The Motivic Anabelian Geometry of Local Heights on Abelian Varieties

Abstract

We study the problem of describing local components of Néron–Tate height functions on abelian varieties over characteristic 0 0 local fields as functions on spaces of torsors under various realisations of the 2 2 -step unipotent motivic fundamental group of the G m \mathbb G_m -torsor corresponding to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the Q ℓ \mathbb Q_\ell - and Q p \mathbb Q_p -pro-unipotent étale realisations when the base field is p p -adic, and in terms of the R \mathbb R -pro-unipotent Betti–de Rham realization when the base field is archimedean. In the course of proving the p p -adic instance of these theorems, we develop a new technique for studying local nonabelian 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 pro-unipotent generalization of the Bloch–Kato exponential sequence under minimal assumptions.