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arXiv · 2005.06669

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

John Cullinan·Meagan Kenney·John Voight
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Abstract

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 curves with extra level structure when the parametrizing modular curve arises from the quotient by a torsion-free group of genus zero.

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John Cullinan, Meagan Kenney, John Voight. 2020-05-13. On a probabilistic local-global principle for torsion on elliptic curves. https://arxiv.org/abs/2005.06669

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