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Gauss sums for \(U(2n+1,q^2)\). (English) Zbl 1036.11527

The author evaluates the Gauss sums \(\sum_{\omega\in G} \chi(\det\omega) \lambda' (\operatorname {tr}\omega)\) for the unitary and special unitary groups \(G= U(2n+1, q^2)\) and \(G= \text{SU} (2n+1, q^2)\) over the finite field \(F_{q^2}\); here \(\chi\) is a multiplicative character and \(\lambda'\) is a nontrivial additive character of \(F_{q^2}\). For the expression obtained \(U(2n+1, q^2)\) is a polynomial in \(q\) whose coefficients involve powers of twisted Kloosterman sums and averages of the usual Gauss sums.

MSC:

11T23 Exponential sums
11T24 Other character sums and Gauss sums
20G40 Linear algebraic groups over finite fields
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