# The ring of algebraic integers of the number field generated by torsion points on an elliptic curve

(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 $\mathbf Q(\zeta_n)$)

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[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!

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Thanks for your enlightening input. If you don't mind, could you paste your response to my question? or make it an entirely new question? so other people can respond to "the correct question"? –  Anonymous Mar 8 '10 at 21:21
I think people will probably see this answer and respond to it accordingly. Let's wait a few hours and see if that's actually the case. –  Pete L. Clark Mar 8 '10 at 21:25
To expand a little on Pete's answer: imagine a 2-torsion point in the curve y^2=f(x). It's of the form (a,0) with a a root of f(x). Now f(x) can be any cubic with distinct roots, so Q(a) can be (for example) any degree 3 field, and in this case a can be any element of it that isn't in Q. –  Kevin Buzzard Mar 8 '10 at 21:26
@Kevin: I think that comment may be worthy of an answer in and of itself, possibly augmented by some remarks about rings of integers in cubic fields (they can already be complicated, right?). –  Pete L. Clark Mar 8 '10 at 21:34
@Kevin: thanks for your answer. Obviously I don't know anything, but maybe it will be better if I adjoin all the n-torsion points to Q and find the ring of integers in that field instead? –  Anonymous Mar 8 '10 at 21:46

Abelian extensions of complex quadratic number fields are generated by division points of certain elliptic functions (which I guess you can translate into the language of torsion points on elliptic curves with complex multiplication - see Pete's answer). Their rings of integers were studied in

• Ph. Cassou-Noguès, M.J. Taylor, Elliptic functions and rings of integers, Progress in Mathematics, Birkhäuser 1987.

The fact that the answer requires a whole book already suggests that things are not as easy as for cyclotomic fields.

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Franz's reference reminded me that there is an entire school (Universite Bordeaux I?) of people who study relations between elliptic curves, rings of integers and Galois module structure. It happens that I have hung out a bit with some of these people, but so far they haven't passed on their deep knowledge of this subject (or even their Francophoneness) to me. Nevertheless I found the following interesting paper of Cassou-Noguès and Taylor which came out soon after their book:

Cassou-Noguès, Ph.(F-BORD); Taylor, M. J.(4-UMIST) A note on elliptic curves and the monogeneity of rings of integers. J. London Math. Soc. (2) 37 (1988), no. 1, 63--72.

I recommend especially the very well written introduction to this paper. It contains the intriguing sentence:

"These results have led us to believe that the rings of integers of all ray class fields of K are monogenic over the ring of integers of the Hilbert class field of K."

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@Pete: thank you. I couldn't access the link since I don't have the barcode password. But thanks anyway. –  Anonymous Mar 9 '10 at 5:26
–  Pete L. Clark Mar 9 '10 at 6:30
Hey Pete you're stealing my journal! ;-) –  Kevin Buzzard Mar 9 '10 at 10:39
Reason why I gave him +1. –  Chandan Singh Dalawat Mar 9 '10 at 13:35
*grin * –  Kevin Buzzard Mar 9 '10 at 15:05

For further work on monogeneity questions, you might want to have a look at some of the papers of Reinhard Schertz (I'm afraid that I don't have precise references right now). He apparently also has a new book out entitled "Complex Multiplication" or something like this, which I've not seen, but which probably also discusses some of his work on this topic.

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