Takuya Yanagisawa | Searcharxiv

arXiv2026

On finiteness properties of the unramified Iwasawa module of a $\mathbb{Z}_p$-extension with restricted ramification

In the present article, we consider $\mathbb{Z}_p$-extensions defined by restrictions on ramifications at $p$-adic primes. We study the finiteness, the existence of non-trivial finite submodules, and the triviality of the unramified Iwasawa modules of these $\mathbb{Z}_p$-extensions. In addition, we give a new sufficient condition for Greenberg's Generalized Conjecture with some non-trivial concrete examples.

math.NT↗

arXiv2025

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

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 $λ$-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 $ν$-invariants, of a certain infinite family of $\mathbb{Z}_p$-extensions over a quartic CM-field.

math.NT↗