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

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.