Israel Journal of Mathematics

, Volume 120, Issue 2, pp 449–466 | Cite as

Holomorphy of Rankin tripleL-functions; special values and root numbers for symmetric cubeL-functions

  • Henry H. Kim
  • Freydoon Shahidi


In this paper we prove the holomorphy of Rankin tripleL-functions for three cusp forms on GL(2) on the entire complex plane, if at least one of them is non-monomial. We conclude the paper by proving the equality of our root numbers for the third and the fourth symmetric powerL-functions with those of Artin through the local Langlands correspondence. We also revisit Deligne’s conjecture on special values of symmetric cubeL-functions.


Modular Form Parabolic Subgroup Eisenstein Series Cusp Form Root Number 
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  1. [B]
    D. Blasius,Critical values of certain tensor product L-functions, Inventiones Mathematicae90 (1987), 181–188.MathSciNetCrossRefMATHGoogle Scholar
  2. [BHK1]
    C. Bushnell, G. Henniart and P. Kutzko,Correspondance de Langlands locale pour GL n et conducteurs de pairs, preprint, 1997.Google Scholar
  3. [BHK2]
    C. Bushnell, G. Henniart and P. Kutzko,Local Rankin-Selberg convolutions for GLn:Explicit conductor formula, Journal of the American Mathematical Society (to appear).Google Scholar
  4. [C-Sh]
    W. Casselman and F. Shahidi,On irreducibility of standard modules for generic representations, Annales Scientifiques de l’École Normale Supérieure31 (1998), 561–589.MathSciNetCrossRefMATHGoogle Scholar
  5. [D]
    P. Deligne,Valeurs de fonctions L et périodes d’intǵrales, Proceedings of Symposia in Pure Mathematics II33 (1979), 313–346.MathSciNetCrossRefGoogle Scholar
  6. [Ga]
    P. Garrett,Decomposition of Eisenstein series, Rankin triple products, Annals of Mathematics125 (1987), 209–235.MathSciNetCrossRefMATHGoogle Scholar
  7. [GH]
    P. Garrett and M. Harris,Special values of triple product L-functions, American Journal of Mathematics115 (1993), 159–238.MathSciNetCrossRefMATHGoogle Scholar
  8. [Ge-Ja]
    S. Gelbart and H. Jacquet,A relation between automorphic representations of GL(2)and GL(3), Annales Scientifiques de l’École Normale Supérieure11 (1978), 471–552.MathSciNetMATHGoogle Scholar
  9. [HK]
    M. Harris and S. Kudla,The central critical value of a triple product L-function, Annals of Mathematics133 (1991), 605–672.MathSciNetCrossRefMATHGoogle Scholar
  10. [HT]
    M. Harris and R. Taylor,On the geometry and cohomology of some simple Shimura varieties, Preliminary version (1998).Google Scholar
  11. [H1]
    G. Henniart,La conjecture de Langlands locale pour GL(3), Mémoires de la Société Mathématique de France11/12 (1984), 1–186.MATHGoogle Scholar
  12. [H2]
    G. Henniart,Une preuve simple de conjectures de Langlands pour GL(n)sur un corp p-adiques, Inventiones Mathematicae (to appear).Google Scholar
  13. [Ik]
    T. Ikeda,On the location of poles of the triple L-functions, Compositio Mathematica83 (1992), 187–237.MathSciNetMATHGoogle Scholar
  14. [JPSS]
    H. Jacquet, I. Piatetski-Shapiro and J. Shalika,Rankin-Selberg convolutions, American Journal of Mathematics105 (1983), 367–464.MathSciNetCrossRefMATHGoogle Scholar
  15. [Ku]
    P. Kutzko,The Langlands conjecture for GL2 of a local field, Annals of Mathematics112 (1980), 381–412.MathSciNetCrossRefMATHGoogle Scholar
  16. [Ki1]
    H. Kim,The residual spectrum of Sp4, Compositio Mathematica99 (1995), 129–151.MathSciNetGoogle Scholar
  17. [Ki2]
    H. Kim,The residual spectrum of G 2, Canadian Journal of Mathematics48 (1996), 1245–1272.MathSciNetCrossRefMATHGoogle Scholar
  18. [Ki3]
    H. Kim,Langlands-Shahidi method and poles of automorphic L-functions: application to exterior square L-functions, Canadian Journal of Mathematics51 (1999), 835–849.MathSciNetCrossRefMATHGoogle Scholar
  19. [Ki4]
    H. Kim,Langlands-Shahidi method and poles of automorphic L-functions II, Israel Journal of Mathematics117 (2000), 261–284;Correction,118 (2000), 379.MathSciNetCrossRefMATHGoogle Scholar
  20. [Ki-Sh]
    H. Kim and F. Shahidi,Symmetric cube L-functions of GL2 are entire, Annals of Mathematics150 (1999), 645–662.MathSciNetCrossRefMATHGoogle Scholar
  21. [L-La]
    J-P. Labesse and R. P. Langlands,L-indistinguishability for SL(2), Canadian Journal of Mathematics31 (1979), 726–785.MathSciNetCrossRefMATHGoogle Scholar
  22. [La1]
    R. P. Langlands,Euler Products, Yale University Press, 1971.Google Scholar
  23. [La2]
    R. P. Langlands,On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics544, Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  24. [La3]
    R. P. Langlands,On the classification of irreducible representations of real algebraic groups, inRepresentation Theory and Harmonic Analysis on Semisimple Lie Groups (P. J. Sally, Jr. and D. A. Vogan, eds.), Mathematical Surveys and Monographs31 (1989), 101–170.MathSciNetCrossRefMATHGoogle Scholar
  25. [Li]
    J. S. Li,Some results on the unramified principal series of p-adic groups, Mathematische Annalen292 (1992), 747–761.MathSciNetCrossRefMATHGoogle Scholar
  26. [M-W1]
    C. Moeglin and J.-L. Waldspurger,Spectral decomposition and Eisenstein series, une paraphrase de l’Ecriture, Cambridge Tracts in Mathematics, Vol. 113, Cambridge University Press, 1995.Google Scholar
  27. [M-W2]
    C. Moeglin and J.-L. Waldspurger,Le spectre résiduel de GL(n), Annales Scientifiques de l’École Normale Supérieure22 (1989), 605–674.MathSciNetMATHGoogle Scholar
  28. [Mu]
    G. Muić,The unitary dual of p-adic G 2, Duke Mathematical Journal90 (1997), 465–493.MathSciNetCrossRefMATHGoogle Scholar
  29. [O]
    T. Orloff,Special values and mixed weight triple products, Inventiones Mathematicae90 (1987), 169–180.MathSciNetCrossRefMATHGoogle Scholar
  30. [PS-Ra]
    I. Piatetski-Shapiro and S. Rallis,Rankin triple L-functions, Compositio Mathematica64 (1987), 31–115.MathSciNetMATHGoogle Scholar
  31. [Ram1]
    D. Ramakrishnan,On the coefficients of cusp forms, Mathematical Research Letters4 (1997), 295–307.MathSciNetCrossRefMATHGoogle Scholar
  32. [Ram2]
    D. Ramakrishnan,Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), preprint (1998).Google Scholar
  33. [Sh1]
    F. Shahidi,A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Annals of Mathematics132 (1990), 273–330.MathSciNetCrossRefMATHGoogle Scholar
  34. [Sh2]
    F. Shahidi,Fourier transforms of intertwining operators and Plancherel measures for GL(n), American Journal of Mathematics106 (1984), 67–111.MathSciNetCrossRefMATHGoogle Scholar
  35. [Sh3]
    F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certain L-functions, Annals of Mathematics127 (1988), 547–584.MathSciNetCrossRefMATHGoogle Scholar
  36. [Sh4]
    F. Shahidi,Automorphic L-functions: a survey, inAutomorphic Forms, Shimura Varieties, and L-functions (L. Clozel and J. Milne, eds.), Vol. II (Ann Arbor, MI, 1988), Academic Press, New York, 1990, pp. 415–437.Google Scholar
  37. [Sh5]
    F. Shahidi,Third symmetric power L-functions for GL(2), Compositio Mathematica70 (1989), 245–273.MathSciNetMATHGoogle Scholar
  38. [Sh6]
    F. Shahidi,Local coefficients as Artin factors for real groups, Duke Mathematical Journal52 (1985), 973–1007.MathSciNetCrossRefMATHGoogle Scholar
  39. [Sh7]
    F. Shahidi,Symmetric power L-functions, inElliptic Curves and Related Topics (H. Kisilevsky and R. Murty, eds.), CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, NJ, 1994, pp. 159–182.Google Scholar
  40. [S]
    G. Shimura,On the periods of modular forms, Mathematische Annalen229 (1977), 211–221.MathSciNetCrossRefMATHGoogle Scholar
  41. [T]
    J. Tate,Number theory background, Proceedings of Symposia in Pure Mathematics II33 (1970), 3–26.Google Scholar
  42. [Z]
    D. Zagier,Modular forms whose Fourier coefficients involve zeta functions of quadratic fields, inModular Forms of One Variable VI, Lecture Notes in Mathematics627, Springer-Verlag, New York, 1977, pp. 105–169.CrossRefGoogle Scholar
  43. [Za]
    S. Zampera,The residual spectrum of type G 2, Journal de Mathématiques Pures et Appliquées76 (1997), 805–835.MathSciNetCrossRefMATHGoogle Scholar
  44. [Zh]
    Y. Zhang,The holomorphy and nonvanishing of normalized intertwining operators, Pacific Journal of Mathematics180 (1997), 385–398.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2000

Authors and Affiliations

  1. 1.Department of MathematicsSouthern Illinois UniversityCarbondaleUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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