A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911845143543808 |
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| author | Silverman, Joseph H. |
| author_facet | Silverman, Joseph H. |
| contents | Let $\mathbb{F}$ be the function field of a curve over an algebraically closed field with $\operatorname{char}(\mathbb{F})\ne2,3$, and let $E/\mathbb{F}$ be an elliptic curve. Then for all finite extensions $\mathbb{K}/\mathbb{F}$ and all non-torsion points $P\in{E(\mathbb{K})}$, the $\mathbb{F}$-normalized canonical height of $P$ is bounded below by \[ \hat{h}_E(P) \ge \frac{1}{10500\cdot h_{\mathbb{F}}(j_E)^{2}\cdot [\mathbb{K}:\mathbb{F}]^{2}}. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields Silverman, Joseph H. Number Theory Primary: 11G05, Secondary: 11R58, 14G40 Let $\mathbb{F}$ be the function field of a curve over an algebraically closed field with $\operatorname{char}(\mathbb{F})\ne2,3$, and let $E/\mathbb{F}$ be an elliptic curve. Then for all finite extensions $\mathbb{K}/\mathbb{F}$ and all non-torsion points $P\in{E(\mathbb{K})}$, the $\mathbb{F}$-normalized canonical height of $P$ is bounded below by \[ \hat{h}_E(P) \ge \frac{1}{10500\cdot h_{\mathbb{F}}(j_E)^{2}\cdot [\mathbb{K}:\mathbb{F}]^{2}}. \] |
| title | A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields |
| topic | Number Theory Primary: 11G05, Secondary: 11R58, 14G40 |
| url | https://arxiv.org/abs/2402.14771 |