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Estimating class numbers over metabelian extensions

Volume 180 / 2017

Antonio Lei Acta Arithmetica 180 (2017), 347-364 MSC: Primary 11R29; Secondary 11R23, 11R20. DOI: 10.4064/aa170216-27-4 Published online: 28 September 2017

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 $d=2$.

Authors

  • Antonio LeiDépartement de mathématiques et de statistique
    Université Laval
    Pavillon Alexandre-Vachon
    1045 avenue de la Médecine
    Québec QC, Canada G1V 0A6
    e-mail

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