Mild pro-2-groups and 2-extensions of Q with restricted ramification

TL;DR

This paper extends the theory of mild pro-2-groups to the case p=2, constructing Galois groups with controlled ramification and linking them to fundamental groups of arithmetic curves.

Contribution

It generalizes the concept of mild groups to p=2 using Lazard's mixed Lie algebras and constructs specific Galois groups with desired ramification properties.

Findings

Constructed Galois groups that are mild for p=2

Established convergence to the maximal pro-2 Galois group of Q

Extended results to fundamental groups of arithmetic curves

Abstract

Using the mixed Lie algebras of Lazard, we extend the results of the first author on mild groups to the case p=2. In particular, we show that for any finite set S_0 of odd rational primes we can find a finite set S of odd rational primes containing S_0 such that the Galois group of the maximal 2-extension of Q unramified outside S is mild. We thus produce a projective system of such Galois groups which converge to the maximal pro-2-quotient of the absolute Galois group of $\Q$ unramified at 2 and $\infty$. Our results also allow results of Alexander Schmidt on pro-p-fundamental groups of marked arithmetic curves to be extended to the case p=2 over a global field which is either a function field of odd characteristic or a totally imaginary number field.