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1. CMB 2007 (vol 50 pp. 71)
Polynomials for Kloosterman Sums 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.
Categories:11L05, 11T24 |
2. CMB 2003 (vol 46 pp. 344)
Gauss and Eisenstein Sums of Order Twelve Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite
field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p} $ be
the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive
integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by
$$
g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}}
$$
Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of
$g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguity
present in a previous evaluation. Here I generalize Evans' result to give
a complete determination of the sum $g_{r}(12)$.
Categories:11L05, 11T24 |
3. CMB 2001 (vol 44 pp. 22)
Gauss Sums of Orders Six and Twelve Precise, elegant evaluations are given for Gauss sums of orders six
and twelve.
Categories:11L05, 11T24 |