On the surjectivity of $\mathfrak{p}$-adic Galois representations attached to Drinfeld modules of rank $2$

Abstract: Let $\mathbb{F}{q}$ be the finite field with $q\geq 5$ elements and $A:=\mathbb{F}{q}[T]$. For a class of $\mathfrak{p} \in \mathrm{Spec}(A) \setminus {(0)}$, but fixed, we produce infinitely many Drinfeld $A$-modules of rank $2$, for which the associated $\mathfrak{p}$-adic Galois representation is surjective. This result is a variant of the work of~[Ray24] for $\mathfrak{p}=(T)$. We also show that for a class of $\mathfrak{l}=(l) \in \mathrm{Spec}(A)$, where $l$ is a monic polynomial, the $\mathfrak{p}$-adic Galois representation, attached to the Drinfeld $A$-module $\varphi_{T}=T+g_{1}\tau-l^{q-1}\tau^2$ with $g_{1} \in A \setminus \mathfrak{l}$, is surjective for all $\mathfrak{p} \in \mathrm{Spec}(A)\setminus{(0)}$. This result generalizes the work of [Zyw11] from $\mathfrak{l}=(T), g_1=1$.