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We consider the polynomials of the form P(x) = xk - γTr(x) over 픽 qn for n ≥ 2. We show that P(x) is not a permutation of 픽qn in the case gcd(k, qn - 1) > 1. Our proof uses an absolutely irreducible curve over 픽qn and the number of rational points on it. [ABSTRACT FROM AUTHOR]

For an odd prime $$p$$ and an even integer $$n$$ with $$\gcd (n,p) > 1$$ , we consider quadratic functions from $${\mathbb F}_{p^{n}}$$ to $${\mathbb F}_p$$ of codimension $$k$$ . For various values of $$k$$ , we obtain classes of quadratic functions giving rise to maximal and minimal Artin-Schreier curves over $${\mathbb F}_{p^{n}}$$ . We completely classify all maximal and minimal curves obtained from quadratic functions of codimension $$2$$ and coefficients in the prime field $${\mathbb F}_p$$ . The results complement our results obtained earlier for the case $$\gcd (n,p) = 1$$ . The arguments are more involved than for the case $$\gcd (n,p) = 1$$ . [ABSTRACT FROM AUTHOR]

For an odd prime p and an even integer n with gcd (n,p)=1, we consider quadratic functions from Fpn to Fp of codimension k. For various values of k, we obtain classes of quadratic functions giving rise to maximal and minimal Artin-Schreier curves over Fpn. We completely classify all maximal and minimal curves obtained from quadratic functions of codimension 2 and coefficients in the prime field Fp. [ABSTRACT FROM AUTHOR]

Abstract: Let be an algebraically closed field of characteristic , and let be a curve over of genus . Assume that and that admits a non-singular plane model. The following result is proven: if has more than automorphisms, then is birationally equivalent to a Hermitian curve. [Copyright &y& Elsevier]

A fundamental problem in the theory of curves over finite fields is to determine the sets M( g):= { N ∈ ℕ / there is a curve over $$ \mathbb{F}_q $$ of genus g with exactly N rational points}. A complete description of M( g) is out of reach. So far, mostly bounds for the numbers N( g):= max M( g) have been studied. In particular, Elkies et al. proved that there is a constant γ > 0 such that for any g ≥ 0 there is some N ∈ M( g) with N ≥ γ g. This implies that lim inf N( g)/ g > 0, and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially and show that there are constants α, β > 0 such that for all g ≥ 0, the whole interval [0, α g − β] ∩ ℕ is contained in M( g). [ABSTRACT FROM AUTHOR]

Abstract: A fundamental problem in the theory of curves over finite fields is to determine the sets there is a curve over of genus g with exactly N rational points}. A complete description of is out of reach. So far, mostly bounds for the numbers have been studied. In particular, Elkies et al. proved that there is a constant , such that for any there is some with . This implies that , and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially: there are constants such that for all , the whole interval is contained in . [Copyright &y& Elsevier]