Israel Journal of Mathematics

, Volume 175, Issue 1, pp 349–362 | Cite as

Seventh power moments of Kloosterman sums

  • Ronald Evans


Evaluations of the n-th power moments S n of Kloosterman sums are known only for n ⩽ 6. We present here substantial evidence for an evaluation of S 7 in terms of Hecke eigenvalues for a weight 3 newform on ΓO(525) with quartic nebentypus of conductor 105. We also prove some congruences modulo 3, 5 and 7 for the closely related quantity T 7, where T n is a sum of traces of n-th symmetric powers of the Kloosterman sheaf.


Dirichlet Character Symmetric Power Quadratic Character Power Moment Congruence Modulo 
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  1. [1]
    H. T. Choi and R. J. Evans, Congruences for sums of powers of Kloosterman sums, International Journal of Number Theory 3 (2007), 105–117.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. J. Evans, Twisted hyper-Kloosterman sums over finite rings of integers, in Number Theory for the Millennium, Proc. Millennial Conf. Number Theory (Urbana, IL, May 21–26, 2000) (M. A. Bennett et al., eds.), Vol. I, A. K. Peters, Natick, MA, 2002, pp. 429–448.Google Scholar
  3. [3]
    L. Fu and D. Wan, L—functions for symmetric products of Kloosterman sums, Journal für die Reine und Angewandte Mathematik 589 (2005), 79–103.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    K. Hulek, J. Spandaw, B. van Geemen and D. van Straten, The modularity of the Barth-Nieto quintic and its relatives, Advances in Geometry 1 (2001), 263–289.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol 17, American Mathematical Society, Providence, RI, 1997.zbMATHGoogle Scholar
  6. [6]
    N. M. Katz, Gauss sums, Kloosterman Sums, and Monodromy Groups, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1988.zbMATHGoogle Scholar
  7. [7]
    N. M. Katz, Email correspondence, 2005–2006.Google Scholar
  8. [8]
    R. Livné, Motivic orthogonal two-dimensional representations of Gal(\( \bar Q \)/Q), Israel Journal of Mathematics 92 (1995), 149–156.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    C. Peters, J. Top and M. van der Vlugt, The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes, Journal für die Reine und Angewandte Mathematik 432 (1992), 151–176.zbMATHGoogle Scholar
  10. [10]
    SAGE Mathematical Software,
  11. [11]
    W. Stein, The Modular Forms Database,

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of Mathematics, 0112University of California at San DiegoLa JollaUSA

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