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So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.

I was reading Silverman's "Arithmetic of Elliptic Curves" and it seems that most of its treatment is over fields.

My question is, does it make sense to define an elliptic curve over a ring (eg: a noncommutative ring)? If not, why not (where would the "construction" fail)? Is it simply not an object of much interest?

Edit: Apparently the question of elliptic curves over noncommutative rings is considered to some extent in this.

http://user.math.uzh.ch/fontein/diplom-fontein.pdf

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A commutative ring, yes. This is treated to some extent in Silverman's second book; for the more general story of "abelian schemes," which is what you're really after, I might look at Milne's articles in the volume Arithmetic Geometry edited by Cornell and Silverman.

As for noncommutative rings, I'm afraid I have no idea -- I'm not even sure what "construction" would be in a position to fail!

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  • $\begingroup$ So, I googled "noncommutative torus" "abelian variety" and sure enough there were results, including a paper by Manin. $\endgroup$
    – Rob Harron
    Commented Apr 28, 2012 at 4:32
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    $\begingroup$ @Rob: "noncommutative torus," as I understand it, doesn't refer to an elliptic curve over a noncommutative ring, but to a noncommutative deformation of the ring of functions on a torus. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 28, 2012 at 16:48
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    $\begingroup$ I agree with Qiaochu here. As I recall, Manin's noncommutative tori are meant (e.g.) to stand in relation to real quadratic fields as CM elliptic curves do to imaginary quadratic fields. $\endgroup$
    – JSE
    Commented Apr 29, 2012 at 4:28
  • $\begingroup$ I guess the more fundamental question would be is it even possible to define curves over noncommutative rings? $\endgroup$
    – Eugene
    Commented May 12, 2012 at 0:35
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Elliptic curves can be defined over arbitrary base schemes $S$. In particular, for every (commutative!) ring $R$ one can talk about elliptic curves over (the spectrum of) $R$. Loosely speaking, what one gets is a family $E$ of elliptic curves parametrized by the points of $S$. One then proves the existence of the group law ($E$ can be given the structure of an $S$-group scheme), and goes from there. E.g., locally over $S$, $E$ can be put into Weierstrass form.

In the book Arithmetic Moduli of Elliptic Curves by Katz and Mazur, an elliptic curve over $S$ is defined as a proper smooth morphism $f : E \rightarrow S$ of finite presentation, with a section $0 : S \rightarrow E$, such that all geometric fibers of $f$ are integral (equivalently, connected) curves of genus one.

What can be done for noncommutative $R$ I don't know. It seems to me that you have to say what you mean by an elliptic curve over a noncommutative ring. One can't simply replace 'field' in 'elliptic curve over a field' by the name of some other algebraic structure and expect it to make sense, I guess.

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  • $\begingroup$ What you are referring to in Deligne–Rapoport are generalized elliptic curves, not elliptic curves. $\endgroup$
    – Rob Harron
    Commented Apr 28, 2012 at 14:20
  • $\begingroup$ Yes you're right. My mistake, I'll make an edit. $\endgroup$
    – R.P.
    Commented Apr 28, 2012 at 14:26
  • $\begingroup$ why do you require it to be of finite presentation? If you are proper you are qcqs and if you are smooth you are locally of finite presentation. I guess it might be so that the definition sounds similar for nodal curves. $\endgroup$
    – user158636
    Commented Jul 10, 2020 at 15:15
  • $\begingroup$ I did not require anything, I was simply stating the Katz-Mazur definition. It's been a while now since I forced myself to think of any scheme-theoretic questions whatsoever, so I am afraid I'm unable to comment on your question any further... $\endgroup$
    – R.P.
    Commented Jul 10, 2020 at 16:37

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