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Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

Published online by Cambridge University Press:  17 April 2009

Noburo Ishii
Noburo Ishii
Affiliation:
Department of Mathematics and Information Science, Osaka Prefecture University, 1–1 Gakuen-cho, Sakai, Osaka, 599–8531, Japan e-mail: ishii@mi.cias.osakafu-u.ac.jp
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Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 125 - 142
Copyright
Copyright © Australian Mathematical Society 2004

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