Anticyclotomic Iwasawa Theory of Cm Elliptic Curves

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zp-extension of the CM field, where p is a prime of good, ordinary reduction for E. When the complex L-function of E vanishes to even order, Rubin’s proof the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule. 0. Introduction and statement of results Let K be an imaginary quadratic field of class number one, and let E/Q be an elliptic curve with complex multiplication by the maximal order OK of K. Let ψ denote the K-valued grossencharacter associated to E, and fix a rational prime p > 3 at which E has good, ordinary reduction. Write Q p ⊂ Cp for the maximal unramified extension of Qp, and let R0 denote the completion of its ring of integers. If F/K is any Galois extension, then we write Λ(F ) = Zp[[Gal(F/K)]] for the generalised Iwasawa algebra, and we set Λ(F )R0 = R0[[Gal(F/K)]]. Let C∞ and D∞ be the cyclotomic and anticyclotomic Zp-extensions of K, respectively, and let K∞ = C∞D∞ be the unique Z 2 p-extension of K. As p is a prime of ordinary reduction for E, it follows that p splits into two distinct primes pOK = pp ∗ over K. A construction of Katz gives a canonical measure L ∈ Λ(K∞)R0 , the two-variable p-adic L-function, denoted μp∗(K∞, ψp∗) in the text, which interpolates the value at s = 0 of twists of L(ψ, s) by characters of Gal(K∞/K). It is a theorem of Coates [1] that the Pontryagin dual of the Selmer group Selp∗(E/K∞) ⊂ H(K∞, E[p ]) is a torsion Λ(K∞)-module, and a fundamental theorem of Rubin, the two-variable Iwasawa main conjecture, asserts that the characteristic ideal of this torsion module is generated by L. In many cases this allows one to deduce properties of the p-power Selmer group of E over subfields of K∞. For example, Date: Version of November 24, 2003. 2000 Mathematics Subject Classification. 11G05, 11R23, 11G16. The first author is partially supported by NSF grant DMS-0070449. The second author is supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship. 1 2 ADEBISI AGBOOLA AND BENJAMIN HOWARD if we identify Λ(K∞) ∼= Λ(D∞)[[Gal(C∞/K)]] and choose a topological generator γ ∈ Gal(C∞/K), then we may expand L as a power series in (γ − 1) L = L0 + L1(γ − 1) + L2(γ − 1) 2 + · · · with Li ∈ Λ(D∞)R0 . Standard “control theorems” imply that the characteristic ideal of X(D∞) def = Hom (