New York Journal of Mathematics
Volume 28 (2022), 230-249


Robert Slob

Primitive divisors of sequences associated to elliptic curves over function fields

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Published: January 28, 2022.
Keywords: Elliptic divisibility sequences; primitive divisors; Zsigmondy bound; elliptic curves; function fields; elliptic surfaces.
Subject: Primary 11G05; Secondary 11B83, 14H52, 14J27.

We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let k be an algebraically closed field, let C be a nonsingular projective curve over k, and let K denote the function field of C. Suppose E is an ordinary elliptic curve over K and suppose there does not exist an elliptic curve E0 defined over k that is isomorphic to E over K. Suppose P ∈ E(K) is a non-torsion point and Q ∈ E(K) is a torsion point of order r. The sequence of points {nP+Q} ⊂ E(K) induces a sequence of effective divisors {DnP+Q} on C. We provide conditions on r and the characteristic of k for there to exist a bound N such that DnP+Q has a primitive divisor for all n ≥ N. This extends the analogous result of Verzobio in the case where K is a number field.


This work originated from the author's master's thesis, which was written under supervision of Gunther Cornelissen. The author would like to thank him for the valuable conversations during this time. The author would also like to express his gratitude towards Jeroen Sijsling, Matteo Verzobio, and the anonymous referee for useful comments on an earlier draft of this paper.

Author information

Robert Slob:
Institut für Reine Mathematik
Universität Ulm
Helmholtzstrasse 18, 89081 Ulm, Germany