Exponential sums for \({\text{O}}^-(2n,q)\) and their applications (Q2717588)

Let \(q\) be an odd prime power, \(\lambda\) a nontrivial additive character of the finite field \(\mathbb{F}_q\), \(\chi\) a multiplicative character of \(\mathbb{F}_q\), and \(r\) a positive integer. The author evaluates the exponential sums \(\sum_{w\in \text{SO}^-(2n,q)}\lambda((\text{tr } w)^r)\) and \(\sum_{w\in \text{O}^-(2n,q)}\chi(\text{det } w)\lambda((\text{tr } w)^r)\) in terms of certain exponential sums. This extends previous results of the author for the case \(r=1\) [Acta Arith. 80, No. 4, 343-365 (1997; Zbl 0871.11090)]. NEWLINENEWLINENEWLINEAs application the author determines the number of \(w\in \text{SO}^-(2n,q)\) and \(w\in \text{O}^-(2n,q)\), respectively, with \((\text{tr} w)^r=\beta\) for \(\beta \in \mathbb{F}_q\).