Generalised height pairings and the Albanese kernel

TL;DR

This paper investigates the generalised height pairing in the context of rational points on curves, exploring its domain, computability, and connections to Beilinson--Bloch conjectures, with implications for nonabelian Chabauty methods.

Contribution

It establishes conditions under which the generalised height pairing can be computed and explores motivic refinements related to nonabelian cohomology in Chabauty theory.

Findings

If the Albanese kernel of X×X is torsion, the pairing is computable.

The paper links the pairing's domain to Beilinson--Bloch conjectures.

Introduces motivic refinements of nonabelian cohomology varieties.

Abstract

The Chabauty--Coleman--Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekov\'a\v{r}'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.