Algebraic Anabelian Functors

TL;DR

This paper introduces an algebraic anabelian functor that connects algebraic schemes to group homomorphisms, providing a new perspective on anabelian geometry and proving Grothendieck's section conjecture for algebraic schemes.

Contribution

It constructs a full and faithful covariant functor from algebraic schemes to outer homomorphism sets, reformulating anabelian geometry over fields.

Findings

Existence of a canonical algebraic anabelian functor

Reformulation of anabelian geometry over fields

Proof of Grothendieck's section conjecture for algebraic schemes

Abstract

In this paper we will prove that there exists a covariant functor, called algebraic anabelian functor, from the category of algebraic schemes over a given field to the category of outer homomorphism sets of groups. The algebraic anabelian functor, given in a canonical manner, is full and faithful. It reformulates the anabelian geometry over a field. As an application of the anabelian functor, we will also give a proof of the section conjecture of Grothendieck for the case of algebraic schemes.