I am just reading about Iwasawa theory about Coates and Sujatha's book on Iwasawa Theory. I was wondering that since Iwasawa thought about the whole theory from the analogy of curves over finite fields, so what should be the analog of the module $U_\infty$/$C_\infty$ in the curve case (if there is any) where $U_\infty$ is the inverse system of local units and $C_\infty$ is the cyclotomic units.