Permutation Binomials over Finite Fields

Let ${\mathbb{F}}_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in {\mathbb{F}}_q$ for which the binomial $x^n(x^{(q-1)\mathrm{/}r}+a)$ is a permutation polynomial in the cases $r=2$ and $r=3$.