Estimating class numbers over metabelian extensions

Antonio Lei¹

¹ Département de mathématiques et de statistique Université Laval Pavillon Alexandre-Vachon 1045 avenue de la Médecine Québec QC, Canada G1V 0A6

Acta Arithmetica, Tome 180 (2017) no. 4, pp. 347-364

Résumé

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 $d=2$.

Détail

DOI : 10.4064/aa170216-27-4

Keywords: odd prime infty infty p adic lie extension whose galois group form mathbb d rtimes mathbb under certain assumptions ramification structure iwasawa module associated infty infty study asymptotic behaviour size p primary part ideal class groups certain finite subextensions inside infty infty generalizes classical result iwasawa cuoco monsky abelian gives precise formula recent result perbet non commutative