On capitulations and pseudo-null submodules in certain ${\Bbb Z}_p^d$-extensions

TL;DR

This paper explores the relationship between ideal capitulations and pseudo-null submodules in ${ m Z}_p^d$-extensions, extending Ozaki's results from ${ m Z}_p$-extensions to higher-dimensional cases.

Contribution

It generalizes Ozaki's findings to ${ m Z}_p^d$-extensions, providing new insights into the structure of Iwasawa modules and ideal capitulations in these extensions.

Findings

Established a connection between capitulations and pseudo-null submodules in ${ m Z}_p^d$-extensions.

Extended Ozaki's results from ${ m Z}_p$-extensions to higher-dimensional ${ m Z}_p^d$-extensions.

Provided a framework for analyzing Iwasawa modules in multi-dimensional $p$-adic Lie extensions.

Abstract

Let $p$ be a prime number. By a result of Ozaki, the capitulations of ideals in ${\Bbb Z}_p$-extensions and the finite submodules of Iwasawa modules are closely related. In this article, we discuss this relationship in ${\Bbb Z}_p^d$-extensions.