  Let p be a prime number, and let k be an imaginary quadratic field in which p
decomposes into two primes \mathfrak{p} and \bar{\mathfrak{p}}. Let k_\infty be
the unique Z_p-extension of k which is unramified outside of \mathfrak{p}, and
let K_\infty be a finite extension of k_\infty, abelian over k. Let
U_\infty/C_\infty be the projective limit of principal semi-local units modulo
elliptic units. We prove that the various modules of invariants and
coinvariants of U_\infty/C_\infty are finite.
