Gauss sums over some matrix groups

Details Gauss sums over some matrix groups Gauss sums over some matrix groups

2011-05-23

arxiv.org/abs/1105.4513v1

Mathematics

Number Theory

9 pages

Scientific paper

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involves Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds by Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [ Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. (23) 2010, 501--513 ].

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