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

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).

29 pages. Comments welcome!