On the Iwasawa 𝜆-invariants of real abelian fields

Abstract

For a prime number p p and a number field k k , let A ∞ A_\infty denote the projective limit of the p p -parts of the ideal class groups of the intermediate fields of the cyclotomic Z p \mathbb {Z}_p -extension over k k . It is conjectured that A ∞ A_\infty is finite if k k is totally real. When p p is an odd prime and k k is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where p p divides the degree of k k , we also obtain a rather simple criterion.