Uniform bounds for the order of a rational torsion point on CM elliptic curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:31:59Z http://mathoverflow.net/feeds/question/97851 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97851/uniform-bounds-for-the-order-of-a-rational-torsion-point-on-cm-elliptic-curves Uniform bounds for the order of a rational torsion point on CM elliptic curves Adam Harris 2012-05-24T16:44:01Z 2012-08-24T18:58:45Z <p>Let \$K\$ be an imaginary quadratic field and \$E\$ an elliptic curve with CM by the maximal order of \$K\$, such that \$E\$ is defined over the Hilbert class field \$H\$. Is it known whether there is a bound (independent of the degree of \$H\$) on the order of a \$H\$-rational torsion point on \$E\$?</p> http://mathoverflow.net/questions/97851/uniform-bounds-for-the-order-of-a-rational-torsion-point-on-cm-elliptic-curves/105411#105411 Answer by stankewicz for Uniform bounds for the order of a rational torsion point on CM elliptic curves stankewicz 2012-08-24T18:58:45Z 2012-08-24T18:58:45Z <p>Hey, I probably should have answered this one some time ago. It was proved in 1989 by J.L. Parish that the order of an \$H\$-rational torsion point is 1,2,3,4 or 6, and this also can be deduced from work of either Silverberg or Prasad-Yogananda. In any case the statement you want is at the beginning of section VI in the paper below and the proof is done in section V.</p> <p><a href="http://www.sciencedirect.com/science/article/pii/0022314X89900127" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0022314X89900127</a></p>