Reconstructing function fields from rational quotients of mod-$\ell$ Galois groups

TL;DR

This paper advances birational anabelian geometry by demonstrating how to reconstruct higher-dimensional function fields over algebraically closed fields from their mod-$eta$ Galois groups and rational quotients, extending Bogomolov's program.

Contribution

It introduces a method to reconstruct function fields of transcendence degree at least 5 from mod-$eta$ Galois groups, a key step in the global theory for higher dimensions.

Findings

Reconstruction of function fields from Galois groups is possible for transcendence degree ≥ 5.

The approach extends the mod-$eta$ analogue of Bogomolov's program.

Provides a framework for understanding birational properties via Galois groups.

Abstract

In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we show how to reconstruct a function field $K$ of transcendence degree $\geq 5$ over an algebraically closed field, up-to inseparable extensions, from the mod-$\ell$ abelian-by-central Galois group of $K$ endowed with the collection of mod-$\ell$ rational quotients.