Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915478317826048 |
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| author | Kloosterman, Remke |
| author_facet | Kloosterman, Remke |
| contents | Let $A,B$ be nonzero rational numbers. Consider the elliptic curve $E_{A,B}/\mathbb{Q}(t)$ with Weierstrass equation $y^2=x^3+At^6+B$.
An algorithm to determine $\mathrm{rank } E_{A,B}(\mathbb{Q}(t))$ as a function of $(A,B)$ was presented in a recent paper by Desjardins and Naskrecki. We will give a different and shorter proof for the correctness of that algorithm, using a more geometric approach and discuss for which classes of examples this approach might be useful. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces Kloosterman, Remke Number Theory Algebraic Geometry Let $A,B$ be nonzero rational numbers. Consider the elliptic curve $E_{A,B}/\mathbb{Q}(t)$ with Weierstrass equation $y^2=x^3+At^6+B$. An algorithm to determine $\mathrm{rank } E_{A,B}(\mathbb{Q}(t))$ as a function of $(A,B)$ was presented in a recent paper by Desjardins and Naskrecki. We will give a different and shorter proof for the correctness of that algorithm, using a more geometric approach and discuss for which classes of examples this approach might be useful. |
| title | Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2506.19423 |