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Polynomials for Kloosterman Sums

Published online by Cambridge University Press:  20 November 2018

S. Gurak
S. Gurak*
Affiliation:
University of San Diego, San Diego, CA 92110, USA e-mail: gurak@sandiego.edu
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Abstract

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Fix an integer $m>1$, and set ${{\zeta }_{m}}=\exp \left( 2\pi i/m \right)$. Let $\bar{x}$ denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman sums $R\left( d \right)=\sum\limits_{x}{\zeta _{m}^{x+d\bar{x}}},1\le d\le m,\left( d,m \right)=1$, satisfy the polynomial

$${{f}_{m}}\left( x \right)=\underset{d}{\mathop{\prod }}\,\left( x-R\left( d \right) \right)={{x}^{\phi \left( m \right)}}+{{c}_{1}}{{x}^{\phi \left( m \right)-1}}+\cdot \cdot \cdot +{{c}_{\phi \left( m \right)}},$$

where the sum and product are taken over a complete system of reduced residues modulo $m$. Here we give a natural factorization of ${{f}_{m}}\left( x \right)$, namely,

$${{f}_{m}}\left( x \right)=\underset{\sigma }{\mathop{\prod }}\,f_{m}^{\left( \sigma \right)}\left( x \right),$$

where $\sigma$ runs through the square classes of the group $Z_{m}^{*}$ of reduced residues modulo $m$. Questions concerning the explicit determination of the factors $f_{m}^{\left( \sigma \right)}\left( x \right)$ (or at least their beginning coefficients), their reducibility over the rational field $\text{Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.

Keywords

11L0511T24
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 1 , 01 March 2007 , pp. 71 - 84
Copyright
Copyright © Canadian Mathematical Society 2007

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