New evidence for Rémond's generalisation of Lehmer's conjecture
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908417363279872 |
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| author | Checcoli, Sara Dill, Gabriel Andreas |
| author_facet | Checcoli, Sara Dill, Gabriel Andreas |
| contents | In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of Rémond's generalisation of Lehmer's conjecture. Namely, for a finite rank subgroup $Γ$ of an almost split semiabelian variety $G$, we consider the group of rational points of $G$ over a finite extension of the field generated by the saturated closure of $Γ$, i.e. the division closure of the subgroup generated by $Γ$ and all its images under geometric endomorphisms of $G$. We show that this becomes a free group after one quotients out the saturated closure of $Γ$. The proof uses, amongst other ingredients, a criterion of Pottmeyer, which relies on a result of Pontryagin, together with a result from Kummer theory, of which we reproduce a proof by Rémond. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18776 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New evidence for Rémond's generalisation of Lehmer's conjecture Checcoli, Sara Dill, Gabriel Andreas Number Theory Algebraic Geometry 20K15, 11J95, 11R32, 11G50 In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of Rémond's generalisation of Lehmer's conjecture. Namely, for a finite rank subgroup $Γ$ of an almost split semiabelian variety $G$, we consider the group of rational points of $G$ over a finite extension of the field generated by the saturated closure of $Γ$, i.e. the division closure of the subgroup generated by $Γ$ and all its images under geometric endomorphisms of $G$. We show that this becomes a free group after one quotients out the saturated closure of $Γ$. The proof uses, amongst other ingredients, a criterion of Pottmeyer, which relies on a result of Pontryagin, together with a result from Kummer theory, of which we reproduce a proof by Rémond. |
| title | New evidence for Rémond's generalisation of Lehmer's conjecture |
| topic | Number Theory Algebraic Geometry 20K15, 11J95, 11R32, 11G50 |
| url | https://arxiv.org/abs/2506.18776 |