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

2019/04/28 by Bartosz Naskręcki, Naskręcki, Bartosz, Marco Streng +1
Computer Science · Mathematics · #11B39 #11B83 #11G05 #11G07 #14H52 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1904.12393

openalex publication_date 2019/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over\nfunction fields in case the j-invariant of the elliptic curve is constant. In\nmore detail, given an elliptic curve E with a point P of infinite order,\nthe sequence D₁, D₂, \… of denominators of multiples P, 2P,\…\nof P is a strong divisibility sequence in the sense that \gcd(D_m, D_n) =\nD_\gcd(m,n). This is the genus-one analogue of the genus-zero Fibonacci,\nLucas and Lehmer sequences. A number N is called a Zsigmondy bound of the\nsequence if each term D_n with n>N presents a new prime factor. The\noptimal uniform Zsigmondy bound for the genus-zero sequences over \Q\nis 30 by Bilu-Hanrot-Voutier, 2000, but finding such a bound remains an open\nproblem in genus one, both over \Q and over function fields. We prove\nthat the optimal Zsigmondy bound for ordinary elliptic divisibility sequences\nover function fields is 2 if the j-invariant is constant. In the\nsupersingular case, we give a complete classification of which terms can and\ncannot have a new prime factor.\n

Related

• Primitive divisors of sequences associated to elliptic curves over function fields
• On the greatest prime factor of some divisibility sequences
• Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve
• Prime divisors of sequences associated to elliptic curves
• Explicitly bounding perfect powers in elliptic divisibility sequences