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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.08493 |
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| _version_ | 1866914913110196224 |
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| author | Achenjang, Niven |
| author_facet | Achenjang, Niven |
| contents | Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2ζ_B(2)ζ_B(10)$, where $ζ_B$ is the zeta function of the curve $B$. In particular, in the limit as $q=\#k\to\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in "bad" characteristics. This completes the proof that the average rank of elliptic curves, over $\textit{any}$ fixed global field, is finite. Handling the case of characteristic $2$ requires us to develop a new theory of integral models of 2-Selmer elements, dubbed "hyper-Weierstrass curves." |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08493 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2 Achenjang, Niven Number Theory Algebraic Geometry 11G05 Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2ζ_B(2)ζ_B(10)$, where $ζ_B$ is the zeta function of the curve $B$. In particular, in the limit as $q=\#k\to\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in "bad" characteristics. This completes the proof that the average rank of elliptic curves, over $\textit{any}$ fixed global field, is finite. Handling the case of characteristic $2$ requires us to develop a new theory of integral models of 2-Selmer elements, dubbed "hyper-Weierstrass curves." |
| title | The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2 |
| topic | Number Theory Algebraic Geometry 11G05 |
| url | https://arxiv.org/abs/2310.08493 |