Explicit lifts of quintic Jacobi sums and period polynomials for \(\mathbb F_q\)
From the author's introduction: Let \(e\geq 2\) be a positive integer and \(q=p^r\) a prime power such that \(q\equiv 1\pmod e\). Write \(q=ef+1\). Let \(\zeta_p\) be a \(p\)th primitive root of unity, \(\gamma\) a fixed generator of \(\mathbb F_q\). Gaussian periods \(\eta_{0,r},\dots, \eta_{e-1,r}\) of degree \(e\) for \(\mathbb F_q\) are defined by \[ \eta_{i,r}:= \sum_{j=0}^{f-1} \zeta_p^{\text{Tr} (\gamma^{ej+i})}, \] where GR is the trace map \(\text{Tr}:\mathbb F_q\to\mathbb F_p\), and the period polynomial \(P_{e.r}(X)\) of degree \(e\) for \(\mathbb F_q\) is given by \(P_{e,r}(X):= \prod_{i=0}^{e-1} (X-\eta_{i,r})\). We also use the reduced form \(P_{e,r}^*(X):= \prod_{i=0}^{e-1} (X-\eta_{i,r}^*)\), where \(\eta_{i,r}^*= e\eta_{i,r}+1\), since the coefficient of \(X^{e-1}\) of \(P_{e,r}^*(X)\) is vanished. In the classical case \(q=p\), \textit{C. F. Gauss} [Disquisitiones Arithmeticae, Section 358] showed that the period polynomial \(P_{e,1}(X)\) is irreducible over \(\mathbb Q\). However this is not always true for general \(q=p^r\). In 1981, for \(\delta= \gcd(e,(q-1)/(p-1))\), \textit{G. Myerson} [ Acta Arith. 39, 251--264 (1981; Zbl 0393.12028)] showed that the period polynomial \(P_{e,r}(X)\) splits over \(\mathbb Q\) into \(\delta\) factors \[ P_{e,r}(X)= \prod_{k=0}^{\delta-1} P_{e,r}^{(k)}(X), \] where \(P_{e,r}^{(k)}(X)\) is in \(\mathbb Z[X]\) and irreducible or a power of an irreducible polynomial. Note that \(P_{e,r}(X)\) is irreducible over \(\mathbb Q\) if and only if \(p\equiv 1\pmod e\) and \((r,e)= 1\), i.e. \(\delta=1\). The explicit determination of the factors of \(P_{e,r}(X)\), if reducible, is important because it is known that the (exponential) Gauss sum \(g_r(e)\) is one of the roots of \(P_{e,r}^*(X)\) [see \textit{B. C. Berndt, R. J. Evans} and \textit{K. S. Williams}, Gauss and Jacobi sums, New York, NY: John Wiley \& Sons (1998; Zbl 0906.11001)]. Myerson (loc. cit.) determined the factors \(P_{e,r}^{(k)}(X)\) for \(e=2,3,4\). In 2004, \textit{S. J. Gurak} [Period polynomials for \(\mathbb F_q\) fixed small degree, Kisilevsky, Hershy (ed.) et al., Number theory. Papers from the 7th conference of the Canadian Number Theory Association, University of Montreal, Montreal, QC, Canada, May 19--25, 2002. Providence, RI: American Mathematical Society (AMS). CRM Proceedings \& Lecture Notes 36, 127--145 (2004; Zbl 1153.11321)] gave the factors \(P_{e,r}^{(k)}(X)\) for the case \(e\mid 8,12\) [see also \textit{S. J. Gurak}, Period polynomials for \(\mathbb{F}_{p^2}\) of fixed small degree, Jungnickel, Dieter (ed.) et al., Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications \(F_q5\), University of Augsburg, Germany, August 2-6, 1999. Berlin: Springer. 196--207 (2001; Zbl 1006.11069)]. However it seems to be hard to determine the explicit factors \(P_{e,r}^{(k)}(X)\) for general prime degree. In this paper, we give the factors \(P_{e,r}^{(k)}(X)\) in the quintic case \(e=5\) by constructing explicit lifts of quintic Jacobi sums.
scientific article; zbMATH DE number 5161646