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

## TL;DR

This paper proves a new birational anabelian reconstruction theorem for algebraic curves over algebraically closed fields, showing that the function field can be recovered from automorphism groups, extending previous results in the field.

## Contribution

It establishes a birational anabelian reconstruction for curves over algebraically closed fields in arbitrary characteristic using automorphism groups.

## Key findings

- Function field can be reconstructed from automorphism groups
- The result applies to curves over any characteristic
- Advances the understanding of birational anabelian conjectures

## 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.