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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/2403.07476 |
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| _version_ | 1866917611663523840 |
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| author | Dogra, Netan |
| author_facet | Dogra, Netan |
| contents | We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set $X(\mathbb{Q}_2 )_2 $, when $X$ is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at $2$ and $\infty$ for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group $\mathrm{CH}^2 (\mathrm{Jac}(X),1)$. As a result we show that most genus 2 curves in the LMFDB of Mordell--Weil rank 2 with exactly one rational Weierstrass point satsify $\# X(\mathbb{Q}_2 )_2 <\infty $. We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II Dogra, Netan Number Theory We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set $X(\mathbb{Q}_2 )_2 $, when $X$ is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at $2$ and $\infty$ for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group $\mathrm{CH}^2 (\mathrm{Jac}(X),1)$. As a result we show that most genus 2 curves in the LMFDB of Mordell--Weil rank 2 with exactly one rational Weierstrass point satsify $\# X(\mathbb{Q}_2 )_2 <\infty $. We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions). |
| title | 2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.07476 |