A congruence modulo 4 for Kloosterman sums over finite fields of characteristic 3 (Q2844620)

Let \(\mathbb F\) be a finite field of characteristic \(p\) with \(q = p^ m\) elements, and let \(a\) and \(c\) be elements of \(\mathbb F\). The Kloosterman sum \(K(a, c; m)\) is defined as \(\sum _ {x \in\mathbb F*} \omega ^ {\text{Tr}(ax+cx ^ {-1})}\), where \(\omega \) is a complex primitive \(p\)-th root of unity, and Tr is the trace map from \(\mathbb F\) to the prime subfield \(\mathbb F_ p\) of \(\mathbb F\). It is easy to see that \(K(a, c; m)\) is a real algebraic integer lying in the field \(\mathbb Q(\omega)\), which implies that \(K(a, c; m) \in \mathbb Q(\omega + \omega ^ {-1})\). In particular, \(K(a, c; m) \in \mathbb Z\), provided that \(p = 3\). Henceforth, we assume that \(p = 3\) and \(c = 1\), and put \(K(a; m) = K(a, 1; m)\). As an answer to a problem posed by Garaschuk and Lisoněk, the paper under review characterizes those \(a \in \mathbb F\) for which \(K(a; m)\) is congruent to \(\varepsilon\) modulo \(4\), where \(\varepsilon\) equals \(1\) or \(3\). The characterization depends on whether or not \(m\) is even. The analogous problem, for \(\varepsilon = 0\) and \(\varepsilon = 2\) has already been solved by \textit{K. Garaschuk} and \textit{P. Lisoněk} [Finite Fields Appl. 14, No. 4, 1083--1090 (2008; Zbl 1179.11046)].