The divisibility modulo 24 of Kloosterman sums on GF(2m)GF(2^m), mm even

In a recent work by Charpin, Helleseth, and Zinoviev Kloosterman sums $K(a)$ over a finite field \F_{2^m} were evaluated modulo 24 in the case $m$ odd, and the number of those $a$ giving the same value for $K(a)$ modulo 24 was given. In this paper the same is done in the case $m$ even. The key techniques used in this paper are different from those used in the aforementioned work. In particular, we exploit recent results on the number of irreducible polynomials with prescribed coefficients.Comment: 15 pages, submitted; an annoying typo corrected in the abstrac

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

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