On plane arcs contained in cubic curves (Q1604419)

If the group \(H\) of the \({\mathbf F}_q\)-rational points of a non-singular cubic curve has even order, then the coset of a subgroup of \(H\) of index two is an arc (called Zirilli arc) in the Galois plane of order \(q\). The completeness of such an arc has been proved, except for the case \(j = 0\), where \(j\) is the \(j\)-invariant of the underlying cubic curve. The aim of this paper is to settle the completeness problem for the exceptional case and to provide an alternative proof of the known results.