Primitive divisors of sequences associated to elliptic curves over function fields

TL;DR

This paper investigates the existence of a uniform bound for primitive divisors in sequences of divisors associated with points on elliptic curves over function fields, extending known results from number fields.

Contribution

It establishes conditions under which a Zsigmondy bound exists for divisor sequences on elliptic curves over function fields, generalizing previous number field results.

Findings

Existence of a bound N for primitive divisors depending on r and characteristic

Conditions on r and characteristic for the bound to hold

Extension of Verzobio's number field results to function fields

Abstract

We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective curve over $k$, and let $K$ denote the function field of $\mathcal{C}$. Suppose $E$ is an ordinary elliptic curve over $K$ and suppose there does not exist an elliptic curve $E_0$ defined over $k$ that is isomorphic to $E$ over $K$. Suppose $P\in E(K)$ is a non-torsion point and $Q\in E(K)$ is a torsion point of order $r$. The sequence of points $\{nP+Q\}\subset E(K)$ induces a sequence of effective divisors $\{D_{nP+Q}\}$ on $\mathcal{C}$. We provide conditions on $r$ and the characteristic of $k$ for there to exist a bound $N$ such that $D_{nP+Q}$ has a primitive divisor for all $n\geq N$. This extends the analogous result of Verzobio in the case where $K$ is a number field.