Functoriality for the exterior square of and the symmetric fourth of

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

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 as automorphic forms on and the symmetric fourth of cusp forms on as automorphic forms on . 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 and the absolute convergence of the exterior square -functions of . Second, we prove that the fourth symmetric power -functions of cuspidal representations of are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound for Hecke eigenvalues of Maass forms over any number field.

**[AC]**James Arthur and Laurent Clozel,*Simple algebras, base change, and the advanced theory of the trace formula*, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR**1007299****[BR]**Don Blasius and Dinakar Ramakrishnan,*Maass forms and Galois representations*, Galois groups over 𝑄 (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 33–77. MR**1012167**, 10.1007/978-1-4613-9649-9_2**[JS]**H. Jacquet and J. A. Shalika,*On Euler products and the classification of automorphic representations. I*, Amer. J. Math.**103**(1981), no. 3, 499–558. MR**618323**, 10.2307/2374103

H. Jacquet and J. A. Shalika,*On Euler products and the classification of automorphic forms. II*, Amer. J. Math.**103**(1981), no. 4, 777–815. MR**623137**, 10.2307/2374050**[La]**R. P. Langlands,*Automorphic representations, Shimura varieties, and motives. Ein Märchen*, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR**546619****[Ra]**Dinakar Ramakrishnan,*Modularity of the Rankin-Selberg 𝐿-series, and multiplicity one for 𝑆𝐿(2)*, Ann. of Math. (2)**152**(2000), no. 1, 45–111. MR**1792292**, 10.2307/2661379

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Additional Information

**Henry H. Kim**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Email:
henrykim@math.toronto.edu

**Appendix 1 by Dinakar Ramakrishnan**

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

Email:
dinakar@its.caltech.edu

**Appendix 2 by Henry H. Kim and Peter Sarnak**

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

Email:
sarnak@math.princeton.edu

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

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