On the Iwasawa $\lambda$-invariants of real abelian fields

For a prime number $p$ and a number field $k$, let $A_\infty$ denote the projective limit of the $p$-parts of the ideal class groups of the intermediate fields of the cyclotomic $\mathbb{Z} _p$-extension over $k$. It is conjectured that $A_\infty$ is finite if $k$ is totally real. When $p$ is an odd prime and $k$ 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 $p$ divides the degree of $k$, we also obtain a rather simple criterion.