Natural density of rank-$2$ Drinfeld modules with big Galois image

Abstract

In this paper, we compute the natural density of rank-$1$ Drinfeld module over $\mathbb{F}_q[T]$ with surjective adelic Galois representation; and the natural density of rank-$2$ Drinfeld modules over $\mathbb{F}_q[T]$ whose $\mathfrak{l}$-adic Galois image containing the special linear subgroup for finitely many prime ideal $\mathfrak{l}$.