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 over a global field, the sequence D1, D2,… of denominators of multiples P, 2P,… of P is a strong divisibility sequence in the sense that gcd⁡(Dm,Dn)=Dgcd⁡(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 Dn with n>N presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Q is 30 by Bilu-Hanrot-Voutier [2], but finding such a bound remains an open problem in genus one, both over 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.