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

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 $\mathrm{\infty }$ for the mod 2 Bloch–Kato Selmer group associated to the higher Chow group ${\mathrm{C}\mathrm{H}}^2(\mathrm{J}\mathrm{a}\mathrm{c}(X),1)$. 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 $\mathrm{\#}X({\mathbb{Q}}_2{)}_2<\mathrm{\infty }$. We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions).