Steinitz class of Mordell groups of elliptic curves with complex multiplication

Details Steinitz class of Mordell groups of elliptic curves with complex multiplication Steinitz class of Mordell groups of elliptic curves with complex multiplication

1998-09-03

arxiv.org/abs/math/9809208v1

Mathematics

Number Theory

Scientific paper

Let E be an elliptic curve having Complex Multiplication by the full ring O_K of integers of K=Q(\sqrt{-D}), let H=K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an O_K-module, and its structure denpends on its Steinitz class St(E), which is studied here. In partucular, when D is a prime number, it is proved that St(E)=1 if D\equiv 3 (mod 4); and St(E)=[P]^t if D\equiv 1 (mod 4), where [P] is the ideal class of K represented by prime factor P of 2 in K, t is a fixed integer. General structures are also discussed for St(E) and for modules over Dedekind domain. These results develop the results by D. Dummit and W. Miller for D=10 and some elliptic curves to more general D and general elliptic curves.

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