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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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Table of 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."