Maximal singular curves over finite fields from elliptic and hyperelliptic curves

We construct explicit families of maximal singular curves over finite fields. A maximal singular curve is a curve that attains the Aubry-Perret bound, which is the natural extension of the Hasse-Weil-Serre bound on the maximum number of rational points on a smooth curve over a finite field.Starting from a smooth elliptic or hyper elliptic curve $\tilde{C}$ over $\mathbb{F}_{q}$ ($q$ odd), we generate singular curves $C$ with non-split nodes or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöur's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöur theory of trigonal Gorenstein curves. The construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves. Finally, we also determine the gonality of all obtained curves.