The ranks of twists of an elliptic curve in characteristic $3$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909700088397824 |
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| author | Guardieiro, João Paulo |
| author_facet | Guardieiro, João Paulo |
| contents | Starting from the elliptic curve $E: y^2 = x^3 - x$ over $\mathbb{F}_9$, a curve $\mathcal{X}$ over $\mathbb{F}_{3^{2n}}$ and a cyclic cover of $\mathcal{X}$ of degree $m \in \{2,3,4,6\}$, we construct the corresponding $m$-twists over the function field $\mathbb{F}_{3^{2n}}(\mathcal{X})$. We also obtain the Mordell-Weil rank of these twists in terms of the Zeta functions of $\mathcal{X}$ and of suitable Kummer and Artin-Schreier extensions of it. Finally, we also describe the fibers of the elliptic fibration associated to such twists in the case $\mathcal{X} = \mathbb{P}^1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16785 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The ranks of twists of an elliptic curve in characteristic $3$ Guardieiro, João Paulo Algebraic Geometry 14H52, 11G20 (Primary) 14G17, 14H05 (Secondary) Starting from the elliptic curve $E: y^2 = x^3 - x$ over $\mathbb{F}_9$, a curve $\mathcal{X}$ over $\mathbb{F}_{3^{2n}}$ and a cyclic cover of $\mathcal{X}$ of degree $m \in \{2,3,4,6\}$, we construct the corresponding $m$-twists over the function field $\mathbb{F}_{3^{2n}}(\mathcal{X})$. We also obtain the Mordell-Weil rank of these twists in terms of the Zeta functions of $\mathcal{X}$ and of suitable Kummer and Artin-Schreier extensions of it. Finally, we also describe the fibers of the elliptic fibration associated to such twists in the case $\mathcal{X} = \mathbb{P}^1$. |
| title | The ranks of twists of an elliptic curve in characteristic $3$ |
| topic | Algebraic Geometry 14H52, 11G20 (Primary) 14G17, 14H05 (Secondary) |
| url | https://arxiv.org/abs/2507.16785 |