Polynomials for Kloosterman Sums

Gurak, S.

Canadian mathematical bulletin, Tome 50 (2007) no. 1, pp. 71-84

Résumé

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.

Détail

DOI : 10.4153/CMB-2007-007-2

Mots-clés : 11L05, 11T24