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 F-q and does not contain Fq rational points. We are interested in the asymptotic behavior of d(q) for q -> infinity. To the best of the author's knowledge the problem of estimating the asymptotic behavior of d(q) was not considered previously. In this note we establish the following bounds: 1/4 <= lim(q ->infinity) log(q) d(q) <= 1/3. More specifically, for every characteristic p > 3 we construct a sequence of pointless Fermat curves x(dk) + y(dk) + z(dk) = 0. over F (m)(p)(k), such that lim(k-infinity) log (m)(p)(k) d(k) = 1/3. (c) 2006 Elsevier Inc. All rights reserved.