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arXiv · 2403.07476

2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II

Netan Dogra
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Abstract

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).

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Netan Dogra. 2024-03-12. 2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II. https://arxiv.org/abs/2403.07476

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