Imaginary logarithmic classes of abelian number fields

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.