Elliptic divisibility sequences in certain ranks over finite fields

Abstract

We develop techniques first studied by Morgan Ward to characterize sequences which arise from elliptic curves and which contain a zero term. We first define elliptic divisibility sequences over finite fields by noting that they are not the sequences which arise by reduction from integer sequences. After that, we give general terms of these sequences over the finite fields F(p) (p > 3 is a prime) and then we determine elliptic curves and singular curves associated with them.