Boundedness of automorphic 𝐿–functions in vertical strips

Gelbart, Stephen; Shahidi, Freydoon

doi:10.1090/S0894-0347-00-00351-9

Boundedness of automorphic $L$–functions in vertical strips
HTML articles powered by AMS MathViewer

by Stephen Gelbart and Freydoon Shahidi

J. Amer. Math. Soc. 14 (2001), 79-107

DOI: https://doi.org/10.1090/S0894-0347-00-00351-9

Published electronically: September 18, 2000

PDF | Request permission

Abstract:

In this paper we prove the boundedness in vertical strips of finite width for all the $L$–functions that appear in constant terms of Eisenstein series, under a certain natural assumption on local normalized intertwining operators. This result has been quite important in establishing recent cases of Langlands functoriality by Cogdell, Kim, Piatetski-Shapiro, and Shahidi, using converse theorems.

References

James Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), no. 1, 19–84. MR 999488, DOI 10.1016/0022-1236(89)90110-9
James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in $G(\textbf {Q})$, Duke Math. J. 45 (1978), no. 4, 911–952. MR 518111
James Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, 87–121. MR 558260

Endoscopic $L$–functions and a combinatorial identity

51

A. Borel, Automorphic $L$-functions, 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. 27–61. MR 546608
Daniel Bump, David Ginzburg, and Jeffrey Hoffstein, The symmetric cube, Invent. Math. 125 (1996), no. 3, 413–449. MR 1400313, DOI 10.1007/s002220050082
William Casselman and Freydoon Shahidi, On irreducibility of standard modules for generic representations, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 4, 561–589 (English, with English and French summaries). MR 1634020, DOI 10.1016/S0012-9593(98)80107-9
J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $\textrm {GL}_n$, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 157–214. MR 1307299, DOI 10.1007/BF02698889
J. W. Cogdell and I. I. Piatetski-Shapiro, Unitarity and functoriality, Geom. Funct. Anal. 5 (1995), no. 2, 164–173. MR 1334866, DOI 10.1007/BF01895666
J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $\textrm {GL}_n$. II, J. Reine Angew. Math. 507 (1999), 165–188. MR 1670207, DOI 10.1515/crll.1999.507.165

Converse theorems for $GL_{n}$ and their application to liftings

On lifting from classical groups to $GL_{N}$

Paul B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209–235. MR 881269, DOI 10.2307/1971310
Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066, DOI 10.24033/asens.1355
Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis, Explicit constructions of automorphic $L$-functions, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR 892097, DOI 10.1007/BFb0078125
David Ginzburg, Stephen Rallis, and David Soudry, Self-dual automorphic $\textrm {GL}_n$ modules and construction of a backward lifting from $\textrm {GL}_n$ to classical groups, Internat. Math. Res. Notices 14 (1997), 687–701. MR 1460389, DOI 10.1155/S1073792897000457

A local correspondence between supercuspidal representations of $GL_{2n}(F)$ and $\tilde Sp_{2n}(F)$

Roe Goodman and Nolan R. Wallach, Whittaker vectors and conical vectors, J. Functional Analysis 39 (1980), no. 2, 199–279. MR 597811, DOI 10.1016/0022-1236(80)90013-0
Harish-Chandra, Automorphic forms on semisimple Lie groups, Lecture Notes in Mathematics, No. 62, Springer-Verlag, Berlin-New York, 1968. Notes by J. G. M. Mars. MR 0232893, DOI 10.1007/BFb0098434

On the geometry and cohomology of some simple Shimura varieties

Une preuve simple des conjecture de Langlands pour $GL(n)$ sur un corps $p$–adique

139

Tamotsu Ikeda, On the location of poles of the triple $L$-functions, Compositio Math. 83 (1992), no. 2, 187–237. MR 1174424
Hervé Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France 95 (1967), 243–309 (French). MR 271275, DOI 10.24033/bsmf.1654
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, DOI 10.2307/2374103
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, DOI 10.2307/2374103
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
Henry H. Kim, Langlands-Shahidi method and poles of automorphic $L$-functions: application to exterior square $L$-functions, Canad. J. Math. 51 (1999), no. 4, 835–849. MR 1701344, DOI 10.4153/CJM-1999-036-0

Langlands–Shahidi method and poles of automorphic $L$–functions II

117

Lifting from $GL_{4}$ to $GL_{6}$; Exterior square lift

Symmetric cube $L$–functions for $GL_{2}$ are entire

150

Holomorphy of Rankin triple $L$–functions; special values and root numbers for symmetric cube $L$–functions

On the holomorphy of certain $L$–functions

Functorial products for $GL_{2}\times GL_{3}$ and functorial symmetric cube for $GL_{2}$

Philip Kutzko, The Langlands conjecture for $\textrm {Gl}_{2}$ of a local field, Ann. of Math. (2) 112 (1980), no. 2, 381–412. MR 592296, DOI 10.2307/1971151
Robert P. Langlands, Euler products, Yale Mathematical Monographs, vol. 1, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. MR 0419366
Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181, DOI 10.1007/BFb0079929
R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170. MR 1011897, DOI 10.1090/surv/031/03
B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006, DOI 10.1090/mmono/150
C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MR 1361168, DOI 10.1017/CBO9780511470905
C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752, DOI 10.24033/asens.1595
Goran Muić, Some results on square integrable representations; irreducibility of standard representations, Internat. Math. Res. Notices 14 (1998), 705–726. MR 1637097, DOI 10.1155/S1073792898000427

A proof of Casselman–Shahidi’s conjecture for quasi–split classical groups

Goran Muić and Freydoon Shahidi, Irreducibility of standard representations for Iwahori-spherical representations, Math. Ann. 312 (1998), no. 1, 151–165. MR 1645956, DOI 10.1007/s002080050216
Werner Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. (2) 130 (1989), no. 3, 473–529. MR 1025165, DOI 10.2307/1971453
Ravi Raghunathan, A comparison of zeros of $L$-functions, Math. Res. Lett. 6 (1999), no. 2, 155–167. MR 1689206, DOI 10.4310/MRL.1999.v6.n2.a4
Ravi Raghunathan, A converse theorem for Dirichlet series with poles, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 3, 231–235 (English, with English and French summaries). MR 1650237, DOI 10.1016/S0764-4442(98)80138-9

Modularity of the Rankin–Selberg $L$–series, and multiplicity one for $SL(2)$

Zeév Rudnick and Peter Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, DOI 10.1215/S0012-7094-96-08115-6
Gérard Schiffmann, Intégrales d’entrelacement et fonctions de Whittaker, Bull. Soc. Math. France 99 (1971), 3–72 (French). MR 311838, DOI 10.24033/bsmf.1711
Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain $L$-functions, Ann. of Math. (2) 127 (1988), no. 3, 547–584. MR 942520, DOI 10.2307/2007005
Freydoon Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973–1007. MR 816396, DOI 10.1215/S0012-7094-85-05252-4
Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
F. Shahidi, Automorphic $L$-functions: a survey, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 415–437. MR 1044824
Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for $\textrm {GL}(n)$, Amer. J. Math. 106 (1984), no. 1, 67–111. MR 729755, DOI 10.2307/2374430
Freydoon Shahidi, Third symmetric power $L$-functions for $\textrm {GL}(2)$, Compositio Math. 70 (1989), no. 3, 245–273. MR 1002045
Freydoon Shahidi, Symmetric power $L$-functions for $\textrm {GL}(2)$, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 159–182. MR 1260961, DOI 10.1090/crmp/004/11
Freydoon Shahidi, Whittaker models for real groups, Duke Math. J. 47 (1980), no. 1, 99–125. MR 563369
Allan J. Silberger, Special representations of reductive $p$-adic groups are not integrable, Ann. of Math. (2) 111 (1980), no. 3, 571–587. MR 577138, DOI 10.2307/1971110
David Soudry, Rankin-Selberg convolutions for $\textrm {SO}_{2l+1}\times \textrm {GL}_n$: local theory, Mem. Amer. Math. Soc. 105 (1993), no. 500, vi+100. MR 1169228, DOI 10.1090/memo/0500
David Soudry, On the Archimedean theory of Rankin-Selberg convolutions for $\textrm {SO}_{2l+1}\times \textrm {GL}_n$, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 2, 161–224. MR 1318068, DOI 10.24033/asens.1712

Theory of functions

Siniša Žampera, The residual spectrum of the group of type $G_2$, J. Math. Pures Appl. (9) 76 (1997), no. 9, 805–835 (English, with English and French summaries). MR 1485422, DOI 10.1016/S0021-7824(97)89971-0
Yuanli Zhang, The holomorphy and nonvanishing of normalized local intertwining operators, Pacific J. Math. 180 (1997), no. 2, 385–398. MR 1487571, DOI 10.2140/pjm.1997.180.385