Icosahedral Fibres of the Symmetric Cube and Algebraicity

Let us begin with some motivation and consider a continuous irreducible representation ρ with trivial determinant of the absolute Galois group GF (of a number field F ) into GL(2,C) which is icosahedral, i.e., whose image in PGL(2,C) is the alternating group A5. Then it is well known that ρ is rational over K = Q[ √ 5], but not equivalent to its Galois conjugate ρ under the non-trivial automorphism of K. Moreover, one has (see [Kim2], [Wan], for example): (i) sym(ρ) ≃ sym(ρ), and (ii) sym(ρ) ≃ sym(ρ) ⊗ ρ. These two are the signature properties of such representations, and they lend themselves to natural automorphic analogs. Let π be a cuspidal automorphic representation of GL(2,AF ) with central character ω. For every m ≥ 1 one has its symmetric m-th power L-function L(s, π; symm), which is an Euler product over the places v of F , with the v-factors (for finite unramified v of norm qv) being given by