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(Q₂ )₂, when X is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at $2$ and ∞ for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group CH² (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(Q₂ )₂ <∞. We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions).