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

Abstract

In this note we study the symmetric powers of strongly modular icosahedral representations $ρ$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $ρ$, there exists a cuspidal automorphic representation $Π= Π_{\infty} \otimes Π_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (ρ)) = L (s, Π_{f})$. One sees that ${\rm sym}^{5} (ρ)$ is twist equivalent to $ρ' \otimes {\rm sym}^{2} (ρ)$ for another modular icosahedral representation $ρ'$, 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 $π\boxtimes {\rm sym}^{2} (π')$, where $π$ and $π'$ are cuspidal automorphic representations of $GL (2) / F$. We also give a complete structure theory of modular icosahedral representations. As a result, we prove that $L (s, {\rm sym}^{m} (ρ) \otimes χ)$ does not admit any Landau--Siegel zero when it is not divisible by $L$--functions of quadratic characters. In general, there is no such divisibility and and there are no Landau--Siegel zeros for such $L$--functions.