## Abstract

Let K=Q(-q), where q is any prime number congruent to 7 modulo 8, with ring of integers O and Hilbert class field H. Suppose p does not divide [H:K] is a prime number which splits in K, say pO=pp*. Let H infinity=HK infinity, where K infinity is the unique Zp-extension of K unramified outside p. Write M(H infinity) for the maximal abelian p-extension of H infinity unramified outside the primes ...

## Abstract

Let K=Q(-q), where q is any prime number congruent to 7 modulo 8, with ring of integers O and Hilbert class field H. Suppose p does not divide [H:K] is a prime number which splits in K, say pO=pp*. Let H infinity=HK infinity, where K infinity is the unique Zp-extension of K unramified outside p. Write M(H infinity) for the maximal abelian p-extension of H infinity unramified outside the primes above p, and set X(H infinity)=Gal(M(H infinity)/H infinity). In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module X(H infinity). As a consequence, we have that if X(H infinity)=0, the relevant L-values are p-adic units. In addition, the main conjecture for X(H infinity) has implications towards (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak p-adic Leopoldt conjecture.