Kummer-faithful fields with finitely generated absolute Galois group

TL;DR

This paper demonstrates that most algebraic extensions of number fields with finitely generated Galois groups are Kummer-faithful, meaning their Mordell-Weil groups have trivial divisible parts, with implications for abelian Galois groups.

Contribution

It introduces a probabilistic approach showing that almost all such fields are Kummer-faithful, extending understanding of Mordell-Weil groups over these fields.

Findings

Almost all fields with finitely generated Galois groups are Kummer-faithful.

The Mordell-Weil group over these fields has trivial divisible part.

Existence of a Kummer-faithful field with abelian Galois group.

Abstract

This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single element. A probabilistic argument using the Haar measure on the absolute Galois group of a number field shows that almost all such fields are Kummer-faithful, i.e., the Mordell--Weil group of any semiabelian variety over any finite extension of such a field has trivial divisible part. This result implies that there exists a Kummer-faithful field algebraic over a number field whose absolute Galois group is abelian.