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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.14795 |
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| _version_ | 1866917274133200896 |
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| author | Park, Jun-Yong |
| author_facet | Park, Jun-Yong |
| contents | We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields proven by Phillips, and the overarching conjecture of Goldfeld and Katz-Sarnak regarding the ``Distribution of Ranks of Elliptic Curves''. Consequently, we arrive at the quantitative statement which naturally renders even finer conjecture regarding the lower order main terms differing for the number of $E/K$ with $|E(K)| = 1$ and $E(K) = \mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative rank distribution conjecture over $\mathbb{F}_q(t)$ Park, Jun-Yong Number Theory Algebraic Geometry We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields proven by Phillips, and the overarching conjecture of Goldfeld and Katz-Sarnak regarding the ``Distribution of Ranks of Elliptic Curves''. Consequently, we arrive at the quantitative statement which naturally renders even finer conjecture regarding the lower order main terms differing for the number of $E/K$ with $|E(K)| = 1$ and $E(K) = \mathbb{Z}$. |
| title | Quantitative rank distribution conjecture over $\mathbb{F}_q(t)$ |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2409.14795 |