Abstract
For a nontrivial additive character λ and a multiplicative character χ of the finite field withq elements, the ‘Gauss’ sums Σλ(trg) overg∈Sp(2n,q) and Σχ(detg)λ(trg) overg∈GSp(2n, q) are considered. We show that it can be expressed as a polynomial inq with coefficients involving powers of Kloosterman sums for the first one and as that with coefficients involving sums of twisted powers of Kloosterman sums for the second one. As a result, we can determine certain ‘generalized Kloosterman sums over nonsingular matrices’ and ‘generalized Kloosterman sums over nonsingular alternating matrices’, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.
Similar content being viewed by others
References
Carlitz L (1954) Representation by skew forms in a finite field. Arch Math5: 19–31
Hodges JH (1955) Representations by bilinear forms in a finite field. Duke Math J22: 497–509
Hodges JH (1956) Weighted partitions for general matrices over a finite field. Duke Math J23: 545–552
Hodges JH (1956) Exponential sums for skew matrices in a finite field. Arch Math7: 116–121
Hodges JH (1957) Weighted partitions for skew matrices over a finite field. Arch Math8: 116–121
Kim DS (1997) Gauss sums for general and special linear groups over a finite field. Arch Math (to appear)
Kim DS (1998) Gauss sums forO(2n+1,q). Finite Fields and Their Applications (to appear)
Kim DS (1997) Gauss sums forO −(2n,q). Acta Arith80: 343–365
Kim DS, Lee IS (1996) Gauss sums forO +(2n,q). Acta Arith78: 75–89
Lehmer DH, Lehmer E (1960) On the cubes of Kloosterman sums. Acta Arith6: 15–22
Lehmer DH, Lehmer E (1967) The cyclotomy of Kloosterman sums. Acta Arith12: 385–407
Lidl R, Niederreiter H (1987) Finite Fields. Cambridge: Univ Press
Salié (1932) Über die Kloostermanschen Summen S(u,v;q). Math Z34: 91–109
Author information
Authors and Affiliations
Additional information
Dedicated to my father, Chang Hong Kim
Supported in part by Basic Science Research Institute program, Ministry of Education of Korea, BSRI 95-1414 and KOSEF Research Grant 95-K3-0101 (RCAA)
Rights and permissions
About this article
Cite this article
Kim, D.S. Gauss sums for symplectic groups over a finite field. Monatshefte für Mathematik 126, 55–71 (1998). https://doi.org/10.1007/BF01312455
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01312455