Exponential sums for \(O(2n+1,q)\) and their applications (Q2731116)

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 \(\sum_{w\in \text{SO}(2n+1,q)}\lambda((\text{tr }w)^r)\) and \(\sum_{w\in \text{O}(2n+1,q)}\chi(\det w)\lambda((\text{tr }w)^r)\) in terms of certain exponential sums. This extends previous results of the author for the case \(r=1\) [Finite Fields Appl. 4, No. 1, 62-86 (1998; Zbl 0937.11058)]. As an application, the author determines the number of \(w\in\) SO\((2n+1,q)\) and \(w\in\) O\((2n+1,q)\), respectively, with \((\text{tr }w)^r=\beta\) for \(\beta \in \mathbb{F}_q\).