Advertisement

Inventiones mathematicae

, Volume 70, Issue 2, pp 219–288 | Cite as

Kloosterman sums and Fourier coefficients of cusp forms

  • J. -M. Deshouillers
  • H. Iwaniec
Article

Keywords

Fourier Fourier Coefficient Cusp Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bru] Bruggeman, R.W.: Fourier coefficients of cusp forms. Invent. Math.45, 1–18 (1978)Google Scholar
  2. [Bus] Buser, P.: On Cheeger's inequalityλ 1h 2/4. Proc. Symposia, in Pure Math., XXXVI, pp. 29–77 AMS, Providence 1981Google Scholar
  3. [Che] Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Problems in Analysis, A symposium in honour of Bochner, Princeton University Press, Princeton, N.J. 1970 pp. 195–199Google Scholar
  4. [Del] Deligne, P.: La conjecture de Weil I. Publ. Math. I.H.E.S.,43, 273–307 (1974)Google Scholar
  5. [Dl 1] Deshouillers, J.-M., Iwaniec, H.: On the greatest prime factor ofn 2+1 Ann. Inst. Fourier. in press (1982)Google Scholar
  6. [DI2] Deshouillers, J.-M., Iwaniec, H.: An additive divisor problem. J. London Math. Soc.26, (2), 1–14 (1982)Google Scholar
  7. [Di3] Deshouillers, J.-M., Iwaniec, H.: Power mean-values for the Riemann zeta-function. Mathematika in press (1982)Google Scholar
  8. [DI4] Deshouillers, J.-M., Iwaniec, H.: On the Brun-Titchmarsh theorem and the greatest prime factor ofp+a. Proc. Janos Bolyai Soc. Conf. in press (1982)Google Scholar
  9. [EMOT] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions II. New York-Toronto-London: Mc Graw-Hill 1953Google Scholar
  10. [Fad] Faddeev, L.D.: Expansions in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Lobačevskii plane. Trudy Moscov, Mat. Obsc.,17, 323–350 (1967)Google Scholar
  11. [G-R] Gradsztejn, I.S., Ryżyk, I.M.: Tablice całek, sum, szeregów i iloczynow. PWN-Warszawa, 1964Google Scholar
  12. [Gun] Gunning, R.C.: Lectures on modular forms. Princeton University Press, 1962Google Scholar
  13. [H.-B.] Heath-Brown, D.R.: The fourth power moment of the Riemann zeta-function. Proceeding London Math. Soc.38(3), 385–422 (1979)Google Scholar
  14. [Hej] Hejhal, D.A.: Some observations concerning eigenvalues of the Laplacian and DirichletL-series in Recent progress in analytic number theory. London: Academic Press 1981Google Scholar
  15. [Hoo] Hooley, C.: On the greatest prime factor of a quadratic polynomial. Acta Math.117, 281–299 (1967)Google Scholar
  16. [Iwa 1] Iwaniec, H.: Fourier coefficients of cusp forms and the Riemann zeta-function. Sém. Th. Nb. Bordeaux (1979–1980), exposé no18,36 pagesGoogle Scholar
  17. [Iwa 2] Iwaniec, H.: Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums. Proceedings from the Journées Arithmétiques at Exeter in press (1982)Google Scholar
  18. [Iwa 3] Iwaniec, H.: On mean values for Dirichlet's polynomials and the Riemann zeta-function. J. London Math. Soc.22, (2), 39–45 (1980)Google Scholar
  19. [Iwa 4] Iwaniec, H.: On the Brun-Titchmarsh theorem. J. Math. Soc. Japan34, 95–123 (1982)Google Scholar
  20. [Kub] Kubota, T.: Elementary Theory of Eisenstein Series. New-York: John Wiley and Sons 1973Google Scholar
  21. [Kuz1] Kuznietsov, N.V.: Petersson hypothesis for forms of weight zero and Linnik hypothesis (in Russian), Preprint no02, Khab. KHII, Khabarovsk 1977Google Scholar
  22. [Kuz2] Kuznietsov, N.V.: Petersson hypothesis for parabolic forms of weight zero and Linnik hypothesis. Sums of Kloostermann sums. Math. Sbornik111, (153), no3, 334–383 (1980)Google Scholar
  23. [Lin] Linnik, Yu.V.: Additive problems and eigenvalues of the modular operators. Proc. Internat. Congr. Math. Stockholm 270–284 (1962)Google Scholar
  24. [Maa1] Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen. Math. Ann.121, (o2) 141–183 (1949)Google Scholar
  25. [Maa2] Maass, H.: On modular functions of one complex variable. Tata Institute, Bombay, 1964Google Scholar
  26. [Mon] Montgomery, H.L.: Topics in multiplicative number theory. Lect. Notes in Math. vol. 227. Berlin-New-York: Springer 1971Google Scholar
  27. [Pet] Petersson, H.: Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math.58, 169–215 (1932)Google Scholar
  28. [Pro1] Proskurin, N.V.: Summation formulas for generalized Kloosterman sums (in Russian). Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.82, 103–135 (1979)Google Scholar
  29. [Pro2] Proskurin, N.V.: On a hypothesis of Yu.V. Linnik (in Russian). Zap. Naučn. Sem. Leningrad Otdel. Mat. Inst. Steklov.91, 94–118 (1979)Google Scholar
  30. [Pro3] Proskurin, N.V.: The estimations of the eigenvalues of Hecke's operators in the space of parabolic forms of weight zero. Studies in Number Theory, vol. 5, 136–143 (1979). Pub. Math. Inst. SteklovGoogle Scholar
  31. [Ran] Rankin, R.: Modular forms and functions. Cambridge-London-New-York: Cambridge University Press 1977Google Scholar
  32. [Roe1] Roelcke, W.: Über die Wellengleichung bei Grenzkreisgruppen erster Art. Sitz. Ber. Heidl. Akad. der Wiss. (Math. natur. Kl.) 1956, 4. Abh.Google Scholar
  33. [Roe2] Roelcke, W.: Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I. II. Math. Ann.167, 292–337 (1966);168, 261–324 (1967)Google Scholar
  34. [Sel1] Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet's series. J. Indian Mat. Soc.20, 47–87 (1956)Google Scholar
  35. [Sel2] Selberg, A.: On the estimation of Fourier coefficients of modular forms. Proc. Symposia in Pure Math. VIII, A.M.S., Providence 1965, pp. 1–15Google Scholar
  36. [Shi] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton: Princeton University Press 1971Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • J. -M. Deshouillers
    • 1
  • H. Iwaniec
    • 2
  1. 1.Laborative Associé au CNRS N0 226Université de Bordeaux ITalenceFrance
  2. 2.Mathematics Institute Polish Academy of Sciences ulWarszawa

Personalised recommendations