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# Polynomials for Kloosterman Sums

Published:2007-03-01
Printed: Mar 2007
• S. Gurak
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## Abstract

Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$ denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq m$, $(d,m)=1$, satisfy the polynomial $$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1} x^{\phi(m)-1} + \dots + c_{\phi(m)},$$ where the sum and product are taken over a complete system of reduced residues modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely, $$f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$ where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$ of reduced residues modulo $m$. Questions concerning the explicit determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their beginning coefficients), their reducibility over the rational field ${\bf Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.
 MSC Classifications: 11L05 - Gauss and Kloosterman sums; generalizations 11T24 - Other character sums and Gauss sums