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

For a CM-field K and an odd prime number p , let K ~ ′ be a certain multiple Z p -extension of K . In this paper, we study several basic properties of the unramified Iwasawa module X K ~ ′ of K ~ ′ as a Z p 〚 Gal ( K ~ ′ / K ) 〛 -module. Our first main result is a description of the order of a Galois coinvariant of X K ~ ′ 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 λ -invariant and the ideal class group of K , we determine a necessary and sufficient condition such that X K ~ is Z p 〚 Gal ( K ~ / K ) 〛 -cyclic. Here, K ~ is the Z p 2 -extension of K .