Maximal singular curves over finite fields from elliptic and hyperelliptic curves

Abstract

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 hyperelliptic curve $\tilde{C}$ over ${\mathbb{F}}_q$ ($q$ odd), we generate singular curves $C$ with non-split singularities or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöhr's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöhr's theory of trigonal Gorenstein curves. Some of our models attain the Aubry-Perret bound, whenever their normalization is maximal, and the construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves to produce the desired family. Finally, we determine the gonality over ${\mathbb{F}}_q$ of most of the curves we construct.

arXiv comment32 pages, comments very much welcome at any time!!