Remarks on the Symmetric Powers of Cusp Forms on GL(2)

In this paper we prove the following conditional result: Let F be a number field, and π a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^(m)(π) are modular, i.e., define automorphic forms on GL(m + 1)/F. If sym^6(π) is cuspidal, then so are the sym^(m)(π), for all m. Moreover, sym^(6)(π) is Eisensteinian iff sym^(5)(π) is an abelian twist of the functorial product of π with the symmetric square of a cusp form π' on GL(2)/F.