On quadratic extensions of number fields and Iwasawa invariants for basic Z3-extensions

Abstract

Let $Z_3$ be the ring of 3-adic integers. For each number field $F$, let $F_{\infty ,3}$ denote the basic $Z_3$-extension over $F;$ let ${\lambda }_3\left(F\right)$ and ${\mu }_3\left(F\right)$ denote respectively the Iwasawa $\lambda \text{-}$ and $\mu$-invariants of $F_{\infty ,3}/F$. Here a number field means a finite extension over the rational field $Q$ contained in the complex field $C;F\subset C,$. Now let $k$ be a number field. Let $L$-denote the infinite set of totally imaginary quadratic extensions in $C$ over $k$(so that $L$-coincides with the set $L^{-}$ in the text when $k$ is totally real); let $L_{+}$ denote the infinite set of quadratic extensions in $C$ over $k$ in which every infinite place of $k$ splits (so that $L_{+}$ coincides with the set $L^{+}$ in the text when $k$ is totally real). After studying the distribution of certain quadratic extensions over $k$, that of certain cubic extensions over $k$, and the relation between the two distributions, this paper proves that, if $k$ is totally real, then a subset of $\{K\in L_{-}|{\lambda }_3\left(K\right)={\lambda }_3\left(k\right),{\mu }_3\left(K\right)={\mu }_3\left(k\right)\}$ has an explicit positive density in $L$-. The paper also proves that a subset of $\{L\in L_{+}|{\lambda }_3\left(L\right)=$${\mu }_3\left(L\right)=0\}$ has an explicit positive density in $L_{+}$ if 3 does not divide the class number of $k$ but is divided by only one prime ideal of $k$. Some consequences of the above results are added in the last part of the paper.