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 $\phi(a)\colon A\to F\{ \tau\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1\tau+a_2\tau^2$. The Galois action on the torsion of $\phi(a)$ gives rise to a Galois representation $\rho_{ \phi(a)}\colon \operatorname{ Gal}(F^{ \operatorname{ sep}}/F)\to \operatorname{ GL}_2(\widehat{ A})$, where $\widehat{ A}$ is the profinite completion of $A$. We show that the image of $\rho_{ \phi(a)}$ 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 $[\operatorname{ GL}_2(\widehat{ A}):\rho_{ \phi(a)}(\operatorname{ Gal}(F^{ \operatorname{ sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $\rho_{ \phi(a)}$ is surjective for a positive density set of $a\in A^2$.
Categories: math.NT