Vol. 74, No. 1, 1978

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Elliptic curves over complex quadratic fields

Bennett Setzer

Vol. 74 (1978), No. 1, 235–250

This paper concerns elliptic curves defined over complex quadratic fields and having good reduction at all primes. Those fields are characterized which support such curves having a 2-division point defined over the field. The number of isomorphism classes, over the ground field, of these curves is also determined. For curves without a 2-divison point defined over the field, the possible Galois groups of the 2-division field over the rationals are determined. Using class field theory, it is shown that certain complex quadratic fields support no elliptic curves with good reduction everywhere.

Mathematical Subject Classification 2000
Primary: 14G25
Secondary: 14K15, 10D05
Received: 22 September 1976
Revised: 22 June 1977
Published: 1 January 1978
Bennett Setzer