On some anabelian properties of arithmetic curves

TL;DR

This paper extends Neukirch's birational anabelian geometry argument to arithmetic curves, showing how to recover boundary point data from the fundamental group under certain conditions.

Contribution

It generalizes anabelian reconstruction techniques from function fields to number fields, detailing how boundary data can be derived from the fundamental group.

Findings

Reconstruction of boundary decomposition groups from the fundamental group

Identification of the p-part of the cyclotomic character from fundamental group data

Under certain hypotheses, all boundary information can be recovered from the fundamental group

Abstract

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $\pi_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary points inside it, the $p$-part of the cyclotomic character, the number of points on the boundary of all finite etale covers, etc. Under certain finiteness hypothesis on Tate-Shafarevich groups with divisible coefficients, one can reconstruct all this quantities from the fundamental group alone.