Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve

TL;DR

This paper investigates a divisibility sequence associated with points on abelian varieties isogenous to powers of elliptic curves, revealing its connection to elliptic divisibility sequences and expanding understanding of their algebraic properties.

Contribution

It introduces a new divisibility sequence linked to abelian varieties isogenous to elliptic curve powers and explores its relation to elliptic divisibility sequences.

Findings

Establishes a connection between the divisibility sequence and elliptic divisibility sequences.

Provides structural insights into divisibility sequences on abelian varieties.

Enhances understanding of algebraic properties of these sequences.

Abstract

Let $A$ be an abelian variety defined over a number field $K$, $E/K$ be an elliptic curve, and $\phi:A\to E^m$ be an isogeny defined over $K$. Let $P\in A(K)$ be such that $\phi(P)=(Q_1,\dots, Q_m)$ with $\text{Rank}_\mathbb{Z}(\langle Q_1,\dots, Q_m\rangle)=1$. We will study a divisibility sequence related to the point $P$ and show its relation with elliptic divisibility sequences.