Cohomology of varieties over the maximal Kummer extension of a number field
Davide Lombardo, Tamás Szamuely
TL;DR
This paper proves that for a smooth projective variety $X$ over a number field $K$, the odd-degree $G_{K^{\mathrm Kum}}$-invariants of the geometric étale cohomology with $\mathbf{Q}/\mathbf{Z}(j)$-coefficients are finite, implying finite torsion of abelian varieties over the maximal Kummer extension. The argument reduces to controlling $p$-adic and mod-$p$ invariants over solvable extensions of finite class, via a Serre–Wintenberger type theorem and a key group-theoretic lemma that forces reductive images with solvable dense subgroups to be tori. Structural Galois-theoretic lemmas about the intersections of $K^{\mathrm Kum}$, $K^{\mathrm ab}$, and $K^{\mathrm cyc}$ are developed to manage local degree bounds and enable abelian reductions. The results extend prior work by Roessler–Szamuely and Murotani–Ozeki, and they also show that finiteness over solvable extensions is not controlled purely by the Galois group of the extension, through an explicit solvable-example involving CM elliptic curves. The work provides a robust framework for finiteness over broad infinite Galois extensions and clarifies the role of semisimplicity and local monodromy in these finiteness phenomena.
Abstract
Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric étale 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 Rössler 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.