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 $\varphi_{T}=T+g_{1}τ-l^{q-1}τ^2$ 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 $\mathfrak{l}=(T), g_1=1$.

Theorems & Definitions (33)

• Theorem 1.1: Ser72
• Theorem 1.2: PR09
• Definition 2.1: Drinfeld module
• Example 2.2: Carlitz module
• Definition 2.3: Morphism
• Definition 2.4: $j$-invariant
• Theorem 2.5
• Definition 2.6: Stable/Good reduction
• Theorem 3.1: Main Theorem
• Proposition 3.2