Degree bounds for projective division fields associated to elliptic modules with a trivial endomorphism ring

Let k be a global field, let A be a Dedekind domain with Quot(A) = k, and let K be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules M that are defined over K and that have a trivial endomorphism ring, with k = Q, A = Z in the former case and with k a global function field, A its ring of functions regular away from a fixed prime in the latter case, we prove, for any nonzero ideal a . A, best possible estimates in the norm vertical bar a vertical bar for the degree over K of the subfield of the a-division field of M fixed by the scalars.