Hasse-Weil theorem and construction of complete arcs of little cardinality in Galois planes of odd order (Q1330273)

A \(k\)-arc in a projective plane of order \(q\) is a set \(K\) of \(k\) points no three of which are collinear. \(K\) is complete if it is not contained in any \(m\)-arc with \(m > k\). For each odd prime power \(q\) for which 3 divides \(q + 1\) and \(q\) is at least as large as 121, the author constructs a complete \(k\)-arc \(K\) in \(PG(2,q)\) for which \(k\) lies between \((11/24) (q + 1) + 3\) and \((q + 1)/2 + 2\). The points of \(K\) consist of \(K-2\) points on a certain conic together with two points on a line exterior to the conic. The proof of completeness depends on a fairly intricate application of the Hasse-Weil Theorem concerning the number of points of an irreducible algebraic curve in \(PG (2,q)\).