Imaginary logarithmic classes of abelian number fields

TL;DR

This paper investigates the structure of logarithmic class groups of imaginary abelian number fields, extending Iwasawa theory concepts and computing isotypic components related to specific characters.

Contribution

It introduces methods to compute logarithmic class groups' components for imaginary abelian fields and extends Gold's criterion to this context.

Findings

Computed isotypic components of logarithmic class groups for given fields.

Extended Gold's criterion on lambda invariants to imaginary abelian fields.

Analyzed semi-simple and non semi-simple cases with transition formulas.

Abstract

We are interested in classical and logarithmic imaginary classes of abelian number fields in connection with Iwasawa theory. For any given odd prime ${\ell}$ and any imaginary abelian number field K, we compute the isotypic components of the logarithmic ${\ell}$-class group of K associated to imaginary irreducible ${\ell}$-adic characters of its Galois group and we extend to this situation the Gold criterium on lambda invariants. We first study the semi-simple case, thus the non semi-simple case in connection with transition formulas of genus for supercircular fields.