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.

We study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

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where q is any prime number congruent to 7 modulo 8, with ring of integers O and Hilbert class field H. Suppose p [H : K] is a prime number which splits in K, say pO = pp * . Let H∞ = HK∞, where K∞ is the unique Zp-extension of K unramified outside p. Write M (H∞) for the maximal abelian p-extension of H∞ unramified outside the primes above p, and set X(H∞) = Gal(M (H∞)/H∞). In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module X(H∞). As a consequence, we have that if X(H∞) = 0, the relevant L-values are p-adic units. In addition, the main conjecture for X(H∞) has implications towards (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak p-adic Leopoldt conjecture.

We study elliptic curves of the form x 3 + y 3 = 2p and x 3 + y 3 = 2p 2 where p is any odd prime satisfying p ≡ 2 mod 9 or p ≡ 5 mod 9. We first show that the 3-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 2-Selmer group to the 2-rank of the ideal class group of Q( 3√ p) to obtain some examples of elliptic curves with rank one and non-trivial 2-part of the Tate-Shafarevich group.