Cohomology of normic systems and fake Z_p extensions

TL;DR

This paper develops a framework for analyzing Tate cohomology in Galois modules over $Z_p$-extensions, establishing exact sequences to understand class number growth in fake $Z_p$-extensions of dihedral type.

Contribution

It introduces a general approach to study Tate cohomology in a new quotient category, enabling control over unit cohomology and class number growth in specialized extensions.

Findings

Established a five-term exact sequence in a quotient category for Galois modules.

Controlled the behavior of Tate cohomology groups of units along $Z_p$-extensions.

Analyzed class number growth in fake $Z_p$-extensions of dihedral type.

Abstract

We set up a general framework to study Tate cohomology groups of Galois modules along $\mathbb{Z}_p$-extensions of number fields. Under suitable assumptions on the Galois modules, we establish the existence of a five-term exact sequence in a certain quotient category whose objects are simultaneously direct and inverse systems, subject to some compatibility. The exact sequence allows one, in particular, to control the behaviour of the Tate cohomology groups of the units along $\mathbb{Z}_p$-extensions. As an application, we study the growth of class numbers along what we call "fake $\mathbb{Z}_p$-extensions of dihedral type". This study relies on a previous work, where we established a class number formula for dihedral extensions in terms of the cohomology groups of the units.