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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 $E$?

edited May 25 at 13:17

asked May 24 at 16:44

Adam Harris
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CM theory tells you that the maximal abelian extension of $H$ is generated by the coordinates of the torsion points of the curve, and the dictionary is quite explicit. This should surely give you something. Did you try this? If so, what happens? If not, maybe Serre's article in Cassels-Froehlich, or Silverman II, are places to start? – Kevin Buzzard May 24 at 18:15

All I know really at the moment is that if you take $K_1$ and $K_2$ then the degree of $K_1^{ab}\cap K_2^{ab}$ over $\mathbb{Q}^{ab}$ is finite (by this CM theory - Serre I think) but I've no idea how to approach infinitely many. I haven't looked at Serre's article in Cassels so I'll look there thanks. – Adam Harris May 24 at 18:40

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CM theory tells you exactly which Galois actions on the $n$-torsion points are possible so it's easy to check when a vector is invariant. I believe that's what stankewicz's article uses but I don't have access so I'm not sure. – Will Sawin Aug 24 at 20:05

stankewicz · Accepted Answer · 2012-08-24 18:58:45Z UTC

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.

http://www.sciencedirect.com/science/article/pii/0022314X89900127

Uniform bounds for the order of a rational torsion point on CM elliptic curves

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