A number field analogue of the Grothendieck conjecture for curves over finite fields

TL;DR

This paper establishes a new analogy between number fields and function fields over finite fields, providing a number field analogue of the Grothendieck conjecture for hyperbolic curves, extending prior results in the function field case.

Contribution

It introduces a novel analogy between number fields and function fields and proves a number field analogue of the Grothendieck conjecture for hyperbolic curves over finite fields.

Findings

Established a number field analogue of the Grothendieck conjecture for hyperbolic curves.

Extended the analogy between cyclotomic extensions and constant field extensions.

Provided an affirmative answer to a conjecture by Neukirch-Schmidt-Wingberg in a specific case.

Abstract

In the present paper, we provide a new analogy between number fields and 1-dimensional function fields over finite fields from the viewpoint that the maximal cyclotomic extension of a number field is analogous to the constant field extension of a function field to an algebraic closure. Namely, we give a number field analogue of the Grothendieck conjecture for hyperbolic curves over finite fields proved by Tamagawa and Mochizuki, which is an affirmative answer to Conjecture(12.5.3) of ``Cohomology of number fields" by Neukirch-Schmidt-Wingberg in the case where the ramified prime sets are empty.