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Explicit values of multi-dimensional Kloosterman sums for prime powers, II


Author: S. Gurak
Journal: Math. Comp. 77 (2008), 475-493
MSC (2000): Primary 11L05, 11T24
DOI: https://doi.org/10.1090/S0025-5718-07-02011-X
Published electronically: May 14, 2007
MathSciNet review: 2353962
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Abstract: For any integer $ m>1$ fix $ \zeta_{m}={\rm exp}(2 \pi i/m)$, and let $ {\bf Z}_{m}^{*}$ denote the group of reduced residues modulo $ m$. Let $ q=p^{\alpha}$, a power of a prime $ p$. The hyper-Kloosterman sums of dimension $ n>0$ are defined for $ q$ by

$\displaystyle \displaylines{ R(d,q)= \sum_{x_{1}, ..., x_{n} \in Z_{q}^{*}} \ze... ...ots +x_{n} +d(x_{1} \cdots x_{n})^{-1}} \;\;\;\;\;\; (d \in {\bf Z}_{q}^{*}), }$

where $ x^{-1}$ denotes the multiplicative inverse of $ x$ modulo $ q$.

Salie evaluated $ R(d,q)$ in the classical setting $ n=1$ for even $ q$, and for odd $ q=p^{\alpha}$ with $ \alpha >1$. Later, Smith provided formulas that simplified the computation of $ R(d,q)$ in these cases for $ n>1$. Recently, Cochrane, Liu and Zheng computed upper bounds for $ R(d,q)$ in the general case $ n >0$, stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for $ \alpha >1$, relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author's previous determination of these Kloosterman sums using character theory and $ p$-adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.


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  • 1. B.C. Berndt, R.J. Evans and K.S. Williams, Gauss and Jacobi Sums, Wiley-Interscience, New York, (1998). MR 1625181 (99d:11092)
  • 2. Z. Borevich and I. Shafarevich, Number Theory, Academic Press, New York, (1966). MR 0195803 (33:4001)
  • 3. J. Bourgain, ``Exponential sum estimates on subgroups of $ {\bf Z}_{q}^{*}$, $ q$ arbitrary,'' J. Analyse Math. 97 (2005), 317-355.
  • 4. J. Bourgain and M-C. Chang, ``Exponential sum estimates over subgroups and almost subgroups of $ {\bf Z}_{q}^{*}$, where $ q$ is composite with few prime factors'', Geom. Funct. Anal. 16 (2006), 327-366. MR 2231466 (2007d:11093)
  • 5. T. Cochrane and Z. Zheng, ``Pure and mixed exponential sums,'' Acta Arith. 91 no. 3 (1999), 249-278. MR 1735676 (2000k:11093)
  • 6. T. Cochrane, M. Liu and Z. Zheng, ``Upper bounds on n-dimensional Kloosterman sums,'' J. Number Theory 106 (2004), 259-274. MR 2059074 (2005d:11122)
  • 7. P. Deligne, ``Applications de la formula des traces aux sommes trigonometriques'' in Cohomologie etale (SGA 4.5), 168-232, Lecture Notes in Math. 569, Springer-Verlag, Berlin (1977).
  • 8. W. Duke, ``On multiple Salie sums'', Proc. Amer. Math Soc. 114 (1992), 623-625. MR 1077785 (92f:11113)
  • 9. R.J. Evans, ``Twisted Hyper-Kloosterman Sums over finite rings of integers'', in Proc. Millennial Conf. No. Theory, vol I, 429 -449; M.A. Bennett et al. eds, A.K. Peters, Natick, MA (2002). MR 1956239 (2003m:11125)
  • 10. S. Gurak, ``Minimal polynomials for Gauss periods with $ f=2$'', Acta Arith. 121, no. 3 (2006), 233-257. MR 2218343 (2006m:11119)
  • 11. S. Gurak, ``On the minimal polynomial of Gauss periods for prime powers'', Math Comp. 75 (2006), 2021-2035. MR 2240647
  • 12. S. Gurak, ``Explicit values of multi-dimensional Kloosterman sums for prime powers, I'' (to appear)
  • 13. S. Gurak, ``Polynomials for Hyper-Kloosterman sums'' (to appear)
  • 14. D.R. Heath-Brown and S. Konyagan, ``New bounds for Gauss sums derived from $ k$-th powers and for Heilbron's Exponential Sum,'' Quat. J. Math. 51 (2000), 221-235. MR 1765792 (2001h:11106)
  • 15. H. Iwaniec, Topics in classical automorphic forms Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI (1997). MR 1474964 (98e:11051)
  • 16. H.D. Kloosterman, ``On the representation of a number in the form $ ax^{2} +by^{2} + cz^{2} + dt^{2}$'', Acta Math. 49 (1926), 407-464.
  • 17. H. Salie, ``Uber die Kloostermanschen Summen $ S(u,v:q)$'', Math. Z. 34 (1932), 91-109. MR 1545243
  • 18. R.A. Smith, ``On $ n$-dimensional Kloosterman sums'', J. Number Theory 11 (1979), 324-343. MR 544261 (80i:10052)

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Additional Information

S. Gurak
Affiliation: Department of Mathematics, University of San Diego, San Diego, California 92110

DOI: https://doi.org/10.1090/S0025-5718-07-02011-X
Received by editor(s): May 10, 2006
Received by editor(s) in revised form: November 8, 2006
Published electronically: May 14, 2007
Dedicated: In memory of Derick H. Lehmer
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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