×

zbMATH — the first resource for mathematics

On the nonvanishing of the central value of the Rankin-Selberg \(L\)-functions. (English) Zbl 1057.11029
The authors characterize the nonvanishing of the central value of the Rankin-Selberg \(L\)-functions in terms of periods of Fourier-Jacobi type. Let \(\mathbb A\) denote the adele ring of a number field \(k\). Let \(\pi\) be an irreducible unitary cuspidal automorphic representation of \(\text{GL}_n(\mathbb A)\). It is known that the Rankin-Selberg \(L\)-function \(L(s, \pi\times \pi)\) has a simple pole at \(s=1\) if and only if \(\pi^\vee\simeq \pi\). If \(\pi\) is self-dual, then it follows from \[ L(s,\pi\times \pi)=L(s,\pi, \Lambda^2)L(s, \pi, \text{Sym}^2) \] that either \(L(s,\pi, \Lambda^2)\), the exterior square \(L\)-function, has a simple pole at \(s=1\), or \(L(s, \pi, \text{Sym}^2)\), the symmetric square \(L\)-function, has simple pole at \(s=1\). In the first case, \(n\) is even and \(\pi\) is called symplectic; \(\pi\) is called orthogonal in the latter case. The terminology is suggested from the Langlands principle of functoriality. The following theorem is known, due to D. Ginzburg, S. Rallis and D. Soudry for almost everywhere [Ann. Math. (2) 150, No. 3, 807–866 (1999; Zbl 0949.22019), Int. Math. Res. Not. 2001, No. 14, 729–764 (2001; Zbl 1060.11031)], and J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi everywhere [Publ. Math., Inst. Hautes Étud. Sci. 99, 163–233 (2004; Zbl 1090.22010)]. If \(\pi\) is symplectic, the \(n=2r\) and \(\pi\) is the functorial lift from an irreducible generic cuspidal automorphic representation \(\sigma\) of \(\text{SO}_{2r+1}(\mathbb A)\). If \(\pi\) is orthogonal, then if \(n=2\ell\) is even, \(\pi\) is the functorial lift from an irreducible generic cuspidal automorphic representation \(\sigma\) of \(\text{SO}_{2\ell}(\mathbb A)\); and if \(n=2\ell+1\) is odd, \(\pi\) is the functorial lift from an irreducible generic cuspidal automorphic representation \(\sigma\) of \(\text{Sp}_{2\ell}(\mathbb A)\).
One of the main results in the paper is the following
Theorem. Let \(\pi_1\) be an irreducible unitary cuspidal automorphic orthogonal representation of \(\text{GL}_{2\ell+1} (\mathbb A)\), and let \(\pi_2\) be an irreducible unitary cuspidal automorphic symplectic representation of \(\text{GL}_{2r} (\mathbb A)\). Assume that the standard \(L\)-function \(L(\frac{1}{2}, \pi_2)\not =0\). Let \(\sigma\) be an irreducible unitary generic cuspidal automorphic representation of \(\text{Sp}_{2\ell}(\mathbb A)\) which lifts functorially to \(\pi_1\), and let \(\widetilde \tau\) be an irreducible unitary generic cuspidal automorphic representation \(\sigma\) of \(\widetilde {\text{Sp}}_{2r }(\mathbb A)\) which has the \(\psi\)-transfer \(\pi_2\). If the period integrals \[ \mathcal P_{r,r-\ell} (\phi_\ell, \widetilde \phi_r,\varphi_\ell)\;(r\geq \ell), \quad \widetilde {\mathcal P}_{\ell,\ell-r} (\widetilde \phi_r, \phi_\ell, \varphi_r) \;(r\leq \ell) \] attached to \((\sigma,\widetilde \tau, \psi)\) is nonzero for some choice of data, then \(L(\frac{1}{2}, \pi_1\times \pi_2)\not= 0\). \(\tau\) is an irreducible unitary generic cuspidal automorphic representation \(\sigma\) of \(\text{SO}_{2r+1}(\mathbb A)\) which lifts functorially to \(\pi_2\). By the global theta correspondence, we know that if the standard \(L\)-function \(L(\frac{1}{2},\tau)\not=0\), then \(\tau\) is a global theta lift (with respect to a given character \(\psi\)) from an irreducible unitary generic cuspidal automorphic representation \(\widetilde \tau\) of \(\widetilde {\text{Sp}}_{2r}(\mathbb A)\), where \(\widetilde {\text{Sp}}_{2r}\) is the metaplectic double cover of \(\text{Sp}_{2r}\). In this case, \(\pi_2\) is called a \(\psi\)-transfer of \(\widetilde \tau\) from \(\widetilde {\text{Sp}}_{2r}\) to \(\text{GL}_{2r}\). The authors also show that under certain assumption \(L(\frac{1}{2}, \pi_1\times \pi_2)\not= 0\) implies the non-vanishing of certain period integrals of Fourier-Jacobi type.

MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E46 Semisimple Lie groups and their representations
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in \?(\?), Duke Math. J. 45 (1978), no. 4, 911 – 952. · Zbl 0499.10032
[2] James Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, 87 – 121. · Zbl 0499.10033
[3] 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.
[4] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to \?\?_{\?}, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 5 – 30. · Zbl 1028.11029
[5] Cogdell, J.; Kim, H.; Piatetski-Shapiro, I.; Shahidi, F. Functoriality for the classical groups. To appear in Inst. Hautes Études Sci. Publ. Math. · Zbl 1090.22010
[6] Cogdell, J.; Piatetski-Shapiro, I. Remarks on Rankin-Selberg Convolutions. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004. · Zbl 1080.11038
[7] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. · Zbl 0972.17008
[8] Solomon Friedberg and Hervé Jacquet, Linear periods, J. Reine Angew. Math. 443 (1993), 91 – 139. · Zbl 0782.11033
[9] Masaaki Furusawa, On the theta lift from \?\?_{2\?+1} to \~\?\?_{\?}, J. Reine Angew. Math. 466 (1995), 87 – 110. · Zbl 0827.11032
[10] Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis, Explicit constructions of automorphic \?-functions, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. · Zbl 0612.10022
[11] David Ginzburg, Dihua Jiang, and Stephen Rallis, Nonvanishing of the central critical value of the third symmetric power \?-functions, Forum Math. 13 (2001), no. 1, 109 – 132. · Zbl 1034.11033
[12] Ginzburg, D.; Jiang, D.; Rallis, S. Periods of residue representations of \({\mathrm{SO}}_{2l}\). Accepted by Manuscripta Math. 2003.
[13] 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.
[14] David Ginzburg, Stephen Rallis, and David Soudry, \?-functions for symplectic groups, Bull. Soc. Math. France 126 (1998), no. 2, 181 – 244 (English, with English and French summaries). · Zbl 0928.11026
[15] David Ginzburg, Stephen Rallis, and David Soudry, On a correspondence between cuspidal representations of \?\?_{2\?} and \~\?\?_{2\?}, J. Amer. Math. Soc. 12 (1999), no. 3, 849 – 907. · Zbl 0928.11027
[16] David Ginzburg, Stephen Rallis, and David Soudry, Lifting cusp forms on \?\?_{2\?} to \?\?_{2\?}: the unramified correspondence, Duke Math. J. 100 (1999), no. 2, 243 – 266. · Zbl 0991.11024
[17] David Ginzburg, Stephen Rallis, and David Soudry, On explicit lifts of cusp forms from \?\?_{\?} to classical groups, Ann. of Math. (2) 150 (1999), no. 3, 807 – 866. · Zbl 0949.22019
[18] David Ginzburg, Stephen Rallis, and David Soudry, Generic automorphic forms on \?\?(2\?+1): functorial lift to \?\?(2\?), endoscopy, and base change, Internat. Math. Res. Notices 14 (2001), 729 – 764. · Zbl 1060.11031
[19] David Ginzburg, Stephen Rallis, and David Soudry, Endoscopic representations of \~\?\?_{2\?}, J. Inst. Math. Jussieu 1 (2002), no. 1, 77 – 123. · Zbl 1052.11032
[20] Ginzburg, D.; Rallis, S; Soudry, D. On Fourier Coefficients of Automorphic Forms of Symplectic Groups. Manuscripta Math. 111 (2003), no. 1, 1-16. · Zbl 1027.11034
[21] 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. · Zbl 1104.11024
[22] Benedict H. Gross and Dipendra Prasad, On the decomposition of a representation of \?\?_{\?} when restricted to \?\?_{\?-1}, Canad. J. Math. 44 (1992), no. 5, 974 – 1002. · Zbl 0787.22018
[23] Benedict H. Gross and Dipendra Prasad, On irreducible representations of \?\?_{2\?+1}\times \?\?_{2\?}, Canad. J. Math. 46 (1994), no. 5, 930 – 950. · Zbl 0829.22031
[24] Michael Harris, Hodge-de Rham structures and periods of automorphic forms, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 573 – 624. · Zbl 0824.14015
[25] Michael Harris and Stephen S. Kudla, The central critical value of a triple product \?-function, Ann. of Math. (2) 133 (1991), no. 3, 605 – 672. · Zbl 0731.11031
[26] Michael Harris and Stephen S. Kudla, Arithmetic automorphic forms for the nonholomorphic discrete series of \?\?\?(2), Duke Math. J. 66 (1992), no. 1, 59 – 121. · Zbl 0786.11031
[27] Harris, M.; Kudla, S. On a conjecture of Jacquet. Contributions to Automorphic Forms, Geometry, and Number Theory. The Johns Hopkins University Press, 2004. · Zbl 1080.11039
[28] Tamotsu Ikeda, On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series, J. Math. Kyoto Univ. 34 (1994), no. 3, 615 – 636. · Zbl 0822.11041
[29] H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of \?-functions, Geom. Funct. Anal. Special Volume (2000), 705 – 741. GAFA 2000 (Tel Aviv, 1999). · Zbl 0996.11036
[30] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367 – 464. · Zbl 0525.22018
[31] Hervé Jacquet and Stephen Rallis, Symplectic periods, J. Reine Angew. Math. 423 (1992), 175 – 197. · Zbl 0734.11035
[32] 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. · Zbl 1082.11029
[33] Dihua Jiang, \?\(_{2}\)-periods and residual representations, J. Reine Angew. Math. 497 (1998), 17 – 46. · Zbl 0931.11014
[34] Dihua Jiang, Nonvanishing of the central critical value of the triple product \?-functions, Internat. Math. Res. Notices 2 (1998), 73 – 84. · Zbl 0909.11022
[35] Dihua Jiang, On Jacquet’s conjecture: the split period case, Internat. Math. Res. Notices 3 (2001), 145 – 163. · Zbl 0986.11033
[36] 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. · Zbl 1049.11055
[37] 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. · Zbl 1062.11077
[38] Henry H. Kim, Langlands-Shahidi method and poles of automorphic \?-functions. II, Israel J. Math. 117 (2000), 261 – 284. , https://doi.org/10.1007/BF02773573 Henry H. Kim, Correction: ”Langlands-Shahidi method and poles of automorphic \?-functions. II”, Israel J. Math. 118 (2000), 379. · Zbl 1041.11035
[39] Henry H. Kim, Applications of Langlands’ functorial lift of odd orthogonal groups, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2775 – 2796. · Zbl 1060.11033
[40] Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967; Yale Mathematical Monographs, 1. · Zbl 0231.20016
[41] Erez M. Lapid, On the nonnegativity of Rankin-Selberg \?-functions at the center of symmetry, Int. Math. Res. Not. 2 (2003), 65 – 75. · Zbl 1046.11032
[42] Erez Lapid and Stephen Rallis, On the nonnegativity of \?(1\over2,\?) for \?\?_{2\?+1}, Ann. of Math. (2) 157 (2003), no. 3, 891 – 917. · Zbl 1067.11026
[43] Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps \?-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). · Zbl 0642.22002
[44] C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes \?-adiques, Math. Z. 196 (1987), no. 3, 427 – 452 (French). · Zbl 0612.22008
[45] C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de \?\?(\?), Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605 – 674 (French). · Zbl 0696.10023
[46] 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].
[47] I. I. Pjateckij-Šapiro, Euler subgroups, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 597 – 620.
[48] Dinakar Ramakrishnan, Pure motives and automorphic forms, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 411 – 446. · Zbl 0812.11066
[49] Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for \?\?(\?), Amer. J. Math. 106 (1984), no. 1, 67 – 111. · Zbl 0567.22008
[50] Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain \?-functions, Ann. of Math. (2) 127 (1988), no. 3, 547 – 584. · Zbl 0654.10029
[51] J. A. Shalika, The multiplicity one theorem for \?\?_{\?}, Ann. of Math. (2) 100 (1974), 171 – 193. · Zbl 0316.12010
[52] Soudry, D. Langlands functoriality from classical groups to \(GL_n\). To appear in Asterisque. · Zbl 1086.11025
[53] J.-L. Waldspurger, Quelques propriétés arithmétiques de certaines formes automorphes sur \?\?(2), Compositio Math. 54 (1985), no. 2, 121 – 171 (French). J.-L. Waldspurger, Sur les valeurs de certaines fonctions \? automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173 – 242 (French).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.