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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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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 HTML | PDF
J. Amer. Math. Soc. 16 (2003), 139-183 Request permission

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
  • MR Author ID: 324906
  • Email: henrykim@math.toronto.edu
  • Dinakar Ramakrishnan
  • Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
  • MR Author ID: 228519
  • Email: dinakar@its.caltech.edu
  • Peter Sarnak
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • MR Author ID: 154725
  • Email: sarnak@math.princeton.edu
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 139-183
  • MSC (2000): Primary 11F30, 11F70, 11R42
  • DOI: https://doi.org/10.1090/S0894-0347-02-00410-1
  • MathSciNet review: 1937203