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

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

2007-06-14

arxiv.org/abs/0706.2112v5

Mathematics

Number Theory

21 pages; revised version with somewhat more clearer proofs; to appear in Acta Arithmetica

Scientific paper

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.

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