Primitive divisors of sequences associated to elliptic curves over function fields

2021/03/11 by Robert Slob, Slob, Robert
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.06787

openalex publication_date 2021/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

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