On the Symmetric Powers of Cusp Forms on Gl(2) of Icosahedral Type

In this Note, we prove three theorems. Throughout, F will denote a number field with absolute Galois group GF = Gal(F̄ /F ), and the adele ring AF = F∞ × AF,f . When ρ is an irreducible continuous 2– dimensional C representation of GF , one says that it is icosahedral, resp. octahedral, resp. tetrahedral, resp. dihedral when the projective image of ρ(GF ) is A5, resp. S4, resp. A4, resp. D2m for some m ≥ 1. Such ρ is said to be modular if and only if there exists a cuspidal automorphic representation π = π∞ ⊗ πf of GL2(AF ) such that L(s, ρ) = L(s, πf). Modularity is unknown (in general) only when ρ is icosahedral, in which case ρ is rational over Q( √ 5). Denoting by τ the nontrivial automorphism of Q( √ 5), we will say that ρ is strongly modular if both ρ and ρ are modular. When F is totally real and ρ totally odd, which is the primary case of interest, modularity implies strong modularity (see below). In the Theorem below, sym(ρ) denotes the symmetric m-th power of ρ, i.e., the composition of ρ with the symmetric m-th power representation of GL2(C) into GLm+1(C).