On a probabilistic local-global principle for torsion on elliptic curves

2020/05/13 by John Cullinan, Cullinan, John, Meagan Kenney +3 · 3 citations
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2005.06669

openalex publication_date 2020/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let m be a positive integer and let E be an elliptic curve over ℚ with the property that m|#E(\mathbbF_p) for a density 1 set of primes p. Building upon work of Katz and Harron-Snowden, we study the probability that m divides the the order of the torsion subgroup of E(ℚ): we find it is nonzero for all m ∈ \ 1, 2, …, 10, 12, 16\ and we compute it exactly when m ∈ \ 1,2,3,4,5,7 \. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve arises from the quotient by a torsion-free group of genus zero.

Cited by

• Counting elliptic curves with prescribed entanglements
• Selmer groups of families of elliptic curves with an ℓ-isogeny
• Counting elliptic curves over ℚ with bounded naive height

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