The T-adic Galois representation is surjective for a positive density of Drinfeld modules

Let F q be the finite field with q ≥ 5 elements, A : = F q [ T ] and F : = F q ( T ) . Assume that q is odd and take | · | to be the absolute value at ∞ that is normalized by | T | = q . Given a pair w = ( g 1 , g 2 ) ∈ A 2 with g 2 ≠ 0 , consider the associated Drinfeld module ϕ w : A → A { τ } of rank 2 defined by ϕ T w = T + g 1 τ + g 2 τ 2 . Fix integers c 1 , c 2 ≥ 1 and define | w | : = max { | g 1 | 1 c 1 , | g 2 | 1 c 2 } . I show that when ordered by height, there is a positive density of pairs w = ( g 1 , g 2 ) , such that the T -adic Galois representation attached to ϕ w is surjective.