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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.16578 |
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| _version_ | 1866917154259992576 |
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| author | Salami, Sajad Zargar, Arman Shamsi |
| author_facet | Salami, Sajad Zargar, Arman Shamsi |
| contents | The splitting field of an elliptic surface $\mathcal E$ defined over ${\mathbb Q}(t)$ is the smallest subfield $\mathcal K$ of $\mathbb C$ such that ${\mathcal E}({\mathbb C}(t))\cong {\mathcal E}({\mathcal K}(t))$. In this paper, we determine the splitting field ${\mathcal K}_m$ and a set of linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface with generic fiber given by ${\mathcal E}_m: y^2=x^3 +t^{m} +1$ over ${\mathbb Q}(t)$ for positive integers $1\leq m\leq 12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3 +t^{m} +1$ (I) Salami, Sajad Zargar, Arman Shamsi Number Theory Algebraic Geometry 14J27, 11G05 The splitting field of an elliptic surface $\mathcal E$ defined over ${\mathbb Q}(t)$ is the smallest subfield $\mathcal K$ of $\mathbb C$ such that ${\mathcal E}({\mathbb C}(t))\cong {\mathcal E}({\mathcal K}(t))$. In this paper, we determine the splitting field ${\mathcal K}_m$ and a set of linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface with generic fiber given by ${\mathcal E}_m: y^2=x^3 +t^{m} +1$ over ${\mathbb Q}(t)$ for positive integers $1\leq m\leq 12$. |
| title | The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3 +t^{m} +1$ (I) |
| topic | Number Theory Algebraic Geometry 14J27, 11G05 |
| url | https://arxiv.org/abs/2512.16578 |