Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

Abstract

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over $\mathbf{Q}$ is $30$ by Bilu-Hanrot-Voutier, 2000, but finding such a bound remains an open problem in genus one, both over $\mathbf{Q}$ and over function fields. We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is $2$ if the $j$-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.