Monatshefte für Mathematik

, Volume 181, Issue 2, pp 355–403 | Cite as

A multi-variable Rankin–Selberg integral for a product of \(GL_2\)-twisted Spinor L-functions

Article
First Online:
Received:
Accepted:
  • 101 Downloads

Abstract

We consider a new integral representation for \(L(s_1, \Pi \times \tau _1) L(s_2, \Pi \times \tau _2),\) where \(\Pi \) is a globally generic cuspidal representation of \(GSp_4,\) and \(\tau _1\) and \(\tau _2\) are two cuspidal representations of \(GL_2\) having the same central character. As and application, we find a new period condition for two such L functions to have a pole simultaneously. This points to an intriguing connection between a Fourier coefficient of a residual representation on GSO(12) and a theta function on Sp(16). A similar integral on GSO(18) fails to unfold completely, but in a way that provides further evidence of a connection.

Keywords

Rankin–Selberg Integral representation Spinor L-function  Theta correspondence Fourier coefficient 

Mathematics Subject Classification

11F70 11F66 11F27 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Asgari, M., Raghuram, A.: A cuspidality criterion for the exterior square transfer for cusp forms on GL(4). On certain L-functions. Clay Math. Proc., vol. 13, pp. 33–53. Amer. Math. Soc., Providence, RI (2011)Google Scholar
  2. 2.
    Asgari, M., Shahidi, F.: Generic transfer from \(GSp(4)\) to \(GL(4)\). Compos. Math. 142(3), 541–550 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bump, D., Ginzburg, D.: The adjoint L-function of GL(4). J. Reine Angew. Math. 505, 119–172 (1998)MathSciNetMATHGoogle Scholar
  4. 4.
    Casselman, W., Shalika, J.: The unramified principal series of \(p\)-adic groups. II. The Whittaker function. Compos. Math. 41(2), 207–231 (1980)MathSciNetMATHGoogle Scholar
  5. 5.
    Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Sci. Math. 102(2), 307–330 (1978)MathSciNetMATHGoogle Scholar
  6. 6.
    Ginzburg, D.: On the symmetric fourth power \(L\)-function of \(GL_2\). Isr. J. Math. 92(1–3), 157–184 (1995)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ginzburg, D.: A Rankin–Selberg integral for the adjoint L-function of \(Sp_4\). Isr. J. Math. 95, 301–339 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ginzburg, D.: Certain conjectures relating unipotent orbits to automorphic representations. Isr. J. Math. 151, 323–355 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ginzburg, D., Hundley, J.: Multivariable Rankin-Selberg integrals for orthogonal groups. Int. Math. Res. Not. (58):3097-3119 (2004)Google Scholar
  10. 10.
    Ginzburg, D., Hundley, J.: On Spin L-functions for \(GSO_{10}\). J. Reine Angew. Math. 603, 183–213 (2007)MathSciNetMATHGoogle Scholar
  11. 11.
    Ginzburg, D., Hundley, J.: Constructions of global integrals in the exceptional group \(F_4\). Kyushu J. Math. 67(2), 389–417 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ginzburg, D., Hundley, J.: A doubling integral for \(G_2\). Isr. Math J. (2015). doi: 10.1007/s11856-015-1164-x
  13. 13.
    Ginzburg, D., Jiang, D., Soudry, D.: Periods of automorphic forms, poles of L-functions and functorial lifting. Sci. China Math. 53(9), 2215–2238 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ginzburg, D., Rallis, S., Soudry, D.: On Fourier coefficients of automorphic forms of symplectic groups. Manuscr. Math. 111(1), 1–16 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ginzburg, D., Rallis, S., Soudry, D.: Periods, poles of \(L\)-functions and symplectic-orthogonal theta lifts. J. Reine Angew. Math. 487, 85–114 (1997)MathSciNetMATHGoogle Scholar
  16. 16.
    Ginzburg, D., Rallis, S., Soudry, D.: The Descent Map from Automorphic Representations of GL(n) to Classical Groups. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2011)Google Scholar
  17. 17.
    Goldfeld, D., Hundley, J.: Automorphic representations and L-functions for the general linear group, vol. I. With exercises and a preface by Xander Faber. Cambridge Studies in Advanced Mathematics, vol. 129. Cambridge University Press, Cambridge (2011)Google Scholar
  18. 18.
    Harris, M., Kudla, S.: Arithmetic automorphic forms for the nonholomorphic discrete series of \({\rm GSp}(2)\). Duke Math. J. 66(1), 59–121 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Howe, R.: \(\theta \)-series and invariant theory. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977). Part 1, pp. 275–285. Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI (1979)Google Scholar
  20. 20.
    Hundley, J., Sayag, E.: Descent construction for GSpin groups. Preprint. arXiv:1110.6788
  21. 21.
    Jacquet, H.: Fonctions de Whittaker associées aux groupes de Chevalley. Bull. Soc. Math. France 95, 243–309 (1967)MathSciNetMATHGoogle Scholar
  22. 22.
    Jacquet, H., Shalika, J.: Exterior square L-functions. Automorphic forms, Shimura varieties, and L-functions, vol. II (Ann Arbor, MI, 1988), pp. 143–226. Perspect. Math., vol. 11. Academic Press, Boston, MA (1990)Google Scholar
  23. 23.
    King, R.C.: Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups. J. Math. Phys. 12, 1588–1598 (1971)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kudla, S.: Notes on the Local Theta Correspondence. Preprint. http://www.math.toronto.edu/~skudla/ssk.research.html
  25. 25.
    Langlands, R.: Euler Products. Yale University Press (1971). http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/
  26. 26.
    Lapid, E., Mao, Z.: On the asymptotics of Whittaker functions. Representation Theory vol. 13, pp. 63–81 with correction on p. 348 of the same volume (2009)Google Scholar
  27. 27.
    Moeglin, C., Waldspurger, J.-L.: Spectral Decomposition and Eisenstein Series. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  28. 28.
    Prasad, D.: Weil Representation. Howe Duality, and the Theta correspondence. CRM Proceedings and Lecture Notes, vol. 1, pp. 105–127 (1993)Google Scholar
  29. 29.
    Rallis, S.: Langlands’ functoriality and the Weil representation. Am. J. Math. 104(3), 469–515 (1982)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wallach, N.R.: Real Reductive Groups. II. Pure and Applied Mathematics, vol. 132-II. Academic Press Inc, Boston (1992)Google Scholar
  31. 31.
    Soudry, D.: Rankin-Selberg convolutions for \(SO_{2l+1}\times GL_n\): local theory. Memoirs of the American Mathematical Society, vol. 105, no. 500. American Mathematical Society, Providence, RI (1993)Google Scholar
  32. 32.
    Soudry, D.: On the Archimedean theory of Rankin-Selberg convolutions for \(SO_{2l+1}\times GL_n\). Annales scientifiques de l’ É.N.S. 4\(^{e}\) série, tome 28, no. 2, 161–224 (1995)Google Scholar
  33. 33.
    Tate, J.: Fourier analysis in number fields, and Hecke’s zeta-functions. In: Cassels, F. (ed.) 1967 Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 305–347. Thompson, Washington, DC (1965)Google Scholar
  34. 34.
    Waldspurger, J.-L.: La formule de Plancherel pour les groupes p-adiques d’après Harish-Chandra. J. Inst. Math. Jussieu 2(2), 235–333 (2003)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wallach, N.R.: Real Reductive Groups. I. Pure and Applied Mathematics, vol. 132. Academic Press Inc, Boston (1988)Google Scholar

Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Department of Mathematics, 244 Mathematics BuildingUniversity at BuffaloBuffaloUSA
  2. 2.Department of MathematicsUniversity of Toronto, Bahen CentreTorontoCanada

Personalised recommendations