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arXiv · 2601.10298

Kummer-faithful fields with finitely generated absolute Galois group

Takuya Asayama
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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.

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Takuya Asayama. 2026-01-15. Kummer-faithful fields with finitely generated absolute Galois group. https://arxiv.org/abs/2601.10298

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