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

Kim, Henry

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

Functoriality for the exterior square of $\operatorname {GL}_{4}$ and the symmetric fourth of $\operatorname {GL}_{2}$
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by Henry H. Kim; with an appendix by Dinakar Ramakrishnan; with an appendix co-authored by Peter Sarnak

J. Amer. Math. Soc. 16 (2003), 139-183

DOI: https://doi.org/10.1090/S0894-0347-02-00410-1

Published electronically: October 30, 2002

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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 $L$-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 $L$-functions of $GL_{4}$. Second, we prove that the fourth symmetric power $L$-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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