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

@article{Cullinan2020OnAP,
title={On a probabilistic local-global principle for torsion on elliptic curves},
author={John Cullinan and Meagan Kenney and John Voight},
journal={Journal de th{\'e}orie des nombres de Bordeaux},
year={2020}
}
• Published 13 May 2020
• Mathematics
• Journal de théorie des nombres de Bordeaux
Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid\#E(\mathbb{F}_p)$ 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 $E(\mathbb{Q})$: we find it is nonzero for all $m \in \{ 1, 2, \dots, 10, 12, 16\}$ and we compute it exactly when $m \in \{ 1,2,3,4,5,7 \}$. As a supplement, we give an asymptotic count of elliptic…

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