On a Galois property of fields generated by the torsion of an abelian variety
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913574623903744 |
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| author | Checcoli, Sara Dill, Gabriel Andreas |
| author_facet | Checcoli, Sara Dill, Gabriel Andreas |
| contents | In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12138 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a Galois property of fields generated by the torsion of an abelian variety Checcoli, Sara Dill, Gabriel Andreas Number Theory 11J95, 11R32 In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height. |
| title | On a Galois property of fields generated by the torsion of an abelian variety |
| topic | Number Theory 11J95, 11R32 |
| url | https://arxiv.org/abs/2306.12138 |