Gauss sums over some matrix groups

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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
DOI
10.1016/j.jnt.2012.06.010
URI
http://hdl.handle.net/10203/103049
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