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Kummer-faithful fields with finitely generated absolute Galois group

Takuya Asayama

TL;DR

The paper investigates Kummer-faithfulness for algebraic extensions of a number field with finitely generated absolute Galois groups, proving that for number fields $K$, almost all $\sigma\in G_K^e$ yield $\overline{K}(\sigma)$ whose finite extensions are Kummer-faithful with respect to semiabelian varieties. It splits the proof into torally Kummer-faithfulness and AVKF-ness, employing Haar-measure arguments, cyclic-extension structure, and independence results (Perucca–Sgobba–Tronto; Serre) to show measure-zero bad-sets and hence almost-sure KF properties. A key corollary is the abundance of KF fields with abelian Galois groups, and the work analyzes the behavior of composites of KF fields, including constructive counterexamples to naive expectations. Collectively, the results connect probabilistic Galois theory with anabelian-geometric constraints on Mordell–Weil groups, illuminating when rational points and Kummer maps are rigid across algebraic extensions. The methods leverage measure theory on $G_K^e$, Galois-representation independence, and density results (e.g., Zywina) to establish robust KF-ness in a broad arithmetic setting.

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.

Kummer-faithful fields with finitely generated absolute Galois group

TL;DR

The paper investigates Kummer-faithfulness for algebraic extensions of a number field with finitely generated absolute Galois groups, proving that for number fields , almost all yield whose finite extensions are Kummer-faithful with respect to semiabelian varieties. It splits the proof into torally Kummer-faithfulness and AVKF-ness, employing Haar-measure arguments, cyclic-extension structure, and independence results (Perucca–Sgobba–Tronto; Serre) to show measure-zero bad-sets and hence almost-sure KF properties. A key corollary is the abundance of KF fields with abelian Galois groups, and the work analyzes the behavior of composites of KF fields, including constructive counterexamples to naive expectations. Collectively, the results connect probabilistic Galois theory with anabelian-geometric constraints on Mordell–Weil groups, illuminating when rational points and Kummer maps are rigid across algebraic extensions. The methods leverage measure theory on , Galois-representation independence, and density results (e.g., Zywina) to establish robust KF-ness in a broad arithmetic setting.

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.
Paper Structure (14 sections, 23 theorems, 78 equations)

This paper contains 14 sections, 23 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Proof for the torally Kummer-faithfulness
  3. Review of properties of $\overline{K}(\sigma)$
  4. Reduction step
  5. Estimating the measure
  6. Independence and the completion of the proof
  7. Proof for the AVKF-ness
  8. Overview of Jarden--Petersen's strategy
  9. Independence
  10. Computing of the measure of $S_\ell(\mathbf{p})$
  11. Conclusion
  12. Corollaries to Theorem \ref{['thm-KF']}
  13. Existence of Kummer-faithful fields with abelian absolute Galois group
  14. Application to the composite of Kummer-faithful fields

Key Result

Theorem 1.4

Suppose that $K$ is a number field and $e$ a positive integer. For almost all $\sigma \in G_K^e$, any finite extension of $\overline{K}(\sigma)$ is both torally Kummer-faithful and AVKF, and thus, Kummer-faithful.

Theorems & Definitions (42)

  • Definition 1.1: Mochizuki15, HMT
  • Remark 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1: cf. Lang02
  • proof
  • Lemma 2.2
  • ...and 32 more