Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 309–342

Simultaneous nonvanishing of automorphic L-functions at the central point

Article

Abstract

Let g be a holomorphic Hecke eigenform and {uj} an orthonormal basis of even Hecke–Maass forms for \({\textup{SL}(2,\mathbb{Z})}\). Denote L(s, g × uj) and L(s, uj) the corresponding L-functions. In this paper, we give an asymptotic formula for the average of \({L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)}\), from which we derive that there are infinitely many uj’s such that \({L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)\neq0}\).

Mathematics Subject Classification (2000)

11M41 11S40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Conrey J.B., Iwaniec H.: The cubic moment of central values of automorphic L-functions. Ann. Math. 151(3), 1175–1216 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Duke W., Friedlander J., Iwaniec H.: A quadratic divisor problem. Invent. Math. 115, 209–217 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Guo J.: On the positivity of the central critical values of automorphic L-functions for GL(2). Duke Math. J. 83(1), 157–190 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Goldfeld D., Hoffstein J., Lieman D.: An effective zero-free region. Ann. Math. 140(1), 177–181 (1994)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Gradshteyn I.S., Ryzhik I.M.: Table of integrals, series, and products. Translated from the Russian, Sixth edn. Academic Press, New York (2000)Google Scholar
  6. 6.
    Hoffstein J., Lockhart P.: Coefficients of Maass forms and the Siegel zero. Ann. Math. 140(1), 161–181 (1994)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Huxley, M.N.: Area, lattice points, and exponential sums. London mathematical society monograhphs. New Series, 13. Oxford Science Publications, Oxford University Press, New York (1996)Google Scholar
  8. 8.
    Ivić A.: On sums of Hecke series in short intervals. J. Thor. Nombres Bordeaux 13(2), 453–468 (2001)MATHGoogle Scholar
  9. 9.
    Iwaniec H.: The spectral growth of automorphic L-functions. J. Reine Angew. Math. 428, 139–159 (1992)MATHMathSciNetGoogle Scholar
  10. 10.
    Fouvry E., Iwaniec H.: Low-lying zeros of dihedral L-functions. Duke Math. J. 116(2), 189–217 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Iwaniec, H., Kowalski, E.: Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI (2004)Google Scholar
  12. 12.
    Iwaniec H., Sarnak P.: The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros. Israes J. Math. 120(Part A), 155–177 (2000)MATHMathSciNetGoogle Scholar
  13. 13.
    Iwaniec, H., Sarnak, P.: Perspectives on the analytic theory of L-functions. Geom. Funct. Analysis, special vol. 1–37 (2000)Google Scholar
  14. 14.
    Jutila, M.: The fourth moment of central values of Hecke series. In: Number theory (Turku, 1999), pp. 167–177. de Gruyter, Berlin (2001)Google Scholar
  15. 15.
    Katk S., Sarnak P.: Heegner points, cycles and Maass forms. Israel J. Math. 84(1-2), 193–227 (1993)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kowalski E., Michel P., VanderKam J.: Mollification of the fourth moment of automorphic L-functions and arithmetic applications. Invent. Math. 142(1), 95–151 (2000)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kowalski E., Michel P., VanderKam J.: Rankin-Selberg L-functions in the level aspect. Duke Math. J. 114, 123–191 (2002)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kuznetsov N.V.: Petersson’s conjecture for cusp forms of weight zero and Linnik’s conjecture. Sums of Kloosterman sums. Math. USSR Sbornik 29, 299–342 (1981)CrossRefGoogle Scholar
  19. 19.
    Lapid, E.: On the nonnegativity of Rankin-Selberg L-functions at the center of symmetry vol. 2, pp. 65–75. International mathematics research notice (2003)Google Scholar
  20. 20.
    Lau, Y., Liu, J., Ye, Y.: A new bound \({k^{\frac{2}{3}+\varepsilon}}\) for Rankin-Selberg L-functions for Hecke congruence subgroups. IMRP international mathematics research papers 2006, Art. ID 35090, 78 pp. (2010)Google Scholar
  21. 21.
    Li X.: The central value of the Rankin-Selberg L-functions. Geom. Funct. Anal. 18(5), 1660–1695 (2009)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Li, X.: Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions (to appear Ann. Math.)Google Scholar
  23. 23.
    Luo W.: Nonvanishing of L-values and the Weyl law. Ann. Math. 154(2), 477–502 (2001)MATHCrossRefGoogle Scholar
  24. 24.
    Luo, W., Rudnick, Z., Sarnak, P.: On the generalized Ramanujan conjecture for GL(n). Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996). In: Proceedings of Symposia in Pure Mathematics, vol. 66, Part 2, pp. 301–310. American Mathematics Society, Providence, RI, (1999)Google Scholar
  25. 25.
    Motohashi Y.: Spectral theory of the Riemann zeta-function. Cambridge Tracts in Mathematics, no. 127. Cambridge University Press, Cambridge (1997)Google Scholar
  26. 26.
    Ramakrishnan, D., Rogawski, J.: Average values of modular L-series via the relative trace formula. Pure Appl. Math. Q. 4(part 3):701–735 (2005)Google Scholar
  27. 27.
    Soundararajan K.: Nonvanishing of quadratic Dirichlet L-functions at \({s=\frac{1}{2}}\). Ann. Math. 152(2), 447–488 (2000)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Tithmarsh, E.C.: The theory of the Riemann zeta-function. Second edn. Oxford University Press, New York (1986) (Edited and with a preface by D. R. Heath-Brown)Google Scholar
  29. 29.
    Wilton J.R.: A note on Ramanujan’s arithmetical function τ(n). Proc. Camb. Phil. Soc. 25, 121–129 (1929)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China

Personalised recommendations