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Title: Heights via anabelian geometry and local Bloch-Kato Selmer sets
Author: Betts, Luke Alexander
ISNI:       0000 0004 7653 7672
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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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 Ql and Qp-pro-unipotent etale realisations when the base field is p-adic, and in terms of the 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.
Supervisor: Not available Sponsor: Wang Scholarship
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available