A note on plane pointless curves (Q946895)

Let \(d(q)\) denote the minimal degree of a smooth projective plane curve over the finite field \(\mathbb{F}_q\), with no \(\mathbb{F}_q\)-rational points. In this paper it is proved that \(1/4\leq \underline{\lim}_{q\to\infty}d(q)\leq1/3\), if the characteristic is \(p>3\). The lower bound is an immediate consequence of the Hasse-Weil bound. To prove the upper bound, the paper exhibits a sequence of pointless Fermat curves \(x^{d_k}+y^{d_k}+z^{d_k}=0\) over \(\mathbb{F}_{p^{m_k}}\), with \(m_k\) strictly increasing and \(\lim_{k\to\infty}d_k=1/3\).