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- W2950233102 abstract "In this note we study the symmetric powers of strongly modular icosahedral representations $rho$ of ${rm Gal} (bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $rho$, there exists a cuspidal 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 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 $pi'$ 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 $L (s, {rm sym}^{m} (rho) otimes chi)$ does not admit any Landau--Siegel zero when it is not divisible by $L$--functions of quadratic characters. In general, there is no such divisibility and and there are no Landau--Siegel zeros for such $L$--functions." @default.
- W2950233102 created "2019-06-27" @default.
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- W2950233102 date "2003-01-09" @default.
- W2950233102 modified "2023-09-27" @default.
- W2950233102 title "On the symmetric powers of cusp forms on $GL (2)$ of icosahedral type" @default.
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