The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules

Abstract

Let $\mathbb{F}_q$ be the finite field with $q\geq 5$ elements, $A:=\mathbb{F}_q[T]$ and $F:=\mathbb{F}_q(T)$. Assume that $q$ is odd and take $|\cdot|$ to be the absolute value at $\infty$ that is normalized by $|T|=q$. Given a pair $w=(g_1, g_2)\in A^2$ with $g_2\neq 0$, consider the associated Drinfeld module $\phi^w: A\rightarrow A\{\tau\}$ of rank $2$ defined by $\phi_T^w=T+g_1\tau+g_2\tau^2$. Fix integers $c_1, c_2\geq 1$ and define $|w|:=max\{|g_1|^{\frac{1}{c_1}}, |g_2|^{\frac{1}{c_2}}\}$. I show that when ordered by height, there is a positive density of pairs $w=(g_1, g_2)$, such that the $T$-adic Galois representation attached to $\phi^w$ is surjective.