Polynomials for Kloosterman Sums

Abstract

AbstractFix an integer m > 1, and set ζm = exp(2πi/m). Let denote the multiplicative inverse of x modulo m. The Kloosterman sums , satisfy the polynomialwhere the sum and product are taken over a complete system of reduced residues modulo m. Here we give a natural factorization of fm(x), namely,where σ runs through the square classes of the group of reduced residues modulo m. Questions concerning the explicit determination of the factors (or at least their beginning coefficients), their reducibility over the rational field Q and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.