On Elliptic Sequences over Commutative Rings

TL;DR

This paper introduces elliptic sequences over commutative rings, classifies them over fields, and shows most are related to elliptic divisibility sequences, providing an algebraic perspective without complex analysis.

Contribution

It defines elliptic sequences over rings, classifies them over fields, and proves standard elliptic divisibility sequences are elliptic algebraically, avoiding complex analysis.

Findings

Most elliptic sequences over fields are dilated multiples of standard EDSs.

Standard EDSs satisfy elliptic relations algebraically, without complex analysis.

The classification into three types helps understand the structure of elliptic sequences.

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.