(Warning: a student asking)
Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of $\mathbf Q(a,b)$? I'm curious about the answer for general elliptic curves, but I'm not sure whether such an answer is possible.


(This question is motivated by the nice description of the rings of integers of cyclotomic fields $\mathbf Q(\zeta\_n)$) 



The ring of algebraic integers of the number field generated by torsion points on an elliptic curve
[Comment: what follows is not really an answer, but rather a focusing of the question.]


In general, there is not such a nice description even of the number field $\mathbb{Q}(a,b)$ -- typically it will be some non-normal number field whose normal closure has Galois group $\operatorname{GL}\_2(\mathbb{Z}/n\mathbb{Z})$, where $n$ is the order of the torsion point. 


In order to maintain the analogy you mention above, you would do well to consider the special case of an elliptic curve with **complex multiplication**, say by the maximal order of an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-N})$, necessarily of class number one since you want the elliptic curve to be defined over $\mathbb{Q}$. In this case, the 
field $K(P)$ will be -- up to a multiquadratic extension -- the anticyclotomic part of the $n$-ray class field of $K$. 


And now it is a great question exactly what the rings of integers of these very nice number fields are. One might even venture to hope that they will be integrally generated by the x and y coordinates of these torsion points on CM elliptic curves (certainly there are well-known integrality properties for torsion points, although I'm afraid I'm blanking on an exact statement at the moment; I fear there may be some problems at 2...).


I'm looking forward to a real answer to this one!


