Primitive divisors of sequences associated to elliptic curves over 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 $n\ge N$. This extends the analogous result of Verzobio in the case where $K$ is a number field.

13 pages. New version: in the old version, we assumed in Theorem 1 that the torsion point $Q$ had prime order unequal to the characteristic of $k$. It was pointed out by a colleague that some small adaptations allow us to extend the main theorem to where $Q$ is a torsion point of arbitrary order (as long as the characteristic of $k$ and the order of $Q$ are not too small, see Table 1)

References in corpus (1)

• Algebraic divisibility sequences over function fields