Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields
Takahiro Murotani, Yoshiyasu Ozeki
TL;DR
This paper extends Ribet’s finiteness theorem for abelian varieties to broader infinite extensions of a number field $K$ by adjoining roots from a finitely generated subgroup of $K^{\times}$, via carefully constructed fields $M$ and $M'$, and connects these finiteness phenomena to étale cohomology invariants. The authors develop a robust Kummer-theoretic toolkit (inspired by Kubo–Taguchi) to control Galois actions on cohomology and use it to prove finiteness and vanishing results for $H^i_{ ext{ét}}(X_{\bar{K}},\mathbb{Q}/\mathbb{Z}(j))^{G_M}$ and related torsion questions for abelian varieties. They introduce and study TKND-AVKF fields, providing criteria and new examples, including a sharp CM-type criterion: $K^{\mathrm{ab}}$ is AVKF (and TKND) exactly when $K$ does not contain a CM field; otherwise AVKF fails. The work also supplies a Kummer-type construction framework for TKND-AVKF fields, yielding broad classes of base fields suitable for anabelian geometry and linking torsion finiteness to deep structural properties of the underlying field towers.
Abstract
It is a theorem of Ribet that an abelian variety defined over a number field $K$ has only finitely many torsion points with values in the maximal cyclotomic extension field $K^{\mathrm{cyc}}$ of $K$. Recently, Rössler and Szamuely generalized Ribet's theorem in terms of the étale cohomology with $\mathbb{Q}/\mathbb{Z}$-coefficients of a smooth proper variety. In this paper, we show that the same finiteness holds even after replacing $K^{\mathrm{cyc}}$ with the field obtained by adjoining to $K$ all roots of all elements of a certain subset of $K$. Furthermore, we give some new examples of TKND-AVKF fields; the notion of TKND-AVKF is introduced by Hoshi, Mochizuki and Tsujimura, and TKND-AVKF fields are expected as one of suitable base fields for anabelian geometry.