In this paper, for a totally real number field $k$ we show the ideal class group of $k{\left({\cup }_{n>0}{\mu }_n\right)}^{+}$ is trivial. We also study the $p$-component of the ideal class group of the cyclotomic ${𝐙}_p$-extension.

We explore the question of how big the image of a Galois representation attached to a $\Lambda$-adic modular form with no complex multiplication is and show that for a “generic” set of $\Lambda$-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set$${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series $\overline{f(T,{\theta }_{i,j})}$ are linearly independent over a certain...

Let $f$ be a primitive cusp form of weight at least 2, and let ${\rho }_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of ${\rho }_f$ to a decomposition group at $p$ is “upper triangular”. If in addition $f$ has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members of a non-CM...

For the cyclotomic ${\mathbb{Z}}_2$-extension $k_{\infty }$ of an imaginary quadratic field $k$, we consider the Galois group $G\left(k_{\infty }\right)$ of the maximal unramified pro-$2$-extension over $k_{\infty }$. In this paper, we give some families of $k$ for which $G\left(k_{\infty }\right)$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G\left(k_{\infty }\right)$ becomes a nonabelian metacyclic pro-$2$-group. We also calculate Iwasawa theoretically the Galois groups of $2$-class field towers of certain cyclotomic $2$-extensions.