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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
Mathematica
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References:
 [1] V. V. Acharya and S. A. Katre, Cyclotomic numbers of order $$2l$$, $$l$$ an odd prime, Acta Arith. 69 (1995), 51-74. · Zbl 0813.11067 · eudml:206672 [2] N. Anuradha and S. A. Katre, Number of points on the projective curves $$aY^l=bX^l+cZ^l$$ and $$aY^{2l}=bX^{2l}+cZ^{2l}$$ defined over finite fields, $$l$$ an odd prime, J. Number Theory 77 (1999), 288-313. · Zbl 0955.11017 · doi:10.1006/jnth.1999.2382 [3] L. D. Baumert, W. H. Mills and R. L. Ward, Uniform cyclotomy, J. Number Theory 14 (1982), 67-82. · Zbl 0475.12026 · doi:10.1016/0022-314X(82)90058-0 [4] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums , Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1998. · Zbl 0906.11001 [5] L. E. Dickson, Cyclotomy, higher congruences and Waring’s problem, Amer. J. Math. 57 (1935), 391-424. · Zbl 0012.01203 · doi:10.2307/2371217 [6] R. J. Evans, Pure Gauss sums over finite fields, Mathematika. 28 (1981), 239-248. · Zbl 0475.10032 · doi:10.1112/S0025579300010299 [7] C. F. Gauss, Disquisitiones Arithmeticae , Section 358. [8] S. Gurak, Period polynomials for $$F_ {p^ 2}$$ of fixed small degree, in Finite fields and applications ( Augsburg , 1999) 196-207, Springer, Berlin. · Zbl 1006.11069 · doi:10.1007/978-3-642-56755-1_16 [9] S. Gurak, Period polynomials for $$\mathbf{F}_ q$$ of fixed small degree, in Number theory , 127-145, Amer. Math. Soc., Providence, 2004. · Zbl 1153.11321 [10] K. Hashimoto and A. Hoshi, Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations, Math. Comp. 74 (2005), 1519-1530. · Zbl 1082.11069 · doi:10.1090/S0025-5718-05-01750-3 [11] K. Hashimoto and A. Hoshi, Geometric generalization of Gaussian period relations with application to Noether’s problem for meta-cyclic groups, Tokyo J. Math. 28 (2005), 13-32. · Zbl 1081.12002 · doi:10.3836/tjm/1244208276 [12] A. Hoshi, Multiplicative quadratic forms on algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no.4, 71-75. · Zbl 1099.11017 · doi:10.3792/pjaa.79.71 · euclid:pja/1116443656 [13] S. A. Katre and A. R. Rajwade, Unique determination of cyclotomic numbers of order five, Manuscripta Math. 53 (1985), 65-75. · Zbl 0577.12014 · doi:10.1007/BF01174011 · eudml:155089 [14] S. A. Katre and A. R. Rajwade, Complete solution of the cyclotomic problem in $$\mathbf{F}_q^{*}$$ for any prime modulus $$l$$, $$q=p^\alpha$$, $$p\equiv 1$$ (mod $$l$$), Acta Arith. 45 (1985), 183-199. · Zbl 0525.12015 · eudml:205966 [15] G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith. 39 (1981), 251-264. · Zbl 0393.12028 · eudml:205766 [16] F. Thaine, Properties that characterize Gaussian periods and cyclotomic numbers, Proc. Amer. Math. Soc. 124 (1996), 35-45. · Zbl 0862.11059 · doi:10.1090/S0002-9939-96-03108-5 [17] F. Thaine, On the coefficients of Jacobi sums in prime cyclotomic fields, Trans. Amer. Math. Soc. 351 (1999), 4769-4790. · Zbl 0944.11036 · doi:10.1090/S0002-9947-99-02223-0 [18] F. Thaine, Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers, Math. Comp. 69 (2000), 1653-1666. · Zbl 0989.11055 · doi:10.1090/S0025-5718-99-01142-4 [19] F. Thaine, Jacobi sums and new families of irreducible polynomials of Gaussian periods, Math. Comp. 70 (2001), 1617-1640. · Zbl 0989.11056 · doi:10.1090/S0025-5718-01-01312-6 [20] F. Thaine, On Gaussian periods that are rational integers, Michigan Math. J. 50 (2002), 313-337. · Zbl 1027.11081 · doi:10.1307/mmj/1028575736 [21] F. Thaine, Cyclic polynomials and the multiplication matrices of their roots, J. Pure Appl. Algebra 188 (2004), 247-286. · Zbl 1041.11069 · doi:10.1016/j.jpaa.2003.07.004 [22] P. V. Wamelen, Jacobi sums over finite fields, Acta Arith. 102 (2002), 1-20. · Zbl 1047.11116 · doi:10.4064/aa102-1-1 [23] S. Wolfram, The Mathematica book , Fourth ed., Wolfram Media, Inc., Cambridge Univ. Press, Cambridge-New York, 1999. · Zbl 0924.65002
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