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

Kim, Henry ¹

¹ Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Journal of the American Mathematical Society, Tome 16 (2003) no. 1, pp. 139-183

Résumé

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.

Détail

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

Bibliographie

[1] Arthur, James, Clozel, Laurent Simple algebras, base change, and the advanced theory of the trace formula 1989

[2] Blasius, Don, Ramakrishnan, Dinakar Maass forms and Galois representations 1989 33 77

[3] Rádl, Franz Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen Math. Z. 1939 429 446

[4] Langlands, R. P. Automorphic representations, Shimura varieties, and motives. Ein Märchen 1979 205 246

[5] Ramakrishnan, Dinakar Modularity of the Rankin-Selberg 𝐿-series, and multiplicity one for 𝑆𝐿(2) Ann. of Math. (2) 2000 45 111

[6] Bump, Daniel, Duke, W., Hoffstein, Jeffrey, Iwaniec, Henryk An estimate for the Hecke eigenvalues of Maass forms Internat. Math. Res. Notices 1992 75 81

[7] Deligne, P. Cohomologie étale 1977

[8] Deshouillers, J.-M., Iwaniec, H. Kloosterman sums and Fourier coefficients of cusp forms Invent. Math. 1982/83 219 288

[9] Jacquet, H., Shalika, J. A. On Euler products and the classification of automorphic representations. I Amer. J. Math. 1981 499 558

[10] Kim, Henry H. Langlands-Shahidi method and poles of automorphic 𝐿-functions: application to exterior square 𝐿-functions Canad. J. Math. 1999 835 849

[11] Kim, Henry H. Langlands-Shahidi method and poles of automorphic 𝐿-functions. II Israel J. Math. 2000 261 284

[12] Luo, W., Rudnick, Z., Sarnak, P. On Selberg’s eigenvalue conjecture Geom. Funct. Anal. 1995 387 401

[13] Luo, Wenzhi, Rudnick, Zeév, Sarnak, Peter On the generalized Ramanujan conjecture for 𝐺𝐿(𝑛) 1999 301 310

[14] Rudnick, Zeév, Sarnak, Peter Zeros of principal 𝐿-functions and random matrix theory Duke Math. J. 1996 269 322

[15] Selberg, Atle On the estimation of Fourier coefficients of modular forms 1965 1 15

[16] Shahidi, Freydoon A proof of Langlands’ conjecture on Plancherel measures Ann. of Math. (2) 1990 273 330

[17] Shahidi, Freydoon On the Ramanujan conjecture and finiteness of poles for certain 𝐿-functions Ann. of Math. (2) 1988 547 584

[18] Artin, E., Tate, J. Class field theory 1968

[19] Buhler, Joe P. Icosahedral Galois representations 1978

[20] Cogdell, J. W., Piatetski-Shapiro, I. I. Converse theorems for 𝐺𝐿_{𝑛}. II J. Reine Angew. Math. 1999 165 188

[21] Cogdell, J. W., Kim, H. H., Piatetski-Shapiro, I. I., Shahidi, F. On lifting from classical groups to 𝐺𝐿_{𝑁} Publ. Math. Inst. Hautes Études Sci. 2001 5 30

[22] Gelbart, Stephen, Jacquet, Hervé A relation between automorphic representations of 𝐺𝐿(2) and 𝐺𝐿(3) Ann. Sci. École Norm. Sup. (4) 1978 471 542

[23] Gérardin, P., Labesse, J.-P. The solution of a base change problem for 𝐺𝐿(2) (following Langlands, Saito, Shintani) 1979 115 133

[24] Harris, Michael The local Langlands conjecture for 𝐺𝐿(𝑛) over a 𝑝-adic field, 𝑛<𝑝 Invent. Math. 1998 177 210

[25] Henniart, Guy Une conséquence de la théorie du changement de base pour 𝐺𝐿(𝑛) 1990 138 142

[26] Henniart, Guy Une preuve simple des conjectures de Langlands pour 𝐺𝐿(𝑛) sur un corps 𝑝-adique Invent. Math. 2000 439 455

[27] Henniart, Guy On the local Langlands conjecture for 𝐺𝐿(𝑛): the cyclic case Ann. of Math. (2) 1986 145 203

[28] Jacquet, H., Piatetskii-Shapiro, I. I., Shalika, J. A. Rankin-Selberg convolutions Amer. J. Math. 1983 367 464

[29] Jacquet, Hervé, Piatetski-Shapiro, Ilja Iosifovitch, Shalika, Joseph Automorphic forms on 𝐺𝐿(3). I Ann. of Math. (2) 1979 169 212

[30] Jacquet, Hervé, Shalika, Joseph Exterior square 𝐿-functions 1990 143 226

[31] Jacquet, Hervé, Shalika, Joseph A lemma on highly ramified 𝜀-factors Math. Ann. 1985 319 332

[32] Rádl, Franz Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen Math. Z. 1939 429 446

[33] Kim, Henry H. Langlands-Shahidi method and poles of automorphic 𝐿-functions: application to exterior square 𝐿-functions Canad. J. Math. 1999 835 849

[34] Kim, Henry H. Langlands-Shahidi method and poles of automorphic 𝐿-functions. II Israel J. Math. 2000 261 284

[35] Kim, Henry H. Langlands-Shahidi method and poles of automorphic 𝐿-functions. II Israel J. Math. 2000 261 284

[36] Kim, Henry H., Shahidi, Freydoon Holomorphy of Rankin triple 𝐿-functions Israel J. Math. 2000 449 466

[37] Kudla, Stephen S. The local Langlands correspondence: the non-Archimedean case 1994 365 391

[38] Labesse, J.-P., Langlands, R. P. 𝐿-indistinguishability for 𝑆𝐿(2) Canadian J. Math. 1979 726 785

[39] Langlands, Robert P. Euler products 1971

[40] Langlands, Robert P. On the functional equations satisfied by Eisenstein series 1976

[41] Langlands, Robert P. Base change for 𝐺𝐿(2) 1980

[42] Langlands, R. P. On the classification of irreducible representations of real algebraic groups 1989 101 170

[43] Luo, Wenzhi, Rudnick, Zeév, Sarnak, Peter On the generalized Ramanujan conjecture for 𝐺𝐿(𝑛) 1999 301 310

[44] Luo, W., Rudnick, Z., Sarnak, P. On Selberg’s eigenvalue conjecture Geom. Funct. Anal. 1995 387 401

[45] Mœglin, C., Waldspurger, J.-L. Spectral decomposition and Eisenstein series 1995

[46] Mœglin, C., Waldspurger, J.-L. Le spectre résiduel de 𝐺𝐿(𝑛) Ann. Sci. École Norm. Sup. (4) 1989 605 674

[47] Prasad, Dipendra, Ramakrishnan, Dinakar On the global root numbers of 𝐺𝐿(𝑛)×𝐺𝐿(𝑚) 1999 311 330

[48] Ramakrishnan, Dinakar Modularity of the Rankin-Selberg 𝐿-series, and multiplicity one for 𝑆𝐿(2) Ann. of Math. (2) 2000 45 111

[49] Ramakrishnan, Dinakar On the coefficients of cusp forms Math. Res. Lett. 1997 295 307

[50] Rogawski, Jonathan D. Representations of 𝐺𝐿(𝑛) and division algebras over a 𝑝-adic field Duke Math. J. 1983 161 196

[51] Shahidi, Freydoon A proof of Langlands’ conjecture on Plancherel measures Ann. of Math. (2) 1990 273 330

[52] Shahidi, Freydoon On certain 𝐿-functions Amer. J. Math. 1981 297 355

[53] Shahidi, Freydoon On the Ramanujan conjecture and finiteness of poles for certain 𝐿-functions Ann. of Math. (2) 1988 547 584

[54] Shahidi, Freydoon Fourier transforms of intertwining operators and Plancherel measures for 𝐺𝐿(𝑛) Amer. J. Math. 1984 67 111

[55] Shahidi, Freydoon Twisted endoscopy and reducibility of induced representations for 𝑝-adic groups Duke Math. J. 1992 1 41

[56] Shahidi, Freydoon On multiplicativity of local factors 1990 279 289

[57] Shahidi, Freydoon Local coefficients as Artin factors for real groups Duke Math. J. 1985 973 1007

[58] Tadić, Marko Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case) Ann. Sci. École Norm. Sup. (4) 1986 335 382

[59] Tunnell, Jerrold Artin’s conjecture for representations of octahedral type Bull. Amer. Math. Soc. (N.S.) 1981 173 175

[60] Zhang, Yuanli The holomorphy and nonvanishing of normalized local intertwining operators Pacific J. Math. 1997 385 398