Primitive divisors of sequences associated to elliptic curves with complex multiplication

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 $g\ne 0$ and $f$ so that $f(P)=g(Q)$. This is a generalization of previous results on elliptic divisibility sequences.

Minor changes. Final version of the paper. Published in Research in Number Theory

References in corpus (1)

• Primitive divisors of sequences associated to elliptic curves

Cited by in corpus (1)

• Some effectivity results for primitive divisors of elliptic divisibility sequences