Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

Abstract

Let \F_q ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in \F_q[x] with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in \F_{q^3} with prescribed trace and norm over \F_q improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over \F_{2^r} divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case $r$ odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field \F_{3^r}, first proved by Katz and Livne, is given.Comment: 21 pages; revised version with somewhat more clearer proofs; to appear in Acta Arithmetic