Global Solvably Closed Anabelian Geometry

Abstract

In this paper, we study the pro-&#931; anabelian geometry of hyperbolic curves, where &#931; is a nonempty set of prime numbers, over Galois groups of “solvably closed extensions” of number fields — i.e., infinite extensions of number fields which have no nontrivial abelian extensions. The main results of this paper are, in essence, immediate corollaries of the following three ingredients: (a) classical results concerning the structure of Galois groups of number fields; (b) an anabelian result of Uchida concerning Galois groups of solvably closed extensions of number fields; (c) a previous result of the author concerning the pro-&#931; anabelian geometry of hyperbolic curves over nonarchimedean local fields.</p