Steinitz Class of Mordell-weil Groups of Elliptic Curves with Complex Multiplication
نویسندگان
چکیده
Let E be an elliptic curve having Complex Multiplication by the ring OK of integers of K = Q( √−D), let H = K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an OK-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.
منابع مشابه
Steinitz Class of Mordell Groups of Elliptic Curves With Complex Multiplication
Let E be an elliptic curve having Complex Multiplication by the full ring OK of integers of K = Q( √ −D), let H = K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an OK-module, and its Steinitz class St(E) is studied. When D is a prime number, it is proved that St(E) = 1 if D ≡ 3 (mod 4); and St(E) = [P]t if p ≡ 1 (mod 4), where [P] is the ideal class of K represente...
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