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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2309.08472 |
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| _version_ | 1866912979264471040 |
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| author | Trip, Manoy T. |
| author_facet | Trip, Manoy T. |
| contents | Consider a genus 2 curve defined over $\mathbb{Q}$ given by an affine equation of the form $y^2 = f(x)$ for some polynomial $f$ of degree 5, and let $p$ be an odd prime. Extending work of Perrin-Riou for elliptic curves, we construct a naive $p$-adic height function on a finite index subgroup of the Jacobian $J$ of this curve, using the explicit embedding of $J$ in $\mathbb{P}^8$ and the associated formal group described by Grant. We use the naive height to construct a global height $h_p: J(\mathbb{Q}) \rightarrow \mathbb{Q}_p$ using a limit construction analogous to Tate's construction of the Néron-Tate height, and show that it is quadratic. We then compare $h_p$ to a $p$-adic height constructed in a different way by Bianchi and show that they are equal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08472 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A naive p-adic height on the Jacobians of curves of genus 2 Trip, Manoy T. Number Theory 11G50 (Primary) 11G30 (Secondary) Consider a genus 2 curve defined over $\mathbb{Q}$ given by an affine equation of the form $y^2 = f(x)$ for some polynomial $f$ of degree 5, and let $p$ be an odd prime. Extending work of Perrin-Riou for elliptic curves, we construct a naive $p$-adic height function on a finite index subgroup of the Jacobian $J$ of this curve, using the explicit embedding of $J$ in $\mathbb{P}^8$ and the associated formal group described by Grant. We use the naive height to construct a global height $h_p: J(\mathbb{Q}) \rightarrow \mathbb{Q}_p$ using a limit construction analogous to Tate's construction of the Néron-Tate height, and show that it is quadratic. We then compare $h_p$ to a $p$-adic height constructed in a different way by Bianchi and show that they are equal. |
| title | A naive p-adic height on the Jacobians of curves of genus 2 |
| topic | Number Theory 11G50 (Primary) 11G30 (Secondary) |
| url | https://arxiv.org/abs/2309.08472 |