ANALOGUES OF IWASAWA'S mu=0 CONJECTURE AND THE WEAK LEOPOLDT CONJECTURE FOR A NON-CYCLOTOMIC Z(2)-EXTENSION

摘要

Let K = Q(root-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-infinity of K which is unramified outside p. Let H be the Hilbert class field of K, and write H-infinity = HK infinity. Let M(H-infinity) be the maximal abelian 2-extension of H-infinity which is unramified outside the primes above p, and put X(H-infinity) = Gal(M(H-infinity)/H-infinity). We prove that X(H-infinity) 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(infinity) of K-infinity with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell-Weil group E(J(infinity)) modulo torsion of certain elliptic curves E with complex multiplication by O.