Galois coinvariants of the unramified Iwasawa modules of multiple Zp-extensions

For a CM-field K and an odd prime number p, let (K) over tilde' be a certain multiple Z(p)-extension of K. In this paper, we study several basic properties of the unramified Iwasawa module X-(K) over tilde' of (K) over tilde' as a Z(p)[Gal((K) over tilde'/K)-module. Our first main result is a description of the order of a Galois coinvariant of X-(K) over tilde' in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic Z(p)-extension of K under a certain assumption on the splitting of primes above p. The second result is that if K is an imaginary quadratic field and if p does not split in K, then, under several assumptions on the Iwasawa lambda-invariant and the ideal class group of K, we determine a necessary and sufficient condition such that X-(K) over tilde is Z(p)[Gal((K) over tilde /K)]-cyclic. Here, (K) over tilde is the Z(p)(2)-extension of K.