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), 139183
MSC (2000):
Primary 11F30, 11F70, 11R42
Published electronically:
October 30, 2002
MathSciNet review:
1937203
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 PiatetskiShapiro to analytic properties of certain functions obtained by the LanglandsShahidi 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
(90m:22041)
 [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
(90m:11078), http://dx.doi.org/10.1007/9781461396499_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
(82m:10050a), http://dx.doi.org/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 (82m:10050b), http://dx.doi.org/10.2307/2374050
 [La]
R.
P. Langlands, Automorphic representations, Shimura varieties, and
motives. Ein Märchen, Automorphic forms, representations and
𝐿functions (Proc. Sympos. Pure Math., Oregon State Univ.,
Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer.
Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
(83f:12010)
 [Ra]
Dinakar
Ramakrishnan, Modularity of the RankinSelberg 𝐿series,
and multiplicity one for 𝑆𝐿(2), Ann. of Math. (2)
152 (2000), no. 1, 45–111. MR 1792292
(2001g:11077), http://dx.doi.org/10.2307/2661379
References to Appendix 1
 [AC]
 J. Arthur and L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Ann. Math. Studies 120, Princeton, NJ, 1989. MR 90m:22041
 [BR]
 D. Blasius and D. Ramakrishnan, Maass forms and Galois representations, in Galois groups over , ed. by Y. Ihara, K. Ribet, and J.P. Serre, Academic Press, NY, 1990, pp. 3377. MR 90m:11078
 [JS]
 H. Jacquet and J.A. Shalika, Euler products and the classification of automorphic forms I & II, Amer. J of Math. 103 (1981), 499558 & 777815. MR 82m:10050a; MR 82m:10050b
 [La]
 R.P. Langlands, Automorphic representations, Shimura varieties and motives. Ein Märchen, Proc. Symp. Pure Math. 33, ed. by A. Borel and W. Casselman, part 2, AMS, 1979, pp. 205246. MR 83f:12010
 [Ra]
 D. Ramakrishnan, Modularity of the RankinSelberg series, and Multiplicity One for SL, Annals of Mathematics 152 (2000), 45111. MR 2001g:11077
References to Appendix 2 
 [BDHI]
 D. Bump, W. Duke, J. Hoffstein and H. Iwaniec, An estimate for the Hecke eigenvalues of Maass forms, IMRN 4 (1992), 7581. MR 93d:11047
 [De]
 P. Deligne, SGA 4 , Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, SpringerVerlag, 1977. MR 57:3132
 [DI]
 J.M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Inv. Math. 70 (1982/83), 219288. MR 84m:10015
 [JS]
 H. Jacquet and J. Shalika, On Euler products and the classification of automorphic forms II, Amer. J. Math. 103 (1981), 777815. MR 82m:10050b
 [Ki1]
 H. Kim, LanglandsShahidi method and poles of automorphic functions: application to exterior square functions, Can. J. Math. 51 (1999), 835849. MR 2000f:11058
 [Ki2]
 , LanglandsShahidi method and poles of automorphic functions II, Israel J. Math. 117 (2000), 261284. MR 2001i:11059a
 [Ki3]
 , Functoriality for the exterior square of and symmetric fourth of , the main paper.
 [KiSh]
 H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177197.
 [LRS]
 W. Luo, Z. Rudnick, and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), 387401. MR 96h:11045
 [LRS2]
 , On the generalized Ramanujan conjecture for , in Automorphic forms, automorphic representations, and arithmetic, Proc. Sympos. Pure Math., vol. 66, Part 2, Amer. Math. Soc., Providence, RI, 1999, pp. 301310. MR 2000e:11072
 [RS]
 Z. Rudnick and P. Sarnak, Zeros of principal functions and random matrix theory, A celebration of John F. Nash, Jr., Duke Math. J. 81 (1996), 269322. MR 97f:11074
 [Se]
 A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math., vol. 8, 1965, pp. 115. MR 32:93
 [Sh1]
 F. Shahidi, A proof of Langlands conjecture on Plancherel measures; complementary series for adic groups, Annals of Math. 132 (1990), 273330. MR 91m:11095
 [Sh2]
 , On the Ramanujan conjecture and finiteness of poles for certain functions, Ann. of Math. 127 (1988), 547584. MR 89h:11021
References 
 [AT]
 E. Artin and J. Tate, Class field theory, W.A. Benjamin, 1967. MR 36:6383
 [As]
 M. Asgari, Local functions for split spinor groups, Can. J. Math. 54 (2002), 673693.
 [Bu]

J. Buhler, Icosahedral Galois Representations, Springer Lecture Notes 654, 1978. MR 58:22019  [Ch]
 J.P. Jeff Chen, Local factors, Central characters, and Representations of general linear groups over nonarchimedean fields, Doctoral dissertation, Yale University (1996).
 [CoPS1]
 J.W. Cogdell and I.I. PiatetskiShapiro, Converse theorems for II, J. Reine Angew. Math. 507 (1999), 165188. MR 2000a:22029
 [CoPS2]
 , Converse theorems for and their application to liftings, preprint.
 [CKPSS]
 J.W. Cogdell, H. Kim, I.I. PiatetskiShapiro, and F. Shahidi, On lifting from classical groups to , Publ. Math. IHES 93 (2001), 530. MR 2002i:11048
 [GeJ]
 S. Gelbart and H. Jacquet, A relation between automorphic representations of and , Ann. Scient. Éc. Norm. Sup. 11 (1978), 471552. MR 81e:10025
 [GeSh]
 S. Gelbart and F. Shahidi, Boundedness of automorphic functions in vertical strips, J. of AMS 14 (2001), 79107.
 [GL]
 P. Gérardin and J.P. Labesse, Base change problem for , Proceedings of Symposia in Pure Mathematics, vol. 33, part 2, 1979, pp. 115133. MR 82e:10047
 [H]
 M. Harris, The local Langlands conjecture for over a adic field, , Inv. Math. 134 (1998), 177210. MR 99j:22024
 [HT]
 M. Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, 2001.
 [He1]
 G. Henniart, Une conséquence de la théorie du changement de base pour . Analytic number theory (Tokyo, 1988), Lecture Notes in Math., 1434, Springer, Berlin, 1990, pp. 138142. MR 91i:22021
 [He2]
 , Une preuve simple des conjectures de Langlands pour sur un corps adique, Inv. Math. 139 (2000), 439455. MR 2001e:11052
 [He3]
 , On the local Langlands conjecture for : the cyclic case, Ann. of Math. 123 (1986), 145203. MR 87k:11132
 [JPSS]
 H. Jacquet, I.I. PiatetskiShapiro, and J. Shalika, RankinSelberg convolutions, Amer. J. Math. 105 (1983), 367464. MR 85g:11044
 [JPSS2]
 H. Jacquet, I.I. PiatetskiShapiro, and J. Shalika, Automorphic forms on , Ann. of Math. 109 (1979), 213258. MR 80i:10034b
 [JS]
 H. Jacquet and J. Shalika, Exterior square functions, in Automorphic forms, Shimura varieties, and functions, Vol. II (Ann Arbor, MI, 1988), Academic Press, 1990, pp. 143226. MR 91g:11050
 [JS2]
 , A lemma on highly ramified factors, Math. Ann. 271 (1985), 319332. MR 87i:22048
 [JS3]
 , On Euler products and the classification of automorphic forms I,II, Amer. J. Math. 103 (1981), 499558; 777815. MR 82m:10050a; MR 82m:10050b
 [Ki1]
 H. Kim, LanglandsShahidi method and poles of automorphic functions: application to exterior square functions, Can. J. Math. 51 (1999), 835849. MR 2000f:11058
 [Ki2]
 , LanglandsShahidi method and poles of automorphic functions II, Israel J. Math. 117 (2000), 261284. MR 2001i:11059a
 [Ki3]
 , Correction to: LanglandsShahidi method and poles of automorphic functions II, Israel J. Math. 118 (2000), 379. MR 2001i:11059b
 [Ki4]
 , On local functions and normalized intertwining operators, preprint.
 [Ki5]
 , Examples of nonnormal quintic automorphic induction, preprint.
 [KiSa]
 H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, Appendix 2 to this paper.
 [KiSh]
 H. Kim and F. Shahidi, Holomorphy of Rankin triple functions; special values and root numbers for symmetric cube functions, Israel J. Math. 120 (2000), 449466. MR 2002c:11055
 [KiSh2]
 , Functorial products for and functorial symmetric cube for , to appear in Ann. of Math..
 [KiSh3]
 , Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177197.
 [Ku]
 S. Kudla, The local Langlands correspondence: the nonarchimedean case, in Proceedings of Symposia in Pure Mathematics, vol. 55, part 2, American Mathematical Society, 1994, pp. 365391. MR 95d:11065
 [LLa]
 JP. Labesse and R.P. Langlands, indistinguishability for , Can. J. Math. 31 (1979), 726785. MR 81b:22017
 [La1]
 R.P. Langlands, Euler Products, Yale University Press, 1971. MR 54:7387
 [La2]
 , On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., Vol. 544, SpringerVerlag, 1976. MR 58:28319
 [La3]
 , Base Change for , Annals of Math. Studies, vol. 96, Princeton, 1980. MR 82a:10032
 [La4]
 , On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie groups (P.J. Sally, Jr. and D.A. Vogan, ed.), Mathematical Surveys and Monographs, vol. 31, AMS, 1989, pp. 101170. MR 91e:22017
 [LuRSa]
 W. Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for , Proceedings of Symposia in Pure Mathematics, vol. 66, part 2, 1999, pp. 301310. MR 2000e:11072
 [LuRSa2]
 , On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), 387401. MR 96h:11045
 [MW1]
 C. Moeglin and J.L. Waldspurger, Spectral Decomposition and Eisenstein series, une paraphrase de l'Ecriture, Cambridge tracts in mathematics, vol. 113, Cambridge University Press, 1995. MR 97d:11083
 [MW2]
 C. Moeglin and J.L. Waldspurger, Le spectre résiduel de , Ann. Scient. Éc. Norm. Sup. 22 (1989), 605674. MR 91b:22028
 [PRa]
 D. Prasad and D. Ramakrishnan, On the global root numbers of , Proceedings of Symposia in Pure Mathematics, vol. 66, part 2, 1999, pp. 311330. MR 2000f:11060
 [Ra1]
 D. Ramakrishnan, Modularity of the RankinSelberg series, and multiplicity one for , Ann. of Math. 152 (2000), 45111. MR 2001g:11077
 [Ra2]
 , On the coefficients of cusp forms, Math. Research Letters 4 (1997), 295307. MR 98e:11064
 [Ra3]
 , A descent criterion for isobaric representations, Appendix 1 to this paper.
 [Ro]
 J. Rogawski, Representations of and division algebras over a adic field, Duke Math. J. 50 (1983), 161196. MR 84j:12018
 [Sh1]
 F. Shahidi, A proof of Langlands conjecture on Plancherel measures; complementary series for adic groups, Annals of Math. 132 (1990), 273330. MR 91m:11095
 [Sh2]
 , On certain functions, Amer. J. Math 103 (1981), 297355. MR 82i:10030
 [Sh3]
 , On the Ramanujan conjecture and finiteness of poles for certain functions, Ann. of Math. 127 (1988), 547584. MR 89h:11021
 [Sh4]
 , Fourier transforms of intertwining operators and Plancherel measures for , Amer. J. of Math. 106 (1984), 67111. MR 86b:22031
 [Sh5]
 , Twisted endoscopy and reducibility of induced representations for adic groups, Duke Math. J. 66, No. 1 (1992), 141. MR 93b:22034
 [Sh6]
 , On multiplicativity of local factors, In: Festschrift in honor of I.I. PiatetskiShapiro, Part II, Israel Math. Conf. Proc. 3, Weizmann, Jerusalem, 1990, pp. 279289. MR 93e:11144
 [Sh7]
 , Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 9731007. MR 87m:11049
 [Ta]
 M. Tadic, Classification of unitary representations in irreducible representations of general linear group (nonArchimedean case), Ann. Sci. Ec. Norm. Sup. 19 (1986), 335382. MR 88b:22021
 [Tu]
 J. Tunnell, Artin's conjecture for representations of octahedral type, Bulletin AMS 5, no. 2 (1981), 173175. MR 82j:12015
 [Zh]
 Y. Zhang, The holomorphy and nonvanishing of normalized intertwining operators, Pac. J. Math. 180 (1997), 385398. MR 98k:22076
Similar Articles
Retrieve articles in Journal of the American Mathematical Society
with MSC (2000):
11F30,
11F70,
11R42
Retrieve articles in all journals
with MSC (2000):
11F30,
11F70,
11R42
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:
http://dx.doi.org/10.1090/S0894034702004101
PII:
S 08940347(02)004101
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
