Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

2025/05/20 by Maarten Derickx, Michael Stoll, Derickx, Maarten +1
Computer Science · Mathematics · #11G05 #11G18 #14G05 #14G25 #14G35 #Algebraic Geometry (math.AG) #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.14109

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotics of the set S(d) of possible prime orders of K-rational points on elliptic curves over number fields K of degree d as d tends to infinity. Assuming some conjectures on the sparsity of newforms of weight 2 and prime level with unexpectedly high analytic rank, we show that max S(d) ≤ 3d + 1 for sufficiently large even d and max S(d) = o(d) for odd d.

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