Most Elliptic Curves over Global Function Fields are Torsion Free

TL;DR

This paper proves that over certain global function fields, most elliptic curves have maximal Galois action on their torsion points, implying they are typically torsion free.

Contribution

It establishes that for a broad class of global function fields, the set of elliptic curves with maximal Galois representations on torsion points has density one.

Findings

Most elliptic curves over these fields have maximal Galois action on torsion points.

The set of such elliptic curves has density 1.

Maximal Galois representations occur generically in this setting.

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 $\rho_{E,n}$ is as large as possible for all $n$, has density $1$.