A note on plane pointless curves

Let d(q) denote the minimal degree of a smooth projective plane curve that is defined over the finite field and does not contain rational points. We are interested in the asymptotic behavior of d(q) for q→∞. To the best of the authorʹs knowledge the problem of estimating the asymptotic behavior of d(q) was not considered previously. In this note we establish the following bounds: More specifically, for every characteristic p>3 we construct a sequence of pointless Fermat curves such that limk→∞logpmkdk=1/3.