Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields

TL;DR

This paper extends Ribet's finiteness theorem for torsion points of abelian varieties over maximal cyclotomic fields to more general fields obtained by adjoining roots, and introduces new examples of TKND-AVKF fields relevant for anabelian geometry.

Contribution

It generalizes Ribet's theorem to fields formed by adjoining roots of elements, and provides new examples of TKND-AVKF fields for use in anabelian geometry.

Findings

Finiteness of torsion points extends to certain root-extended fields.

New examples of TKND-AVKF fields are constructed.

The results support the broader applicability of finiteness theorems in arithmetic geometry.

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, R\"ossler and Szamuely generalized Ribet's theorem in terms of the \'etale 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.