Gauss sums over some matrix groups
In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involve classical Gauss sums (resp, Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups, i.e., the groups of upper triangular matrices. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds of Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [R. Ferguson, C. Hoffman, F. Luca, A. Ostafe, I.E. Shparlinski, Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. 23 (2010) 501-513]. (C) 2012 Elsevier Inc. All rights reserved.
- Publisher
- ACADEMIC PRESS INC ELSEVIER SCIENCE
- Issue Date
- 2012-12
- Language
- English
- Article Type
- Article
- Keywords
SPECIAL LINEAR-GROUPS; EXPONENTIAL-SUMS; FINITE-FIELD; SYMPLECTIC GROUPS; O+(2N,Q); O-(2N,Q)
- Citation
JOURNAL OF NUMBER THEORY, v.132, no.12, pp.2967 - 2976
- ISSN
- 0022-314X
- Appears in Collection
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