Drinfeld modules and subfields of division fields

Let A = double-struck F_q[T], where double-struck F_q is a finite field, let Q = double-struck F_q(T), and let F be a finite extension of Q. Consider φ a Drinfeld A-module over F of rank r. We write r = hed, where E is the center of D :=End _{F{combining macron}}(φ)⊗Q, e = [E : Q] and d = [D : E]1/2. For m ∈ A, let F (φ[m]) be the field obtained by adjoining to F the m-division points φ[m] of φ, and let F (φ[m])′ be the subfield of F (φ[m]) fixed by the scalar elements of Gal(F (φ[m])/F) ⊆ GL_r(A/mA). In this paper, when r ≥ 3 and h ≥ 2, we study the splitting of the primes ℘ of F of degree x in the fields F (φ[m])′ and obtain an asymptotic formula which counts them.