A class of Drinfeld $A$-modules of rank $3$ with surjective Galois representations

Let $q=p^e\ge 7$ be an odd prime power, and set $A:={\mathbb{F}}_q[T]$. In this article, we construct an infinite two-parameter family of Drinfeld $A$-modules of rank $3$ such that, for every non-zero prime ideal $\mathfrak{l}$ of $A$, the associated mod-$\mathfrak{l}$, $\mathfrak{l}$-adic, and adelic Galois representations are surjective. These results generalise the specific example, constructed only for primes $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, in~\cite{Che22}.