On the minimal polynomial of Gauss periods for prime powers

Gurak, S.

doi:10.1090/S0025-5718-06-01885-0

On the minimal polynomial of Gauss periods for prime powers
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Abstract:

For a positive integer $m$, set $\zeta _{m}=\exp (2\pi i/m)$ and let $\textbf {Z}_{m}^{*}$ denote the group of reduced residues modulo $m$. Fix a congruence group $H$ of conductor $m$ and of order $f$. Choose integers $t_{1},\dots ,t_{e}$ to represent the $e=\phi (m)/f$ cosets of $H$ in $\textbf {Z}_{m}^{*}$. The Gauss periods \[ \displaylines { \theta _{j} =\sum _{x \in H} \zeta _{m}^{t_{j}x} \;\;\; (1 \leq j \leq e) }\] corresponding to $H$ are conjugate and distinct over $\textbf {Q}$ with minimal polynomial \[ \displaylines { g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. }\] To determine the coefficients of the period polynomial $g(x)$ (or equivalently, its reciprocal polynomial $G(X)=X^{e}g(X^{-1}))$ is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case $m=p$, an odd prime, with $f >1$ fixed. In this setting, it is known for certain integral power series $A(X)$ and $B(X)$, that for any positive integer $N$ \[ \displaylines { G(X) \equiv A(X)\cdot B(X)^{\frac {p-1}{f}} \;\;\;(\textrm {mod}\;X^{N}) }\] holds in $\textbf {Z}[X]$ for all primes $p \equiv 1(\textrm {mod}\; f)$ except those in an effectively determinable finite set. Here we describe an analogous result for the case $m=p^{\alpha }$, a prime power ($\alpha > 1$). The methods extend for odd prime powers $p^{\alpha }$ to give a similar result for certain twisted Gauss periods of the form \[ \displaylines { \psi _{j} = i^{*} \sqrt {p} \sum _{x \in H} (\frac {t_{j}x}{p}) \zeta _{p^{\alpha }}^{t_{j}x} \;\;(1 \leq j \leq e),} \] where $(\frac { }{p})$ denotes the usual Legendre symbol and $i^{*}= i^{\frac {(p-1)^{2}}{4}}$.

References

Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1625181
A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
Richard P. Brent, On computing factors of cyclotomic polynomials, Math. Comp. 61 (1993), no. 203, 131–149. MR 1205459, DOI 10.1090/S0025-5718-1993-1205459-2
L.E. Dickson, Elementary Theory of Equations, Wiley, New York.
Carl Friedrich Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn.-London, 1966. Translated into English by Arthur A. Clarke, S. J. MR 0197380
S. Gupta and D. Zagier, On the coefficients of the minimal polynomials of Gaussian periods, Math. Comp. 60 (1993), no. 201, 385–398. MR 1155574, DOI 10.1090/S0025-5718-1993-1155574-7
S. Gurak, Minimal polynomials for Gauss circulants and cyclotomic units, Pacific J. Math. 102 (1982), no. 2, 347–353. MR 686555, DOI 10.2140/pjm.1982.102.347
S. Gurak, Minimal polynomials for circular numbers, Pacific J. Math. 112 (1984), no. 2, 313–331. MR 743988, DOI 10.2140/pjm.1984.112.313
S. Gurak, "Minimal polynomials for Gauss periods with f=2," Acta Arith. 121 (2006), 233–257.
S. Gurak, "Explicit evaluation of multi-dimensional Kloosterman sums for prime powers" (to appear).
Helmut Hasse, Vorlesungen über Zahlentheorie, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksightigung der Anwendungsgebiete. Band LIX, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1950 (German). MR 0051844, DOI 10.1007/978-3-642-52795-1
Jürgen Neukirch, Class field theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 280, Springer-Verlag, Berlin, 1986. MR 819231, DOI 10.1007/978-3-642-82465-4
Hans Salié, Über die Kloostermanschen Summen $S(u,v;q)$, Math. Z. 34 (1932), no. 1, 91–109 (German). MR 1545243, DOI 10.1007/BF01180579