Most Elliptic Curves over Global Function Fields are Torsion Free

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 $1$.