Trace of Frobenius endomorphism of an elliptic curve with complex multiplication
Abstract
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^2Dv^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 DMISRR025,2002.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2004
 arXiv:
 arXiv:math/0401289
 Bibcode:
 2004math......1289I
 Keywords:

 Mathematics  Number Theory;
 11G05 (Primary);
 14H52 (Secondary)
 EPrint:
 20 pages