Primitive divisors of sequences associated to elliptic curves with complex multiplication

TL;DR

This paper investigates primitive divisors in sequences derived from points on elliptic curves with complex multiplication, extending known results to a broader class of sequences associated with endomorphisms.

Contribution

It generalizes previous results on elliptic divisibility sequences by studying primitive divisors in sequences linked to endomorphisms of elliptic curves with complex multiplication.

Findings

For all but finitely many endomorphisms, the associated ideal has a primitive divisor.

The results apply when P is a non-torsion point and certain endomorphisms relate P and Q.

Extends classical elliptic divisibility sequence results to endomorphism-based sequences.

Abstract

Let $P$ and $Q$ be two points on an elliptic curve defined over a number field $K$. For $\alpha\in \text{End}(E)$, define $B_\alpha$ to be the $\mathcal{O}_K$-integral ideal generated by the denominator of $x(\alpha(P)+Q)$. Let $\mathcal{O}$ be a subring of $\text{End}(E)$, that is a Dedekind domain. We will study the sequence $\{B_\alpha\}_{\alpha\in \mathcal{O}}$. We will show that, for all but finitely many $\alpha\in \mathcal{O}$, the ideal $B_\alpha$ has a primitive divisor when $P$ is a non-torsion point and there exist two endomorphisms $g\neq 0$ and $f$ so that $f(P)=g(Q)$. This is a generalization of previous results on elliptic divisibility sequences.