Cohomology of varieties over the maximal Kummer extension of a number field
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909980835184640 |
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| author | Lombardo, Davide Szamuely, Tamás |
| author_facet | Lombardo, Davide Szamuely, Tamás |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomology of varieties over the maximal Kummer extension of a number field Lombardo, Davide Szamuely, Tamás Algebraic Geometry Number Theory 11G10, 14K15 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. |
| title | Cohomology of varieties over the maximal Kummer extension of a number field |
| topic | Algebraic Geometry Number Theory 11G10, 14K15 |
| url | https://arxiv.org/abs/2512.18759 |