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Let K=Q(√(-q)), where q is any prime number congruent to 7 modulo 8, and let O be the ring of integers of K. The prime 2 splits in K, say 2O=pp∗, and there is a unique Z_2-extension K_∞ of K which is unramified outside p. Let H be the Hilbert class field of K, and write H_∞=HK_∞. Let M(H_∞) be the maximal abelian 2-extension of H_∞ which is unramified outside the primes above p, and put X(H_∞)=Gal(M(H_∞)/H_∞). We prove that X(H_∞) is always a finitely generated Z_2-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak p-adic Leopoldt conjecture for the compositum J_∞ of K_∞ with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell–Weil group E(J_∞) modulo torsion of certain elliptic curves E with complex multiplication by O.

The paper uses Iwasawa theory at the prime 𝑝=2 to prove non-vanishing theorems for the value at 𝑠=1 of the complex 𝐿-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field 𝐾 = ℚ(√(-q)), where 𝑞 is any prime ≡ 7 mod 8. Our results establish some broad generalizations of the non-vanishing theorem first proven by Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell–Weil group and the Tate–Shafarevich group of the corresponding elliptic curves over the Hilbert class field of 𝐾. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to 𝐾 of the particular elliptic curve with complex multiplication introduced by Gross.