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A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic

Martin Lüdtke

Abstract

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover $K|k$ from its absolute Galois group alone, we prove that it can be recovered from the pair $(\mathrm{Aut}(\overline{K}|k),\mathrm{Aut}(\overline{K}|K))$, consisting of the absolute Galois group of $K$ and the larger group of field automorphisms fixing only the base field.

A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic

Abstract

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields of varieties of dimension over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover from its absolute Galois group alone, we prove that it can be recovered from the pair , consisting of the absolute Galois group of and the larger group of field automorphisms fixing only the base field.

Paper Structure

This paper contains 4 sections, 21 equations.

Table of Contents

  1. Introduction
  2. Injectivity
  3. A Galois-Type Correspondence for Transcendental Field Extensions
  4. Proof of Main Theorem

Theorems & Definitions (12)

  • proof
  • proof
  • proof : Proof of Theorem \ref{['thm-injectivity']}
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['thm-main-theorem']} $\Rightarrow$ Theorem \ref{['thm-function-field-main-theorem']}
  • proof
  • proof
  • proof
  • ...and 2 more