Duke Mathematical Journal

Cuspidality of symmetric powers with applications

Henry H. Kim and Freydoon Shahidi

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The purpose of this paper is to prove that the symmetric fourth power of a cusp form on ${\rm GL}(2)$, whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the Ramanujan-Petersson and Sato-Tate conjectures. In particular, we establish the bound $q\sp {1/9}\sb v$ for unramified Hecke eigenvalues of cusp forms on ${\rm GL}(2)$. Over an arbitrary number field, this is the best bound available at present.

Article information

Duke Math. J., Volume 112, Number 1 (2002), 177-197.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F30: Fourier coefficients of automorphic forms 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]


Kim, Henry H.; Shahidi, Freydoon. Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), no. 1, 177--197. doi:10.1215/S0012-9074-02-11215-0. https://projecteuclid.org/euclid.dmj/1087575125

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