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

JOURNAL OF NUMBER THEORY

卷:213

Article

Naskrecki, Bartosz¹  Streng, Marco² 

[1] Adam Mickiewicz Univ, Fac Math & Comp Sci, Uniwersytetu Poznanskiego 4, PL-61614 Poznan, Poland

[2] Leiden Univ, Math Inst, POB 9512, NL-2300 RA Leiden, Netherlands

关键词: Elliptic divisibility sequences;    Elliptic surfaces;    Primitive divisors;    Function fields;    Constant j-invariant;   

DOI  :  10.1016/j.jnt.2019.12.002

来源: Elsevier

【 摘 要 】

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 Pof infinite order over a global field, the sequence D-1, D-2, ... of denominators of multiples P, 2P,... of Pis a strong divisibility sequence in the sense that gcd(D-m, D-n) = Dgcd('m,n). This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number Nis called a Zsigmondy bound of the sequence if each term Dnwith n > Npresents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Qis 30by Bilu-Hanrot-Voutier[2], but finding such a bound remains an open problem in genus one, both over Qand 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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free