Jacobi sums over finite fields (Q2781446)

Let \(e > 2\) be an integer, \( \zeta_{e}\) a primitive \(e\)th root of unity, \(K={ \mathbb Q}( \zeta_{e})\), and \(p\) a prime number not dividing \(e\). Let \(r\) be an integer such that \(p^{r} \equiv 1 (\bmod\, e)\) and \(r_{0}\) the least positive integer such that \(p^{r_{0}} \equiv 1 (\bmod\, e)\). Let \({ \mathbb F}_{q}\) be the finite field with \(q=p^{r}\) elements, \( \gamma\) a generator of the cyclic group \(F_{q}^{*}\), and \( \chi: { \mathbb F}_{q}^{*} \rightarrow { \mathbb Q}( \zeta_{e})\) a multiplicative character of \({ \mathbb F}_{q}\) such that \( \chi( \gamma)= \zeta_{e}\) and \( \chi(0)=0\). For two integers \(m\) and \(n\) the Jacobi sum \(J( \chi^{m}, \chi^{n})\) is defined by NEWLINE\[NEWLINE J( \chi^{m}, \chi^{n})= \sum_{ \alpha \in { \mathbb F}_{q}} \chi^{m}(\alpha) \chi^{n}(1 - \alpha). NEWLINE\]NEWLINE The author treats two important problems in the theory of Jacobi sums. The first problem is that of giving a Diophantine system (of equations and congruences) whose unique solution determines a particular Jacobi sum. The other problem is that of giving algorithms for fast computation of Jacobi sums.NEWLINENEWLINEThe first result giving Diophantine systems for the coefficients of Jacobi sums with \(e=2^{k}\), \(m=1\), \(n=e/2\) and \(r=1\) was that of \textit{R. J. Evans} [Acta Arith. 39, 281--294 (1981; Zbl 0472.10006)]. Diophantine systems for all \(e\) equal to a prime or twice a prime and any \(m\), \(n\) and \(r\), but with \(r_{0}=1\), was later given by \textit{V. V. Acharya} and \textit{S. A. Katre} [Acta Arith. 69, 51--74 (1995; Zbl 0813.11067)]. The main result of the author generalizes these results in that it allows any \(e\), \(m\), \(n\), \(r\), \(r_{0}\) and provides existence of a unique polynomial NEWLINE\[NEWLINE H=a_{0}+a_{1}x+a_{2}x^{2}+ \cdots +a_{e-1}x^{e-1} \in { \mathbb Z}[x] NEWLINE\]NEWLINE such that NEWLINE\[NEWLINE H( \zeta_{e})=J( \chi^{m}, \chi^{n}). NEWLINE\]NEWLINE The coefficients of \(H(x)\) are uniquely determined by a system of Diophantine equations with only a finite number of solutions and by several congruence conditions ruling out all but one of the finite number of solutions.NEWLINENEWLINEIn conclusion the author gives some numerical examples on the computation of Jacobi sums and offers an algorithm which gives a possibility to compute any Jacobi sum recursively. The algorithm uses lattice reduction techniques to quickly find the value of a Jacobi sum up to a root of unity and also a result of the paper to find the correct root of unity. The algorithms compute any Jacobi sum faster than just naively summing the defined series in the case where \(q > \phi(e)^{ \phi(e)}\).