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Cuspidality of symmetric powers with applications. (English) Zbl 1074.11027

This paper establishes a valuable result in the theory of automorphic forms. It proves a criterion for cuspidality of the fourth symmetric powers of cusp forms on \(\text{GL}_2\).
The \(m\)th symmetric power lifting is an instance of Langlands functoriality. It is associated to the \(m\)-th symmetric representation \(\text{Sym}^m: \text{GL}_2(\mathbb{C}) \to \text{GL}_{m+1}(\mathbb{C})\). If \(\pi = \bigotimes_v \pi_v\) is a cuspidal automorphic representation of \(\text{GL}_2(\mathbb{A})\), where \(\mathbb{A}\) is the ring of adeles of a number field \(F\), then by the local Langlands correspondence, \(\text{Sym}^m(\pi_v)\) is well defined for every \(v\). By the functoriality principle, \(\text{Sym}^m(\pi) = \bigotimes_v \text{Sym}^m(\pi_v)\) is an automorphic representation of \(\text{GL}_{m+1}(\mathbb{A})\). This was proved for \(m=2\) by S. Gelbart and H. Jacquet [Ann. Sci. Éc. Norm. Supér., IV. Sér. 11, No. 4, 471–542 (1978; Zbl 0406.10022)], for \(m=3\) by H. H. Kim and F. Shahidi [Ann. Math. (2) 155, No. 3, 837–893 (2002; Zbl 1040.11036)], and for \(m=4\) by H. H. Kim [J. Am. Math. Soc. 16, No. 1, 139–183 (2003; Zbl 1018.11024)].
In this paper, the authors give a criterion for cuspidality of \(\text{Sym}^4(\pi)\). Let \(A^4(\pi) = \text{Sym}^4(\pi) \otimes \omega_\pi^{-1}\), where \(\omega_\pi\) is the central character of \(\pi\). The authors prove \(A^4(\pi)\) is not cuspidal if and only if \(\pi\) is of dihedral, tetrahedral or octahedral type.
As a consequence, the authors prove a number of results toward the Ramanujan-Petersson and Sato-Tate conjectures.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F30 Fourier coefficients of automorphic forms
11R42 Zeta functions and \(L\)-functions of number fields
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