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 {D_nP+Q} on C. We provide conditions on r and the characteristic of k for there to exist a bound N such that D_nP+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.