An analogue of the Washington-Sinnott theorem for elliptic curves with complex multiplication I

Abstract

In this paper, we generalize Washington's theorem on the stabilization of the p-part of the ideal class groups in the cyclotomic Z q -extension of an abelian number field for distinct primes p and q. We fix an imaginary quadratic field K and a split prime q of K lying above q and let K ∞ / K denote the Z q -extension which is unramified outside q. We show that if F / K is a finite abelian extension, and if F ∞ = F K ∞ , then, for a prime p satisfying certain conditions, the p-part of the class groups stabilize in the Z p -extension F ∞ / F . We also show that if E / K is an elliptic curve with complex multiplication by the ring of integers of K, then under the same conditions on a prime p, the p-part of the Selmer groups for E / F n stabilize as F n runs over the finite layers of F ∞ / F .