Finiteness properties of torsion fields of abelian varieties
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909235862831104 |
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| author | Gajda, Wojciech Petersen, Sebastian |
| author_facet | Gajda, Wojciech Petersen, Sebastian |
| contents | Let $A$ be an abelian variety defined over a field $K.$ We study finite generation properties of the profinite group $\mathrm{Gal}(Ω/K)$ and of certain closed normal subgroups thereof, where $Ω$ is the torsion field of $A$ over $K$. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via étale cohomology. We apply this in order to give an independent proof and generalizations of a recent result of Checcoli and Dill about small exponent subfields of $Ω/K$ in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finiteness properties of torsion fields of abelian varieties Gajda, Wojciech Petersen, Sebastian Number Theory Algebraic Geometry 15K05, 14K15, 11G10 Let $A$ be an abelian variety defined over a field $K.$ We study finite generation properties of the profinite group $\mathrm{Gal}(Ω/K)$ and of certain closed normal subgroups thereof, where $Ω$ is the torsion field of $A$ over $K$. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via étale cohomology. We apply this in order to give an independent proof and generalizations of a recent result of Checcoli and Dill about small exponent subfields of $Ω/K$ in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property. |
| title | Finiteness properties of torsion fields of abelian varieties |
| topic | Number Theory Algebraic Geometry 15K05, 14K15, 11G10 |
| url | https://arxiv.org/abs/2401.05805 |