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 $E$?
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.