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

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 $K^{\mathrm{c}\mathrm{y}\mathrm{c}}$ of $K$. Recently, Rössler and Szamuely generalized Ribet’s theorem in terms of the étale cohomology with $\mathbb{Q}\mathrm{/}\mathbb{Z}$-coefficients of a smooth proper variety. In this paper, we show that the same finiteness holds even after replacing $K^{\mathrm{c}\mathrm{y}\mathrm{c}}$ 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.