Tate-Shafarevich results for quartic twists in characteristic $2$
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929353635397632 |
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| author | Borges, Herivelto Guardieiro, João Paulo Salgado, Cecília Top, Jaap |
| author_facet | Borges, Herivelto Guardieiro, João Paulo Salgado, Cecília Top, Jaap |
| contents | The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular elliptic curve defined over $\mathbb{F}_2$ using the function field of a maximal curve $C$ that admits an order 4 automorphism. For such elliptic curves we provide a rank formula for its Mordell-Weil group in terms of the genera of $C$ and of another curve covered by $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13408 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tate-Shafarevich results for quartic twists in characteristic $2$ Borges, Herivelto Guardieiro, João Paulo Salgado, Cecília Top, Jaap Algebraic Geometry Number Theory 14H52, 11G20 (Primary), 14G15, 14G17, 14H05 (Secondary) The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular elliptic curve defined over $\mathbb{F}_2$ using the function field of a maximal curve $C$ that admits an order 4 automorphism. For such elliptic curves we provide a rank formula for its Mordell-Weil group in terms of the genera of $C$ and of another curve covered by $C$. |
| title | Tate-Shafarevich results for quartic twists in characteristic $2$ |
| topic | Algebraic Geometry Number Theory 14H52, 11G20 (Primary), 14G15, 14G17, 14H05 (Secondary) |
| url | https://arxiv.org/abs/2405.13408 |