Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂

Kim, Henry; Ramakrishnan, with Appendix 1 by Dinakar; Sarnak, Appendix 2 by Henry H. Kim and Peter

doi:10.1090/S0894-0347-02-00410-1

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Functoriality for the exterior square of $GL_{4}$ and the symmetric fourth of $GL_{2}$

Authors: Henry H. Kim, Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Henry H. Kim and Peter Sarnak
Journal: J. Amer. Math. Soc. 16 (2003), 139-183
MSC (2000): Primary 11F30, 11F70, 11R42
Posted: October 30, 2002
MathSciNet review: 1937203
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Abstract: In this paper we prove the functoriality of the exterior square of cusp forms on $GL_{4}$ as automorphic forms on $GL_{6}$ and the symmetric fourth of cusp forms on $GL_{2}$ as automorphic forms on $GL_{5}$ . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of $GL_{4}$ and the absolute convergence of the exterior square -functions of $GL_{4}$ . Second, we prove that the fourth symmetric power -functions of cuspidal representations of $GL_{2}$ are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound $\frac{3}{26}$ for Hecke eigenvalues of Maass forms over any number field.

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References to Appendix 1

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References to Appendix 2

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