Euler Systems of K 2 of Cm Elliptic Curves

We construct certain systems of elements in K2 of some CM elliptic curves. When the classnumber of the field of CM is 1, The image of this system under the regulator map forms an euler system in the sense of Rubin [Ru]. §1. Systems of K2 of CM elliptic curves. Let K be an imaginary quadratic field. Take a Hecke character φ of K of type (1,0) and let fφ be its conductor. Let H = K(fφ) be the ray class field of K modulo fφ. Then by ([dS], lemma 1.4, Ch. 2) there is an elliptic curve E defined over H with complex multiplication by the ring of integers OK of K such that the Hecke character ψ associated to E is of the form ψ = φ ◦NH/K . We assume that E(C) is isomorphic to C/OK . Let F be the conductor of E and write f = NH/K(F). For an integral ideal m of OK E[m] denotes the group of m-torsion points of E(H). For any ideal n of OK we write H(n) for H(E[n]). Let p > 3 be a rational prime which splits in K/Q, say p = pp̄. We assume that p is relatively prime to F. Let a be an integer relatively prime to p. Take a function ga ∈ H(E) such that divga = a (0)− E[a]. Let L be the set of principal prime ideals of OK relatively prime to fp̄a and which split completely in H. Let R be the set of ideals of OK which are divisible only by primes in L. For each m ∈ R we fix a generator xm of E[m] as an OK module so that they satisfy the relation [φ(n)]xmn = xm for any n ∈ R. Here [φ(n)] is φ(n) regarded as an element of End(E). Fix a generator xf of E[f] as an OK module. Since any ideal m ∈ R is prime to f, [φ(m)] is an automorphism of E[f]. For each m ∈ R let ym := xm + [φ(m)] xf ∈ E(H) and take a function sm ∈ H(mf)(E)(the function field of E ⊗H H(mf)) such that