Cohomology of varieties over the maximal Kummer extension of a number field

TL;DR

This paper proves finiteness results for Galois-invariant cohomology classes of varieties over the maximal Kummer extension of a number field, with implications for torsion points on abelian varieties.

Contribution

It establishes the finiteness of Galois-invariant cohomology classes over the maximal Kummer extension, improving previous results and analyzing torsion control over non-abelian extensions.

Findings

Galois-invariant cohomology classes are finite in odd degrees.

Abelian varieties have finite torsion over the maximal Kummer extension.

Finiteness of torsion over non-abelian solvable extensions is not governed by the Galois group.

Abstract

Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the Galois group of the maximal Kummer extension of $K$ in odd degrees. In particular, every abelian variety has finite torsion over the maximal Kummer extension. This improves results by R\"ossler and the second author as well as Murotani and Ozeki. We also show that finiteness of torsion of a given abelian variety over non-abelian solvable extensions of $K$ is not controlled by the Galois group of the extension.