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

Abstract

Let $k$ be a global field, let $A$ be a Dedekind domain with $\text{Quot}(A) = k$, and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and having a trivial endomorphism ring, with $k= \mathbb{Q}$, $A = \mathbb{Z}$ in the former case and $k$ a global function field, $A$ its ring of functions regular away from a fixed prime in the latter case, for any nonzero ideal $\mathfrak{a} \lhd A$ we prove best possible estimates in the norm $|\mathfrak{a}|$ for the degrees over $K$ of the subfields of the $\mathfrak{a}$-division fields of $M$ fixed by scalars.