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Abstract

Let $p$ be an odd prime and $K_{\infty,\infty}/K$ a $p$-adic Lie extension whose Galois group is of the form $\mathbb Z_p^{d-1}\rtimes \mathbb Z_p$. Under certain assumptions on the ramification of $p$ and the structure of an Iwasawa module associated to $K_{\infty,\infty}$, we study the asymptotic behaviour of the size of the $p$-primary part of the ideal class groups over certain finite subextensions inside $K_{\infty,\infty}/K$. This generalizes the classical result of Iwasawa and Cuoco–Monsky in the abelian case and gives a more precise formula than a recent result of Perbet in the non-commutative case when $d=2$.