Asymptotic cohomology of circular units

Let $F$ be a number field, abelian over the rational field, and fix a odd prime number $p$. Consider the cyclotomic $Z_p$-extension $F_{\mathrm{\infty }}\mathrm{/}F$ and denote $F_n$ the $n^{\mathrm{t}\mathrm{h}}$ finite subfield and $C_n$ its group of circular units. Then the Galois groups $G_{m,n}=\text{Gal}(F_m\mathrm{/}{F}_n)$ act naturally on the $C_m$’s (for any $m\ge n>>0$). We compute the Tate cohomology groups ${\text{Hha}}^i(G_{m,n},C_m)$ for $i=-1,0$ without assuming anything else neither on $F$ nor on $p$.