1. On a probabilistic local-global principle for torsion on elliptic curves
- Author
-
Cullinan, John, Kenney, Meagan, and Voight, John
- Subjects
Mathematics - Number Theory ,FOS: Mathematics ,Number Theory (math.NT) - 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., Comment: 51 pages, 2 figures; incorporated final corrections and added/updated references
- Published
- 2022