On automorphisms of some semidirect product groups and ranks of Iwasawa modules
Abstract
Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ does not split. Based on some heuristic, Kundu and Washington asked whether the $\lambda $- and $\mu $-invariants of the anti-cyclotomic $\mathbb {Z}_p$-extension $k_{\infty }^a$ of $k$ are always trivial. Also, if $k_{\infty }^a/k$ is totally ramified, for $n\geq 1$, they showed that the $p$-part of the ideal class group of the $n$th layer of the anti-cyclotomic $\mathbb {Z}_p$-extension of $k$ is not cyclic. Inspired by their paper, we study anti-cyclotomic-like $\mathbb {Z}_p$-extensions, extending both the above question and Kundu–Washington’s result. We show that the values of $\lambda $ of certain anti-cyclotomic-like $\mathbb {Z}_p$-extensions are always even. We also show that the $p$-parts of the ideal class groups of certain anti-cyclotomic-like $\mathbb {Z}_p$-extensions of CM-fields are always non-cyclic.
