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

Author: Henry H. Kim; with an appendix by Dinakar Ramakrishnan; with an appendix co-authored by Peter Sarnak
Journal: J. Amer. Math. Soc. 16 (2003), 139-183
MSC (2000): Primary 11F30, 11F70, 11R42
Published electronically: 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 $L$-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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Additional Information

Henry H. Kim
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Dinakar Ramakrishnan
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Peter Sarnak
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Received by editor(s): August 30, 2001
Received by editor(s) in revised form: September 18, 2002
Published electronically: October 30, 2002
Additional Notes: The first author was partially supported by NSF grant DMS9988672, NSF grant DMS9729992 (at IAS), NSERC grant and by the Clay Mathematics Institute
The second and third authors were partially supported by NSF grants
Article copyright: © Copyright 2002 American Mathematical Society