  We study the problem of describing local components of height functions on
abelian varieties over characteristic $0$ local fields as functions on spaces
of torsors under various realisations of a $2$-step unipotent motivic
fundamental group naturally associated to the defining line bundle. To this
end, we present three main theorems giving such a description in terms of the
$\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent \'etale realisations when the
base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de
Rham realisation when the base field is archimedean.
  In the course of proving the $p$-adic instance of these theorems, we develop
a new technique for studying local non-abelian Bloch--Kato Selmer sets, working
with certain explicit cosimplicial group models for these sets and using
methods from homotopical algebra. Among other uses, these models enable us to
construct a non-abelian generalisation of the Bloch--Kato exponential sequence
under minimal conditions.
