# The motivic anabelian geometry of local heights on abelian varieties

## TL;DR

This paper explores the motivic anabelian geometry of local heights on abelian varieties over various local fields, providing new descriptions and techniques for understanding local height functions through unipotent fundamental groups and Selmer sets.

## Contribution

It introduces a novel approach to describing local height functions via unipotent fundamental groups and develops new methods for studying non-abelian Selmer sets in this context.

## Key findings

- Descriptions of local height functions in terms of unipotent fundamental groups for p-adic and archimedean fields.
- Development of explicit cosimplicial group models for non-abelian Selmer sets.
- Construction of a non-abelian generalization of the Bloch--Kato exponential sequence.

## Abstract

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.