On a two-parameter family of tropical Edwards curves
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913542271139840 |
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| author | Nakamura, Hiroaki Tarmidi, Rani Sasmita |
| author_facet | Nakamura, Hiroaki Tarmidi, Rani Sasmita |
| contents | In this paper, a certain two-parameter family of plane-embeddings of Edwards elliptic curve $E_a: x^2+y^2=a^2(1+x^2y^2)$ is introduced to provide explicitly computed tropical curves corresponding to degeneration in $a\to 1$. Applying the theta uniformization of $E_a$ with the method of ultradiscretization by Kajiwara-Kaneko-Nobe-Tsuda, we give a formula for the coordinate functions that traces the cycle part of the tropical elliptic curve. We also illustrate how one can recover the whole part of the tropical curve as a quotient of the Bruhat-Tits tree after Speyer's algebraic approach in smooth cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02093 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a two-parameter family of tropical Edwards curves Nakamura, Hiroaki Tarmidi, Rani Sasmita Algebraic Geometry 14T20, 14T10, 14K25, 14H52 In this paper, a certain two-parameter family of plane-embeddings of Edwards elliptic curve $E_a: x^2+y^2=a^2(1+x^2y^2)$ is introduced to provide explicitly computed tropical curves corresponding to degeneration in $a\to 1$. Applying the theta uniformization of $E_a$ with the method of ultradiscretization by Kajiwara-Kaneko-Nobe-Tsuda, we give a formula for the coordinate functions that traces the cycle part of the tropical elliptic curve. We also illustrate how one can recover the whole part of the tropical curve as a quotient of the Bruhat-Tits tree after Speyer's algebraic approach in smooth cases. |
| title | On a two-parameter family of tropical Edwards curves |
| topic | Algebraic Geometry 14T20, 14T10, 14K25, 14H52 |
| url | https://arxiv.org/abs/2307.02093 |