## 2-Selmer groups of even hyperelliptic curves over function fields

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Dao Van Thinh
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## Abstract:

In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg’s representation of the group $G=\text {PSO}(2n+2)$ and a Hitchin fibration. Consistent with the result over $\mathbb {Q}$ of Arul Shankar and Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely average number 6.## References

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## Additional Information

**Dao Van Thinh**- Affiliation: BICMR, Peking University, Beijing, China. No. 5, Yiheyuan Road, Haidian District, Beijing 100871
- MR Author ID: 1525870
- ORCID: 0000-0002-8915-1539
- Email: a0123872@u.nus.edu
- Received by editor(s): September 7, 2021
- Received by editor(s) in revised form: October 24, 2022
- Published electronically: April 19, 2023
- Additional Notes: This work was done when the author was supported by NUS Research Scholarship and PKU Boya postdoctoral Fellowship.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 4679-4712 - MSC (2020): Primary 11G30; Secondary 14D05, 14D10
- DOI: https://doi.org/10.1090/tran/8878