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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2407.11679 |
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| _version_ | 1866909257465593856 |
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| author | Azon, Martin |
| author_facet | Azon, Martin |
| contents | We produce an explicit sequence $\left(S_a \right)_{a \geq 1}$ of abelian surfaces over the rational function field $\mathbb{F}_{q}(t)$ whose Tate-Shafarevich groups are finite and large. More precisely, we establish the estimate $\left \arrowvert\mathrm{III}(S_a) \right \arrowvert = H(S_a)^{1 + o(1)}$ as $a \rightarrow \infty$, where $H(S_a)$ denotes the exponential height of $S_a$. Our method is to prove that each $S_a$ satisfies the BSD conjecture, analyse the geometry and arithmetic of its Néron model and give an explicit expression for its $L$-function in terms of Gauss and Kloosterman sums. By studying the relative distribution of the angles associated to these character sums, we estimate the size of the central value of $L(S_a, T)$, hence the order of $\mathrm{III}(S_a)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Abelian surfaces over $\mathbb{F}_{q}(t)$ with large Tate-Shafarevich groups Azon, Martin Number Theory We produce an explicit sequence $\left(S_a \right)_{a \geq 1}$ of abelian surfaces over the rational function field $\mathbb{F}_{q}(t)$ whose Tate-Shafarevich groups are finite and large. More precisely, we establish the estimate $\left \arrowvert\mathrm{III}(S_a) \right \arrowvert = H(S_a)^{1 + o(1)}$ as $a \rightarrow \infty$, where $H(S_a)$ denotes the exponential height of $S_a$. Our method is to prove that each $S_a$ satisfies the BSD conjecture, analyse the geometry and arithmetic of its Néron model and give an explicit expression for its $L$-function in terms of Gauss and Kloosterman sums. By studying the relative distribution of the angles associated to these character sums, we estimate the size of the central value of $L(S_a, T)$, hence the order of $\mathrm{III}(S_a)$. |
| title | Abelian surfaces over $\mathbb{F}_{q}(t)$ with large Tate-Shafarevich groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.11679 |