Polynomials for Kloosterman Sums

Canadian mathematical bulletin

关键词: simplicial cohomology;    semigroup algebra;   

DOI  :  10.4153/CMB-2007-007-2

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Fix an integer $m>1$, and set $zeta_{m}=exp(2pi i/m)$. Let ${ar x}$denote the multiplicative inverse of $x$ modulo $m$. The Kloostermansums $R(d)=sum_{x} zeta_{m}^{x + d{ar x}}$, $1 leq d leqm$, $(d,m)=1$,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 residuesmodulo $m$. 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 ${f Z}_{m}^{*}$of reduced residues modulo $m$. Questions concerning the explicitdetermination of the factors $f_{m}^{(sigma)}(x)$ (or at least theirbeginning coefficients), their reducibility over the rational field${f Q}$ and duplication among the factors are studied. The treatmentis similar to what has been done for period polynomials for finitefields.

【 授权许可】

Unknown