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Explicit lifts of quintic Jacobi sums and period polynomials for \(\mathbb F_q\). (English) Zbl 1116.11093
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 \(q=p\), 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 \(\delta= \gcd(e,(q-1)/(p-1))\), 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 B. C. Berndt, R. J. Evans and 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, 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 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.

MSC:
11T24 Other character sums and Gauss sums
11L05 Gauss and Kloosterman sums; generalizations
11T22 Cyclotomy
Software:
Mathematica
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References:
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