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On the nonvanishing of the central value of the Rankin-Selberg $L$-functions

Author(s): David Ginzburg; Dihua Jiang; Stephen Rallis
Journal: J. Amer. Math. Soc. 17 (2004), 679-722.
MSC (2000): Primary 11F67, 11F70, 22E46, 22E55
Posted: April 1, 2004
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Abstract: We characterize the nonvanishing of the central value of the Rankin-Selberg $L$-functions in terms of periods of Fourier-Jacobi type. This characterization is based on the Langlands philosophy on functoriality and on applications of invariant theory to automorphic representations. The result is the symplectic analog of the Gross-Prasad conjecture.

References:

[A78]
Arthur, J. A trace formula for reductive groups. I. Terms associated to classes in $G(Q)$. Duke Math. J. 45 (1978), no. 4, 911-952. MR 80d:10043

[A80]
Arthur, J. A trace formula for reductive groups. II. Applications of a truncation operator. Compositio Math. 40 (1980), no. 1, 87-121. MR 81b:22018

[BFSP04]
Boecherer, S.; Furusawa, M.; Schulze-Pillot, R. On the global Gross-Prasad conjecture for Yoshida liftings. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004.

[CKPSS01]
Cogdell, J.; Kim, H.; Piatetski-Shapiro, I.; Shahidi, F. On lifting from classical groups to ${\mathrm{GL}}_N$. Inst. Hautes Études Sci. Publ. Math. No. 93 (2001), 5-30. MR 2002i:11048

[CKPSS]
Cogdell, J.; Kim, H.; Piatetski-Shapiro, I.; Shahidi, F. Functoriality for the classical groups. To appear in Inst. Hautes Études Sci. Publ. Math.

[CPS04]
Cogdell, J.; Piatetski-Shapiro, I. Remarks on Rankin-Selberg Convolutions. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004.

[CM93]
Collingwood, D.; McGovern, W. Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. MR 94j:17001
[FJ93]
Friedberg, S.; Jacquet, H. Linear periods. J. Reine Angew. Math. 443 (1993), 91-139. MR 94k:11058
[F95]
Furusawa, M. On the theta lift from ${\rm SO}\sb {2n+1}$ to $\widetilde{\rm Sp}\sb n$. J. Reine Angew. Math. 466 (1995), 87-110. MR 96g:11052

[GPSR87]
Gelbart, S.; Piatetski-Shapiro, I.; Rallis, S. Explicit Constructors of Automorphic $L$-Functions. Springer Lecture Notes, Vol. 1254, 1987. MR 89k:11038

[GJR01]
Ginzburg, D.; Jiang, D.; Rallis, S. Nonvanishing of the central critical value of the third symmetric power $L$-functions. Forum Math. 13 (2001), no. 1, 109-132. MR 2001m:11083

[GJR03]
Ginzburg, D.; Jiang, D.; Rallis, S. Periods of residue representations of ${\mathrm{SO}}_{2l}$. Accepted by Manuscripta Math. 2003.

[GJR]
Ginzburg, D.; Jiang, D.; Rallis, S. Non-vanishing of the Central Value of the Rankin-Selberg L-functions. II. To appear in the volume commemorating the sixtieth birthday of Stephen Rallis.

[GRS98]
Ginzburg, D.; Rallis, S; Soudry, D. $L$-Functions for Symplectic Groups. Bull. Soc. Math. France, 126 (1998), no. 2, 181-244. MR 2000b:22017

[GRS99a]
Ginzburg, D.; Rallis, S; Soudry, D. On the correspondence between cuspidal representations of ${\mathrm{GL}}_{2n}$ and ${\mathop{\widetilde{\rm Sp}}\nolimits}_{2n}$. J. Amer. Math. Soc. 12 (1999), no.3, 849-907. MR 2000b:22018

[GRS99b]
Ginzburg, D.; Rallis, S; Soudry, D. Lifting Cusp Forms on ${\mathrm{GL}}_{2n}$ to ${\mathop{\widetilde{\rm Sp}}\nolimits}_{2n}$: The Unramified Correspondence. Duke Math. J. 100 (1999), no. 2, 243-266. MR 2001a:11085

[GRS99c]
Ginzburg, D.; Rallis, S; Soudry, D. On explicit lifts of cusp forms from ${\mathrm{GL}}_m$to classical groups. Ann. of Math. (2) 150 (1999), no. 3, 807-866. MR 2001b:11040

[GRS01]
Ginzburg, D.; Rallis, S; Soudry, D. Generic automorphic forms on ${\rm SO}(2n+1)$: functorial lift to ${\rm GL}(2n)$, endoscopy, and base change. Internat. Math. Res. Notices 2001, no. 14, 729-764. MR 2002g:11065

[GRS02]
Ginzburg, D.; Rallis, S; Soudry, D. Endoscopic representations of ${\mathop{\widetilde{\rm Sp}}\nolimits}_{2n}$. Journal of the Inst. of Math. Jussieu, 1 (2002), no. 1, 77-123. MR 2004c:11074

[GRS03]
Ginzburg, D.; Rallis, S; Soudry, D. On Fourier Coefficients of Automorphic Forms of Symplectic Groups. Manuscripta Math. 111 (2003), no. 1, 1-16.

[GRS]
Ginzburg, D.; Rallis, S; Soudry, D. Constructions of CAP representations for symplectic groups using the descent method. To appear in the volume commemorating the sixtieth birthday of Stephen Rallis.

[GP92]
Gross, B.; Prasad, D. On the decomposition of a representation of ${\rm SO}\sb n$ when restricted to ${\rm SO}\sb {n-1}$. Canad. J. Math. 44 (1992), no. 5, 974-1002. MR 93j:22031

[GP94]
Gross, B.; Prasad, D. On irreducible representations of ${\rm SO}\sb {2n+1}\times{\rm SO}\sb {2m}$. Canad. J. Math. 46 (1994), no. 5, 930-950. MR 96c:22028

[H94]
Harris, M. Hodge-de Rham structures and periods of automorphic forms. Motives (Seattle, WA, 1991), 573-624, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994. MR 95e:11058

[HK91]
Harris, M.; Kudla, S. The central critical value of a triple product $L$-function. Ann. of Math. (2) 133 (1991), no. 3, 605-672. MR 93a:11043

[HK92]
Harris, M.; Kudla, S. Arithmetic automorphic forms for the nonholomorphic discrete series of ${\rm GSp}(2)$. Duke Math. J. 66 (1992), no. 1, 59-12. MR 93e:22023

[HK04]
Harris, M.; Kudla, S. On a conjecture of Jacquet. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004.

[I94]
Ikeda, Tamotsu. On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series. J. Math. Kyoto Univ. 34 (1994), no. 3, 615-636. MR 95h:11044

[IS00]
Iwaniec, H.; Sarnak, P. Perspectives on the analytic theory of $L$-functions. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part II, 705-741. MR 2002b:11117

[JPSS83]
Jacquet, H.; Piatetski-Shapiro, I.; Shalika, J. Rankin-Selberg convolutions. Amer. J. Math. 105 (1983), no. 2, 367-464. MR 85g:11044

[JR92]
Jacquet, H.; Rallis, S. Symplectic periods. J. Reine Angew. Math. 423 (1992), 175-197. MR 93b:22035

[JLR04]
Jacquet, H.; Lapid, E.; Rallis, S. A spectral identity for skew symmetric matrices. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004.

[Jng98a]
Jiang, D. $G\sb 2$-periods and residual representations. J. Reine Angew. Math. 497 (1998), 17-46. MR 99e:11062

[Jng98b]
Jiang, D. Nonvanishing of the central critical value of the triple product $L$-functions. Internat. Math. Res. Notices 1998, no. 2, 73-84. MR 99a:11060

[Jng01]
Jiang, D. On Jacquet's conjecture: the split period case. Internat. Math. Res. Notices 2001, no. 3, 145-163. MR 2002c:11054

[JS03]
Jiang, D.; Soudry, D. The local converse theorem for ${\mathrm{SO}}(2n+1)$ and applications. Ann. of Math. (2) 157 (2003), no. 3, 743-806.

[JS04]
Jiang, D.; Soudry, D. Generic Representations and the Local Langlands Reciprocity Law for $p$-adic ${\mathrm{SO}}(2n+1)$. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004.

[K00]
Kim, H. Langlands-Shahidi methods and poles of automorphic L-functions II. Israel J. of Math. 117 (2000), 261-284. MR 2001i:11059a

[K02]
Kim, H. Applications of Langlands' functoriality of odd orthogonal groups. Trans. AMS. 354 (2002), no. 7, 2775-2796. MR 2003c:22025

[L71]
Langlands, R. Euler products. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. Yale Mathematical Monographs, 1. Yale University Press, New Haven, Conn.-London, 1971. MR 54:7387

[Lp03]
Lapid, E. On the non-negativity of Rankin-Selberg L-functions at the center of symmetry. Internat. Math. Res. Notices 2003, no. 2, 65-75. MR 2003j:11054

[LR03]
Lapid, E.; Rallis, S. On the non-negativity of $L(\tfrac{1}{2},\pi)$ for ${\mathrm{SO}}_{2n+1}$. Ann. of Math. (2) 157 (2003), no. 3, 891-917. MR 2004d:11039

[MVW87]
Moeglin, C.; Vignéras, M.-F.; Waldspurger, J.-L. Correspondances de Howe sur un corps $p$-adique. (French) Lecture Notes in Mathematics, 1291. Springer-Verlag, Berlin, 1987. MR 91f:11040

[MW87]
Moeglin, C.; Waldspurger, J.-L. Modèles de Whittaker dégénérés pour des groupes $p$-adiques. (French) Math. Z. 196 (1987), no. 3, 427-452. MR 89f:22024

[MW89]
Moeglin, C.; Waldspurger, J.-L. Le spectre résiduel de ${\rm GL}(n)$. (French) Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605-674. MR 91b:22028

[MW95]
Moeglin, C.; Waldspurger, J.-L. Spectral decomposition and Eisenstein series. Cambridge University Press, 1995. MR 97d:11083

[PS71]
Piatetski-Shapiro, I. Euler subgroups. Lie groups and their representations (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971), pp. 597-620. Halsted, New York, 1975. MR 53:10720

[R94]
Ramakrishnan, D. Pure motives and automorphic forms. Motives (Seattle, WA, 1991), 411-446, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994. MR 94m:11134

[Shd84]
Shahidi, F. Fourier transforms of intertwining operators and Plancherel measures for ${\mathrm{GL}}(n)$. Amer. J. Math. 106 (1984), no. 1, 67-111. MR 86b:22031

[Shd88]
Shahidi, F. On the Ramanujan conjecture and the finiteness of poles of certain L-functions. Ann. of Math. (2) 127 (1988), no. 3, 547-584. MR 89h:11021

[Shl74]
Shalika, J. The multiplicity one theorem for ${\rm GL}\sb{n}$. Ann. of Math. (2) 100 (1974), no. 2, 171-193. MR 50:545

[S02]
Soudry, D. Langlands functoriality from classical groups to $GL_n$. To appear in Asterisque.

[W85]
Waldspurger, J.-L. Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie. (French) Compositio Math. 54 (1985), no. 2, 173-242. MR 87g:11061b

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Additional Information:

David Ginzburg
Affiliation: School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat-Aviv, 69978 Israel
Email: ginzburg@post.tau.ac.il

Dihua Jiang
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: dhjiang@math.umn.edu

Stephen Rallis
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: haar@math.ohio-state.edu

DOI: 10.1090/S0894-0347-04-00455-2
PII: S 0894-0347(04)00455-2
Keywords: Special value, $L$-function, period, automorphic form
Received by editor(s): May 8, 2003
Posted: April 1, 2004
Additional Notes: The second author is partially supported by the NSF grant DMS-0098003, the Sloan Research Fellowship, and the McKnight Land-Grant Professorship (University of Minnesota).
Dedicated: Dedicated to Ilya I. Piatetski-Shapiro with admiration on the occasion of his 75th birthday
Copyright of article: Copyright 2004, American Mathematical Society

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