Congruences for Jacobi sums
Let \(\chi\) denote a multiplicative character of order \(k>2\) on a finite field \(F_q\), and write \(q=kf+1\). The author considers Jacobi sums of type \[ J(a,b) = - \sum_{\substack{\alpha \in F_q\\ \alpha \not = 0,-1}} \chi^a(\alpha) \chi^b (-1-\alpha) , \] where \(a\) and \(b\) are integers mod \(k\). He evaluates \(J(a,b)\) modulo \((1- \zeta^a)(1- \zeta^b)(1- \zeta^{-a-b})\) with \(\zeta = e^{2\pi i/k}\), which generalizes results of \textit{K. Iwasawa} [Sympos. Math. 15, 447-459 (1975; Zbl 0324.12007)] and \textit{Y. Ihara} [Ann. Math. (2) 123, 43-106 (1986; Zbl 0595.12003)] for prime power \(k\). The techniques are elementary and self-contained and depend on a result on cyclotomic numbers of order \(k\).
scientific article; zbMATH DE number 1221699
scientific article