On the symmetric powers of cusp forms on $GL(2)$ of icosahedral type

In this note we study the symmetric powers of strongly modular icosahedral representations $\rho$ of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{F}\mathrm{/}F)$, $F$ a number field, and their twisted $L$–functions. We prove that for such $\rho$, there exists a cuspidal automorphic representation $\mathrm{\Pi }={\mathrm{\Pi }}_{\mathrm{\infty }}\otimes {\mathrm{\Pi }}_f$ of $G{L}_6({\mathbb{A}}_F)$ such that $L (s, {\rm sym}^{5} (\rho)) = L (s, \Pi_{f})$\mathrm{.}Oneseesthat${\rm sym}^{5} (\rho)$istwistequivalentto$\rho’ \otimes {\rm sym}^{2} (\rho)$foranothermodularicosahedralrepresentation$\rho’$, and our theorem is a special case of a cuspidality criterion formulated and proved in this paper, which may be of independent interest, for the Kim–Shahidi automorphic tensor product $\pi \boxtimes {\rm sym}^{2} (\pi’)$,where$\pi$and$\pi’$arecuspidalautomorphicrepresentationsof$GL (2) / F$. We also give a complete structure theory of modular icosahedral representations. As a result, we prove that $L (s, {\rm sym}^{m} (\rho) \otimes \chi)$doesnotadmitanyLandau--Siegelzerowhenitisnotdivisibleby$L$–functions of quadratic characters. In general, there is no such divisibility and and there are no Landau–Siegel zeros for such $L$–functions.