Journal of the
Korean Mathematical Society JKMS

Let $k$ be a real abelian field of conductor $f$ and $k_\infty = \cup_{n\ge0} k_n$ be its $\mathbb Z_p$-extension for an odd prime $p$ such that $p\nmid f \varphi(f)$. The aim of this paper is to compute the cohomology groups of circular units. For $m > n \ge 0$, let $G_{m,n}$ be the Galois group Gal$(k_m/k_n)$ and $C_m$ be the group of circular units of $k_m$. Let $l$ be the number of prime ideals of $k$ above $p$. Then, for $m > n \geq 0$, we have
(1) $C_m^{G_{m,n}} = C_n$,
(2) $\widehat{H}^i (G_{m,n},C_m) \simeq (\mathbb Z /p^{m-n}\mathbb Z)^{l-1}$ if $i$ is even,
(3) $\widehat{H}^i(G_{m,n},C_m) \simeq (\mathbb Z/p^{m-n}\mathbb Z)^l$ if $i$ is odd.

Keywords: $\mathbb Z_p$-extension, circular unit, cohomology group

MSC numbers: Primary 11R18, 11R23; Secondary 11R34