Asymptotic cohomology of circular units

Abstract

Let $F$ be a number field, abelian over the rational field, and fix a odd prime number $p$. Consider the cyclotomic $Z_p$-extension $F_{\infty }/F$ and denote $F_n$ the $n^{\mathrm{th}}$ finite subfield and $C_n$ its group of circular units. Then the Galois groups $G_{m,n}=\Gal(F_m/F_n)$ act naturally on the $C_m$'s (for any $m\ge n>>0$). We compute the Tate cohomology groups $\Hha^i(G_{m,n}, C_m)$ for $i=-1,0$ without assuming anything else neither on $F$ nor on $p$.