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 $\ddot{I}•(a):A\to F\{\ddot{I}„\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1\ddot{I}„+a_2\ddot{I}{„}^2$. The Galois action on the torsion of $\ddot{I}•(a)$ gives rise to a Galois representation $\ddot{I}{}_{\ddot{I}•(a)}:\mathrm{Gal}(F^{\mathrm{sep}}/F)\to {\mathrm{GL}}_2(\hat{A})$, where $\hat{A}$ is the profinite completion of $A$. We show that the image of $\ddot{I}{}_{\ddot{I}•(a)}$ 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 $[{\mathrm{GL}}_2(\hat{A}):\ddot{I}{}_{\ddot{I}•(a)}(\mathrm{Gal}(F^{\mathrm{sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $\ddot{I}{}_{\ddot{I}•(a)}$ is surjective for a positive density set of $a\in A^2$.