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

摘要

In this note we study the symmetric powers of strongly modular icosahedralrepresentations $\rho$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, andtheir twisted $L$--functions. We prove that for such $\rho$, there exists acuspidal automorphic representation $\Pi = \Pi_{\infty} \otimes \Pi_{f}$ of$GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (\rho)) = L (s,\Pi_{f})$. One sees that ${\rm sym}^{5} (\rho)$ is twist equivalent to $\rho'\otimes {\rm sym}^{2} (\rho)$ for another modular icosahedral representation$\rho'$, and our theorem is a special case of a cuspidality criterionformulated and proved in this paper, which may be of independent interest, forthe Kim--Shahidi automorphic tensor product $\pi \boxtimes {\rm sym}^{2}(\pi')$, where $\pi$ and $\pi'$ are cuspidal automorphic representations of $GL(2) / F$. We also give a complete structure theory of modular icosahedralrepresentations. As a result, we prove that $L (s, {\rm sym}^{m} (\rho) \otimes\chi)$ does not admit any Landau--Siegel zero when it is not divisible by$L$--functions of quadratic characters. In general, there is no suchdivisibility and and there are no Landau--Siegel zeros for such $L$--functions.