Icosahedral Fibres of the Symmetric Cube and Algebraicity

Abstract

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