2014/09/17 by Alexander B. Ivanov, Ivanov, Alexander
Mathematics · #11R29 #11R42 #14G32 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1409.4909
openalex publication_date 2014/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our main goal in the present article is to explain how one can reconstruct the decomposition subgroups and norms of points on an arithmetic curve inside its fundamental group if the following data are given: the fundamental group, a part of the cyclotomic character and the family of the regulators of the fields corresponding to the generic points of all étale covers of the given curve. The approach is inspired by that of Tamagawa for curves over finite fields but uses Tsfasman-Vlăduţ theorem instead of Lefschetz trace formula. To the authors' knowledge, this is a new technique in the anabelian geometry of arithmetic curves. It is conditional and depends on still unknown properties of arithmetic fundamental groups. We also give a new approach via Iwasawa theory to the local correspondence at the boundary.