Division in modules and Kummer theory

TL;DR

This paper develops a unified module divisibility theory to analyze Kummer extensions from algebraic groups, providing bounds on field extension degrees for elliptic curve division points.

Contribution

It generalizes injective modules and introduces a divisibility framework applicable to Kummer theory, extending prior results to broader classes of elliptic curves.

Findings

Effective bounds for degrees of Kummer extensions from elliptic curve division points

Unified module-theoretic approach to Kummer theory and algebraic groups

Extension of previous bounds to non-CM elliptic curves

Abstract

In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative algebraic groups. With these tools we provide an effective bound for the degree of the field extensions arising from division points of elliptic curves, extending previous results of Javan Peykar for CM curves and of Lombardo and the author for the non-CM case.