DRINFELD MODULES AND SUBFIELDS OF DIVISION FIELDS

Let A = F-q[T], where F-q is a finite field, let Q = F-q(T), and let F be a finite extension of Q. Consider phi a Drinfeld A-module over F of rank r. We write r = hed, where E is the center of D :=End (f) over bar(phi) circle times q, e = [E : Q] and d = [D : E](1/)2. For m epsilon A, let F(phi[m]) be the field obtained by adjoining to F the m-division points phi[m] of phi, and let F(phi[m])' be the subfield of F(phi[m]) fixed by the scalar elements of Gal(F(phi[m])/F) subset of GL(r)(A/mA). In this paper, when r >= 3 and h >= 2, we study the splitting of the primes p of F of degree x in the fields F(phi[m])' and obtain an asymptotic formula which counts them.