The geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

TL;DR

This paper proves new cases of the geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields, extending known results over finite fields and establishing a reduction criterion.

Contribution

It establishes the geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields, including finite fields and fields over the prime field, using new reduction criteria.

Findings

Proved the conjecture over finite fields.

Established the m-step solvable version of the Oda-Tamagawa reduction criterion.

Extended the conjecture to fields finitely generated over the prime field.

Abstract

In this paper, we present some new results on the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bi-anabelian and strong bi-anabelian) geometrically m-step solvable Grothendieck conjecture(s) for affine hyperbolic curves over fields finitely generated over the prime field. First of all, we show the conjecture over finite fields. Next, we show the geometrically m-step solvable version of the Oda-Tamagawa good reduction criterion for hyperbolic curves. Finally, by using these two results, we show the conjecture over fields finitely generated over the prime field.