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

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.

References in corpus (2)

• Analogies between Knots and Primes, 3-Manifolds and Number Rings
• On pro-p fundamental groups of marked arithmetic curves

Cited by in corpus (9)

• Practice of incomplete p-ramification over a number field -- History of abelian p-ramification
• Pro-$p$ groups of positive rank gradient and Hausdorff dimension
• On pro-$p$ link groups of number fields
• On the invariant factors of class groups in towers of number fields
• The Bloch-Kato Conjecture and Galois Theory
• Mild pro-p groups and the Koszulity conjectures
• Dimensions of Zassenhaus filtration subquotients of some pro-$p$-groups
• Combinatorial Techniques in the Galois Theory of $p$-Extensions
• Deficiency of p-Class Tower Groups and Minkowski Units