On Iwasawa's class number formula for $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions

TL;DR

This paper investigates the behavior of Iwasawa modules related to class numbers in certain non-abelian extensions, constructing examples with nonzero Iwasawa invariants and analyzing related matrix determinants.

Contribution

It introduces estimates for the variation of Iwasawa module quotients and constructs $Z_p times Z_p$-extensions with nonzero $$-invariants, advancing understanding of non-abelian Iwasawa theory.

Findings

Estimated the variation of quotients of Iwasawa modules.

Constructed $Z_p times Z_p$-extensions with nonzero $$-invariant.

Calculated determinants of matrices related to $Z_p times Z_p$ groups.

Abstract

Let $p$ be a prime number. In this paper, we estimate the variation of the sizes of quotients of certain finitely generated $p$-torsion Iwasawa modules, which are closely related to class numbers. We also construct some $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions whose Iwasawa $\mu$-invariant is nonzero. At the end of this paper, we calculate the determinants of some matrices that are related to the groups $\mathbb{Z}_p\rtimes\mathbb{Z}_p$.