The m-step Solvable Hom-Form of Birational Anabelian Geometry for Number Fields

TL;DR

This paper advances the understanding of the Grothendieck birational anabelian conjecture for number fields by proving an m-step solvable conditional version with weaker conditions than previous results, and provides unconditional results for rational number fields.

Contribution

It introduces a new m-step solvable conditional framework for the conjecture, improving upon Uchida's conditions, and establishes unconditional cases for rational number fields.

Findings

Proves an m-step solvable conditional version of the conjecture.

Conditions are weaker than Uchida's theorem.

Unconditional results for the case of the rational number field.

Abstract

In 1981, Uchida proved a conditional version of the Hom-form of the Grothendieck birational anabelian conjecture for number fields. In this paper we prove an m-step solvable conditional version of the Grothendieck birational anabelian conjecture for number fields whereby our conditions are slightly weaker than the ones in Uchida's theorem. Furthermore, as in Uchida's work, we show that our result is unconditional in the case where the number field relating to the domain of the given homomorphism is $\mathbb{Q}$