Sums for \(U(2n,q^2)\) and their applications (Q2773340)

Let \(v\) be a complex-valued function on the finite field \(\mathbb{F}_q\), \(v'=v(\text{tr }_{\mathbb{F}_{q^2}|\mathbb{F}_q})\) the lifting of \(v\) to \(\mathbb{F}_{q^2}\), and \(u:\mathbb{F}_{q^2}\rightarrow \mathbb C\) any function. The author evaluates the exponential sums NEWLINE\[NEWLINE\sum_{w\in \text{SU}(2n,q^2)}v'((\text{tr } w))\quad\text{and}\quad \sum_{w\in \text{U}(2n,q^2)}u(\det w)v'(\text{tr } w)NEWLINE\]NEWLINE in terms of certain exponential sums. This extends previous results of the author for the case that \(v'\) is a nontrivial additive character and \(u\) is a multiplicative character of \(\mathbb{F}_q\) [Glasg. Math. J. 40, 79-95 (1998; Zbl 0915.11061)]. As an application, the author determines the number of \(w\in\) U\((2n,q^2)\) with \(\det w=\alpha\) and \(\text{tr }_{\mathbb{F}_{q^2}|\mathbb{F}_q}\text{tr } w=\beta\) for \(\alpha\) with \(\text{N}_{\mathbb{F}_{q^2}|\mathbb{F}_q}(\alpha)=1\) and \(\beta \in \mathbb{F}_q\).