Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

Details Trace of Frobenius endomorphism of an elliptic curve with complex multiplication Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

2004-01-22

arxiv.org/abs/math/0401289v1

Mathematics

Number Theory

20 pages

Scientific paper

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 v such that 4p=u^2-Dv^2. It is well known that square of the trace a_P of Frobenius endomorphism of the reduction of E modulo P is equal to u^2. We determine whether a_P=u or a_P=-u in the case the class number of R is 2 or 3 and D is divided by 3,4 or 5. This article is a revised version of the authors' preprint DMIS-RR-02-5,2002.

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