Weil-Moore anima

TL;DR

The paper introduces the Weil-Moore anima, a refined mathematical object with richer homotopy properties than the Weil group, aiming to improve cohomological analysis in number theory.

Contribution

It constructs the Weil-Moore anima, a new homotopical refinement of the Weil group with nontrivial higher homotopy groups, enhancing cohomological properties.

Findings

Weil-Moore anima has the Weil group as its fundamental group.

It possesses nontrivial higher homotopy groups.

Cohomological properties are improved compared to Weil and Galois groups.

Abstract

The Weil group of a number field is a refinement of its absolute Galois group arising from class field theory. The passage from Galois to Weil is important in several places in number theory. However, we will argue that while from the Galois perspective, a number field is a ``K($\pi$,1)'', from the Weil perspective it is not. Thus we are led to further refine the Weil group, by constructing an object, the Weil-Moore anima, which has the Weil group as its fundamental group, but with nontrivial higher homotopy groups. Our motivation is that the cohomological properties of Weil-Moore anima are in several ways nicer than those of the Weil or Galois groups.