Abstract view
Polynomials for Kloosterman Sums


Published:20070301
Printed: Mar 2007
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
$$f_{m}(x) = \prod_{d} (xR(d)) = x^{\phi(m)} +c_{1}
x^{\phi(m)1} + \dots + c_{\phi(m)},$$
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.