Kummer theory for commutative algebraic groups

Abstract

This dissertation is a collection of four research articles devoted to

the study of Kummer theory for commutative algebraic groups. In number

theory, Kummer theory refers to the study of field extensions generated

by n-th roots of some base field. Its generalization to commutative algebraic groups involves fields generated by the division points of a fixed algebraic group, such as an elliptic curve or a higher dimensional abelian variety. Of particular interest in this dissertation is the degree of such field extensions. In the first two chapter, classical results for elliptic curves are improved by providing explicitly computable bounds and uniform and explicit bounds over the field of rational numbers. In the last two chapters a general framework for the study of similar problems is developed.