Icosahedral Fibres of the Symmetric Cube and Algebraicity

Abstract

For any number field F, call a cusp form \pi on GL(2)/F {\it special icosahedral}, or just s-icosahedral for short, if \pi is not solvable polyhedral, and for a suitable "conjugate" cusp form \pi' on GL(2)/F, sym^3(\pi) is isomorphic to sym^3(\pi'), and the symmetric fifth power L-series of \pi equals the Rankin-Selberg L-function L(s, sym^2(\pi') x \pi) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let \pi be s-icosahedral (of trivial central character). Then \pi_f is algebraic without local components of Steinberg type, \pi_\infty is of Galois type, and \pi_v is tempered everywhere. Moreover, if \pi' is also of trivial central character, it is s-icosahedral as well, and the field of rationality \Q(\pi_f) (of \pi_f) is K:=\Q[\sqrt{5}], with \pi'_f being the Galois conjugate of \pi_f under the non-trivial automorphism of K.