Some Families of Supersingular Artin - Schreier Curves

A curve over finite field is supersingular if its Jacobian is supersingular as an abelian variety. On the one hand, supersingular abelian varieties form the smallest (closed) stratum in the moduli space of abelian varieties, on the other the intersection of Jacobian locus and the stratification of moduli space is little known. Consequently it is very difficult to locate a family of supersingular curve. See [4]. In characteristic 2 some ground-breaking progress has been make in [6][7], where families of supersingular curves are given explicitly using some new sharp slope estimation method. However, that method does not apply easily to cases when characteristic is not 2. In this paper we develop a new method to allow us to prove supersingularity of Artin-Schreier curves in characteristic > 2. To illustrate how our method works, we use it show