A refined version of the geometrically m-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields

TL;DR

This paper advances the understanding of the Grothendieck conjecture for genus 0 curves over finitely generated fields by establishing an equivalence based on geometrically maximal 2-step solvable quotients of their étale fundamental groups.

Contribution

It refines the geometrically 2-step solvable Grothendieck conjecture for genus 0 hyperbolic curves over finitely generated fields, linking isomorphism of curves to fundamental group quotients.

Findings

Two genus 0 hyperbolic curves are isomorphic if their maximal 2-step solvable quotients of étale fundamental groups are isomorphic.

The result holds up to Frobenius twists.

Provides a new criterion for curve isomorphism based on fundamental group quotients.

Abstract

In the present paper, we show a new result on the geometrically $2$-step solvable Grothendieck conjecture for genus $0$ curves over finitely generated fields. More precisely, we show that two genus $0$ hyperbolic curves over a finitely generated field $k$ are isomorphic as $k$-schemes (up to Frobenius twists) if and only if the geometrically maximal $2$-step solvable quotients of their \'etale fundamental groups are isomorphic as topological groups over the absolute Galois group of $k$.