Automorphy of Symm^5(GL(2)) and base change

We prove that for any Hecke eigenform f of level 1 and arbitrary weight there is a self-dual cuspidal automorphic form $\pi$ of $GL_6(\Q)$ corresponding to $\Symm^5 (f)$, i.e., such that the system of Galois representations attached to $\pi$ agrees with the 5-th symmetric power of the one attached to f. We also improve the base change result that we obtained in a previous work: for any newform f, and any totally real number field F (no extra assumptions on f or F), we prove the existence of base change relative to the extension $F/\Q$. Finally, we combine the previous results to deduce that base change also holds for $\Symm^5(f)$: for any Hecke eigenform f of level 1 and any totally real number field F, the automorphic form corresponding to $\Symm^5 (f)$ can be base changed to F.
Comment: 55 pages. Appendices A and B not included (cf. ArXiv preprints 1208.4128, 1209.5105, respect.). We have improved the exposition following the referee's suggestions. Some diagrams showing the congruences involved have been added at some steps for the reader's convenience. The proof of base change in Section 5 has been simplified