On the structure of the Galois group of the maximal pro-pp extension with restricted ramification over the cyclotomic Zp\mathbb{Z}_p-extension

Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic field and $S = \emptyset$ (here $p$ is an odd prime number which does not split in $k$). In the latter part, we treat the case when $k$ is a totally real field and $S \neq \emptyset$.Comment: 20 pages, changed several places, added sentences and reference

Some Questions on the Ideal Class Group of Imaginary Abelian Fields

Let k be an imaginary quadratic field. Assume that the class number of k is exactly an odd prime number p, and p splits into two distinct primes in k. Then it is known that a prime ideal lying above p is not principal. In the present paper, we shall consider a question whether a similar result holds when the class number of k is 2p. We also consider an analogous question for the case that k is an imaginary quartic abelian field.</p

A remark on Greenberg's generalized conjecture for imaginary S3S_3-extensions of Q\mathbb{Q}

Let $K/ \mathbb{Q}$ be an imaginary $S_3$-extension, and $p$ a prime number which splits into exactly three primes in $K$. We give a sufficient condition for the validity of Greenberg's generalized conjecture for $K$ and $p$.Comment: 9 page

On the unramified Iwasawa module of a Zp\mathbb{Z}_p-extension generated by division points of a CM elliptic curve

We consider the unramified Iwasawa module $X (F_\infty)$ of a certain $\mathbb{Z}_p$-extension $F_\infty/F_0$ generated by division points of an elliptic curve with complex multiplication. This $\mathbb{Z}_p$-extension has properties similar to those of the cyclotomic $\mathbb{Z}_p$-extension of a real abelian field, however, it is already known that $X (F_\infty)$ can be infinite in general. In this paper, we mainly consider analogs of weak forms of Greenberg's conjecture for $F_\infty/F_0$.Comment: 13 pages. The contents were largely modified. In particular, one of the main results was improved (Theorem 1.4