Abstract:
Consider the cyclotomic field $k=\mathbb Q(\zeta_\ell)$ for an odd regular prime number $\ell$. Let
$k_\infty$ be the cyclotomic $\mathbb Z_\ell$-extension of $k$. We discuss arithmetic of the
$\mathbb Z_\ell$-extension $K_\infty/k_\infty$, where $K_\infty=k_\infty\cdot K$ and $K=k(\sqrt[\ell]{a})$
for some $a\in\mathbb Z$. Here we assume additionally that $a$ is an $\ell$-th power in $\mathbb Q_\ell$,
and there are exactly three places not over $\ell$ that ramify in $K_\infty/k_\infty$.
It follows from the analogue of the Riemann-Hurwitz formula
that it is the simplest extension non-trivial from the viewpoint of Iwasawa theory.
Let $N$ be the maximal Abelian unramified $\ell$-extension of $K_\infty$ such that all places over
$\ell$ split completely in $N/K_\infty$. Let $T_\ell(K_\infty)=G(N/K_\infty)$ be the Iwasawa module
of the $\mathbb Z_\ell$-extension $K_\infty/K$. Then either $T_\ell(K_\infty)\cong\mathbb Z_\ell^{\ell-1}$
as an $\mathbb Z_\ell$-module, or $T_\ell(K_\infty)$ is a finite group having at most $\ell-1$
generators.
We discuss the structure of $T_\ell(K_\infty)$ as a $\Gamma$-module, where $\Gamma=G(K_\infty/K)$, as well as the
analogy with the Riemann conjecture for a curve over the finite field.
