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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.17933 |
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| _version_ | 1866915741979115520 |
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| author | Kala, Sushant |
| author_facet | Kala, Sushant |
| contents | Let $A$ be an abelian variety defined over a number field $\mathbb{Q}$, and let $\hat{h}$ be the Néron-Tate height on $A(\overline{\mathbb{Q}})$ corresponding to a symmetric ample line bundle on $A$. In this article, we prove that the Néron-Tate height of totally $p$-adic points is bounded below by an absolute constant depending only on $A$ for all but finitely many primes. In other words, if we denote by $\mathbb{Q}^{(p)}$ the maximal algebraic extension of $\mathbb{Q}$ in which $p$ is totally split, then $A(\mathbb{Q}^{(p)})$ satisfies the Bogomolov property for all but finitely many primes. In particular, if $A$ has good reduction at a prime $p$, we obtain the Bogomolov property $A(\Q^{(p)})$. This is the first instance where such a result has been obtained in the good reduction case. In a more general setting, if $A/K$ is an abelian variety and $\mathcal{K}/K$ is an asymptotically positive extension as defined in \cite{AB-SK}, which includes infinite Galois extensions with finite local degree at a non-archimedean place, then $A(\mathcal{K})$ satisfies the Bogomolov property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower bound for the canonical on Abelian varieties over totally $p$-adic extensions Kala, Sushant Number Theory Let $A$ be an abelian variety defined over a number field $\mathbb{Q}$, and let $\hat{h}$ be the Néron-Tate height on $A(\overline{\mathbb{Q}})$ corresponding to a symmetric ample line bundle on $A$. In this article, we prove that the Néron-Tate height of totally $p$-adic points is bounded below by an absolute constant depending only on $A$ for all but finitely many primes. In other words, if we denote by $\mathbb{Q}^{(p)}$ the maximal algebraic extension of $\mathbb{Q}$ in which $p$ is totally split, then $A(\mathbb{Q}^{(p)})$ satisfies the Bogomolov property for all but finitely many primes. In particular, if $A$ has good reduction at a prime $p$, we obtain the Bogomolov property $A(\Q^{(p)})$. This is the first instance where such a result has been obtained in the good reduction case. In a more general setting, if $A/K$ is an abelian variety and $\mathcal{K}/K$ is an asymptotically positive extension as defined in \cite{AB-SK}, which includes infinite Galois extensions with finite local degree at a non-archimedean place, then $A(\mathcal{K})$ satisfies the Bogomolov property. |
| title | Lower bound for the canonical on Abelian varieties over totally $p$-adic extensions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.17933 |