Most elliptic curves over global function fields are torsion free
Volume 202 / 2022
Acta Arithmetica 202 (2022), 21-28
MSC: Primary 11G05; Secondary 11F80, 11N36.
DOI: 10.4064/aa210108-23-8
Published online: 22 November 2021
Abstract
Given an elliptic curve $E$ over a global function field $K$, the Galois action on the $n$-torsion points of $E$ gives rise to a mod-$n$ Galois representation $\rho _{E,n}$. For $K$ satisfying some mild conditions, we show that the set of $E$ for which the image of $\rho _{E,n}$ is as large as possible for all $n$ has density $1$.
