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

Marko Moisio
2008 Acta Arithmetica  
Let _q (q=p^r) be a finite field. In this paper the number of irreducible polynomials of degree m in _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 _q^3 with prescribed trace and norm over _q improving the estimates by Katz in this special case. Moreover, a characterization of
more » ... sterman sums over _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 _3^r, first proved by Katz and Livne, is given.
doi:10.4064/aa132-4-3 fatcat:mtpzlwaktnb6zn3z6pvkw4tdgu