Computing the cardinality of CM elliptic curves using torsion points

Details Computing the cardinality of CM elliptic curves using torsion points Computing the cardinality of CM elliptic curves using torsion points

2002-10-11

arxiv.org/abs/math/0210173v2

Mathematics

Number Theory

Revised and shortened version, including more material using discriminants of curves and division polynomials

Scientific paper

Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants a la Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for computing the cardinality of a given elliptic curve modulo p. We apply our results to the Elliptic Curve Primality Proving algorithm (ECPP).

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