Generalised height pairings and the Albanese kernel

The Chabauty–Coleman–Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekovář’s $p$-adic height pairing which replaces ${\mathbb{G}}_m$ with a higher Chow group. It is unclear both what the domain of definition of this pairing is, and how to compute it. This paper explores the relevance of the Beilinson–Bloch conjectures to this problem. In particular, it is shown that if $X$ is a smooth projective curve and the Albanese kernel of $X\times X$ is torsion, then there is an algorithm to compute the generalised height pairing on a pair of rational points on the Jacobian. This leads to the consideration of certain `motivic refinements’ of the nonabelian cohomology varieties which arise in nonabelian Chabauty.