On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a prime power and let $A:= \mathbb{F}_{q}[T]$. By~\cite{PR09}, the adelic image of the Galois representation attached to a rank $2$ Drinfeld $A$-module $\varphi$ is open, and determining when it is surjective remains a subtle problem. To resolve this question, in this article, we study the $\mathfrak{p}$-adic surjectivity of the Galois representations attached to $\varphi$, where $\mathfrak{p} \in \Omega_A:= \mathrm{Spec}(A) \setminus \{ (0) \}$. There are two directions to investigate this problem: one by fixing the prime $\mathfrak{p}$, and the other by fixing $\varphi$. In the horizontal direction, for a fixed prime $\mathfrak{p} \in \Omega_A$, we give explicit and easily verifiable conditions on Drinfeld $ A$-modules $\varphi$ of rank $2$ which ensure the surjectivity of the $\mathfrak{p}$-adic Galois representation $\rho_{\varphi,\mathfrak{p}}$. This work not only extends the work of~\cite{Ray24} for $\mathfrak{p}=(T)$, but also obtains a variant of~\cite{Ray24} under comparatively simpler conditions in the case $\mathfrak{p}=(T)$. In the vertical direction, we show that for a fixed rank $2$ Drinfeld $A$-module $\varphi$, whose coefficients satisfy certain congruence and valuation conditions, the $\mathfrak{p}$-adic Galois representation $\rho_{\varphi,\mathfrak{p}}$ is surjective for all primes $\mathfrak{p} \in \Omega_A$. This recovers the example of \cite{Zyw11} and yields new examples beyond those considered in \cite{Zyw25}. As a consequence, we obtain the surjectivity of the associated adelic Galois representation.