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Mathematics of Computation

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On the minimal polynomial of Gauss periods for prime powers

Author: S. Gurak
Journal: Math. Comp. 75 (2006), 2021-2035
MSC (2000): Primary 11L05, 11T22, 11T23
Published electronically: July 11, 2006
MathSciNet review: 2240647
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Abstract: For a positive integer $ m$, set $ \zeta_{m}=\exp(2\pi i/m)$ and let $ {\bf 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 $ {\bf Z}_{m}^{*}$. The Gauss periods

$\displaystyle \displaylines{ \theta_{j} =\sum_{x \in H} \zeta_{m}^{t_{j}x} \;\;\; (1 \leq j \leq e) }$

corresponding to $ H$ are conjugate and distinct over $ {\bf Q}$ with minimal polynomial

$\displaystyle \displaylines{ g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. }$

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$

$\displaystyle \displaylines{ G(X) \equiv A(X)\cdot B(X)^{\frac{p-1}{f}} \;\;\;({\rm mod}\;X^{N}) }$

holds in $ {\bf Z}[X]$ for all primes $ p \equiv 1({\rm 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

$\displaystyle \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}}$.

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Additional Information

S. Gurak
Affiliation: Department of Mathematics, University of San Diego, San Diego, California 92110

Received by editor(s): June 2, 2005
Published electronically: July 11, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.