The correlations of finite Desarguesian planes of square order defined by diagonal matrices

Abstract

The present article represents the next step in our ongoing program of classifying the correlations of finite Desarguesian planes. We show that in PG ( 2 , q 2 n ) , the correlations defined by diagonal matrices, with companion automorphism ( q m ), where ( m , 2 n ) = 1 , have the following numbers of absolute points: q 2 n + q n + 2 - q n + 1 + 1 or q 2 n - q n + 1 + q n + 1 or ( q n + 1 ) 2 for n odd; q 2 n - q n + 2 + q n + 1 + 1 or q 2 n + q n + 1 - q n + 1 or ( q n - 1 ) 2 for n even. We also discuss the equivalence classes into which these correlations fall, as well as the configurations of their sets of absolute points.