On Elliptic Sequences over Commutative Rings

Abstract

We define elliptic sequences over a commutative ring as sequences indexed by the (positive) integers satisfying a 4-parameter, highly symmetric family of homogeneous quartic relations among terms which we call elliptic relations. We classify elliptic sequences over a field into three types, and show that most of them are dilated multiples of standard elliptic divisibility sequences (EDSs) which form countably many 4-dimensional families. In particular, we show standard EDSs are elliptic in a purely algebraic way using intricate implications among elliptic relations, without relying on complex analytic theory of Weierstrass functions. We shall use results presented here to give a purely algebraic treatment of division polynomials in a follow-up paper.