Abstract

Let E be an elliptic curve having Complex Multiplication by the ring O_K of integers of K = Q(), let H = K(j(E)) be the Hilbert class field of K. Then the Mordell–Weil group E(H) is an O_K-module. Its Steinitz class St(E) is studied here. In particular, when D is a prime number, St(E) is determined: If D ≡ 3 (mod 4) then St(E) = 1; if D ≡ 1 (mod 4) then St(E) = [P]^t, where P is any prime-ideal factor of 2 in K, [P] the ideal class of K represented by P,  t is a fixed integer. In addition, general structure for modules over Dedekind domain is also discussed. These results develop the results by D. Dummit and W. Miller for D = 10 and specific elliptic curves to more general D and general elliptic curves.

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