be a number field and $E/F$ a Galois extension. Suppose we have a representation $\rho\_E : Gal(\overline{F}/E) \rightarrow GL\_n(\overline{Q}\_p)$. My question is : what are sufficiant conditions so that $\rho\_E$ can be extended to a representation $ Gal(\overline{F}/F) \rightarrow GL\_n(\overline{Q}\_p)$ ?


A necessary condition is that $\rho\_E$ is invariant under $Gal(E/F)$. This paper (last line of page 1)


<http://www.institut.math.jussieu.fr/projets/fa/bpFiles/GaloisPatching_Harris.pdf>


claims that such an extension exists if moreover $\rho\_E$ is irreducible and $E/F$ is cyclic of prime ordre, but I don't know why it is true. 



How can we extend Galois representations ?
Pick a $\sigma \in \Gamma\_F$ such that $\sigma |\_E$ generates $\mathrm{Gal}(E/F)$.
There is an $A \in \mathrm{GL}\_n$ such that for all $x \in \Gamma\_E$, $\rho(\sigma x \sigma^{-1}) = A \rho(x) A^{-1}$, and if $q$ denotes the order of $\mathrm{Gal}(E/F)$, we get that $\rho(\sigma^q x \sigma^{-q}) = A^q \rho(x) A^{-q}$, which by irreducibility and Schur's lemma implies that $\rho(\sigma^q) A^{-q} = \lambda \mathrm{Id}$.
Choose any $\mu$ such that $\mu^q = \lambda$.
Then setting $\rho(\sigma)=\mu A$ is one of the $q$ ways to extend $\rho$.


