Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm
Marko J. Moisio · 2012
Abstract. Let Fq (q = pr) be a finite field. In this note the number of irreducible polynomials of degree m in Fq[x] with prescribed trace and norm coefficients is calculated in certain special cases and general bounds for that number are obtained. As a corollary, sharp bounds are obtained for the number of elements in Fq3 with prescribed trace and norm over Fq improving the estimates by Katz. Next, simple necessary and sufficient conditions are given when a Kloosterman sum over F2r is divisible by three. This result generalizes the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case r odd. Finally, a new elementary proof for the value distribution of a Kloosterman sum over the field F3r, obtained by Katz and Livne, is given. 1.