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

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.

51 pages, 2 figures; incorporated final corrections and added/updated references