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

TL;DR

This paper establishes optimal bounds on the degrees of subfields within division fields of elliptic modules and Drinfeld modules with trivial endomorphism rings over global fields, unifying the approach for both cases.

Contribution

It provides the first unified method to estimate degrees of subfields in division fields for elliptic and Drinfeld modules with trivial endomorphism rings.

Findings

Proves best possible bounds on degrees over $K$ of subfields fixed by scalars.

Establishes estimates in terms of the norm of ideals in the Dedekind domain.

Unifies the treatment of elliptic curves and Drinfeld modules.

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.