• Corpus ID: 119131663

The motivic anabelian geometry of local heights on abelian varieties

@article{Betts2017TheMA,
  title={The motivic anabelian geometry of local heights on abelian varieties},
  author={L. Alexander Betts},
  journal={arXiv: Number Theory},
  year={2017}
}

• L. A. Betts
• Published 15 June 2017
• Mathematics
• arXiv: Number Theory

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.
On the geometric side, we also prove a number of foundational results on local constancy or analyticity of various non-abelian Kummer maps for pro-unipotent etale or Betti--de Rham fundamental groups of arbitrary smooth varieties.