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

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 $k$.