×

Seventh power moments of Kloosterman sums. (English) Zbl 1242.11059

Let \(\mathbb{F}_p\) be a finite field, and write \(\zeta_p=\exp(2\pi i/p)\). Consider the Kloosterman sums \[ K(a)=\sum_{x=1}^{p-1}\zeta_p^{x+a/x},\quad a\in \mathbb{F}_p, \] and their \(n\)-th power moments \[ S_n=\sum_{a=0}^{p-1}K(a)^n,\quad n\in \mathbb{N}. \] This paper presents substantial evidence for an evaluation of \(S_7\) in terms of Hecke eigenvalues for a weight \(3\) newform on \(\Gamma_0(525)\) with quartic nebentypus of conductor \(105\).

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms

Software:

SageMath
PDFBibTeX XMLCite
Full Text: DOI

References:

[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. · Zbl 1127.11054 · doi:10.1142/S1793042107000821
[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. · Zbl 1030.11038
[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. · Zbl 1165.11331 · doi:10.1515/crll.2005.2005.589.79
[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. · Zbl 1014.14021 · doi:10.1515/advg.2001.017
[5] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol 17, American Mathematical Society, Providence, RI, 1997. · Zbl 0905.11023
[6] N. M. Katz, Gauss sums, Kloosterman Sums, and Monodromy Groups, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1988. · Zbl 0675.14004
[7] N. M. Katz, Email correspondence, 2005–2006.
[8] R. Livné, Motivic orthogonal two-dimensional representations of Gal( \( \bar Q \) /Q), Israel Journal of Mathematics 92 (1995), 149–156. · Zbl 0847.11035 · doi:10.1007/BF02762074
[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. · Zbl 0749.14037
[10] SAGE Mathematical Software, www.sagemath.org .
[11] W. Stein, The Modular Forms Database, sage.math.washington.edu .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.