Elliptic curves with rational subgroups of order three

Details Elliptic curves with rational subgroups of order three Elliptic curves with rational subgroups of order three

2005-06-16

arxiv.org/abs/math/0506327v2

Mathematics

Number Theory

Submitted to publication. Corrige version

Scientific paper

In this article we present a characterization of elliptic curves defined over a finite field Fq which possess a rational subgroup of order three. There are two posible cases depending on the rationality of the points in these groups. We show that for finite fields Fq, q= -1 mod 3, all elliptic curves with a point of order 3, they have another rational subgroup whose points are not defined over the finite field. If q = 1 mod 3, this is no true; but there exits a one to one correspondence between curves with points of order 3 and curves with rational subgroups whose points are not rational.

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