On a bound of $p$-ranks of Iwasawa modules of $\mathbb{Z}_p$-extensions over a quartic CM-field

TL;DR

This paper extends results on the boundedness of Iwasawa invariants from imaginary quadratic fields to quartic CM-fields, providing explicit descriptions of invariants in certain infinite $Z_p$-extensions.

Contribution

It establishes new theorems determining Iwasawa invariants, including $ u$-invariants, for infinite $Z_p$-extensions over quartic CM-fields, generalizing previous quadratic cases.

Findings

Determines all Iwasawa invariants for specific $Z_p$-extensions over quartic CM-fields.

Shows boundedness of Iwasawa $lambda$-invariants in these extensions.

Provides explicit formulas for invariants in the studied family.

Abstract

Let $p$ be a prime number. If a number field $k$ has at least one complex place, there are infinitely many $\mathbb{Z}_p$-extensions over $k$, and some authors studied the behavior of Iwasawa invariants of these $\mathbb{Z}_p$-extensions. In particular, Fujii studied the case where $k$ is an imaginary quadratic field and obtained some results on the boundedness of Iwasawa $\lambda$-invariants in a certain infinite family of $\mathbb{Z}_p$-extensions. In the present article, we give analogous theorems in the case where $k$ is a quartic CM-field. One of our main theorems determines all the Iwasawa invariants, including the $\nu$-invariants, of a certain infinite family of $\mathbb{Z}_p$-extensions over a quartic CM-field.