N. Ishii
Published 2004-01-22Version 1
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.
Galois representations attached to elliptic curves with complex multiplication
On elliptic curves with an isogeny of degree 7
On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$
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