Drinfeld modules with maximal Galois action

Abstract

With a fixed prime power $q>1$, define the ring of polynomials $A=\mathbb{F}_q[t]$ and its fraction field $F=\mathbb{F}_q(t)$. For each pair $a=(a_1,a_2) \in A^2$ with $a_2$ nonzero, let $ϕ(a)\colon A\to F\{τ\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1τ+a_2τ^2$. The Galois action on the torsion of $ϕ(a)$ gives rise to a Galois representation $ρ_{ϕ(a)}\colon \operatorname{Gal}(F^{\operatorname{sep}}/F)\to \operatorname{GL}_2(\widehat{A})$, where $\widehat{A}$ is the profinite completion of $A$. We show that the image of $ρ_{ϕ(a)}$ is large for random $a$. More precisely, for all $a\in A^2$ away from a set of density $0$, we prove that the index $[\operatorname{GL}_2(\widehat{A}):ρ_{ϕ(a)}(\operatorname{Gal}(F^{\operatorname{sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $ρ_{ϕ(a)}$ is surjective for a positive density set of $a\in A^2$.