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 D 1 , D 2 , . . . of denominators of multiples P , 2P, . . . of P is a strong divisibility sequence in the sense that gcd(Dm, Dn) = 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 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, 2000, 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.

Common valuations of division polynomials

“…Over function fields, in particular over the function field of a smooth projective curve C over an algebraically closed field k, one can study the set of effective divisors {D nP } where D nP is the divisor of poles of the element x(nP ) for a fixed n, cf. [9], [14], [15], [4].…”

Common valuations of division polynomials

“…Over function fields, in particular over the function field of a smooth projective curve C over an algebraically closed field k one can study the set of effective divisors {D nP } where D nP is the divisor of poles of the element x(nP ) for a fixed n, cf. [IMS + 12], [Nas16b], [NS20], [CR16].…”