Estimating class numbers over metabelian extensions
Tom 180 / 2017
Acta Arithmetica 180 (2017), 347-364
MSC: Primary 11R29; Secondary 11R23, 11R20.
DOI: 10.4064/aa170216-27-4
Opublikowany online: 28 September 2017
Streszczenie
Let 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.