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I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what I've been concentrating on recently which has been learning algebraic geometry from a scheme viewpoint (mainly just for my own interest but also for my masters thesis next year, in which I will develop the basic theory and then apply it to a few as-yet undecided problems).

I'd really like to incorporate elliptic curves into this project by looking at their geometric and possibly arithmetic properties using this modern machinery. However with my current very basic knowledge of elliptic curves it seems like the full power (and elegance!) of schemes, categories etc might be unnecessary unless we look at either a) situations such as elliptic curves over general rings; b) moduli spaces of curves; or c) abelian varieties. Whilst all of these are nice topics I worry they're a bit too far away from actually studying elliptic curves!

So my question is - is there any "interesting" geometric/arithmetic information about elliptic curves over fields with some number-theoretic relation which can be studied most effectively using modern algebraic geometry? Or would it be better to study elliptic curves separately at first and find another application with which to better demonstrate the use of the theory that I develop in the first part of the thesis? Thanks very much in advance, and I'd welcome any reading recommendations.

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    $\begingroup$ The reason why we study moduli spaces is that they tell us about the objects involved in the moduli. The modularity theorem is a great example of this. By the way, to study moduli, one must necessarily study deformations - i.e. the objects over a certain type of ring. $\endgroup$ – RghtHndSd Jan 9 '15 at 0:58
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    $\begingroup$ I'm no expert of algebraic geometry by any measure, but isn't the whole business of cohomology firmly rooted in the Grothendieck-Serre picture of algebraic geometry? E.g. elliptic cohomology is supposedly related to some classification problems in the theory elliptic curves. $\endgroup$ – Espen Nielsen Jan 9 '15 at 1:01
  • $\begingroup$ @EspenNielsen you're right, cohomology is a massive part of the modern algebraic geometry viewpoint! I've not learned about cohomology yet but I have 18 months to work on this thesis and cohomology will play a big role. I'll certainly look at elliptic cohomology when I am more informed, and thanks for the tip $\endgroup$ – Alex Saad Jan 9 '15 at 1:16
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    $\begingroup$ @EspenNielsen: Elliptic cohomology is fairly far removed from anything that the OP is asking about. (It is about a generalized cohomology theory --- in the sense that homotopy theorists use that term --- in which the behaviour of Chern classes is related to the group law on an elliptic curve, just as the behaviour of Chern classes in usual cohomology, resp. K-theory, is related to the additive group law, reap. the multiplicative group law.) $\endgroup$ – tracing Jan 9 '15 at 5:28
  • $\begingroup$ @tracing Thank you for clearing that up for me. I had just heard something along those lines before and parroted it. $\endgroup$ – Espen Nielsen Jan 9 '15 at 8:43
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Schemes play an enormous role in all the modern theory of elliptic curves, and have done so ever since Mazur and Tate proved their theorem that no elliptic curve over $\mathbb Q$ can have a 13-torsion point defined over $\mathbb Q$.

For some additional explanation, you could look at this answer. But bear in mind that theorems on the classification of torsion, while fantastic, are just a tiny part of the theory of elliptic curves, and a tiny part of how schemes are involved. One of the most important theorems about elliptic curves is the modularity theorem, proved by Wiles, Taylor, et. al. twenty or so years ago, which implies FLT. These arguments also depend heavily on modern algebraic geometry.

Also, the proof of the Sato--Tate conjecture.

Also, all current progress on the BSD conjecture.

The underlying point is the the theory of elliptic curves is one of the central topics in modern number theory, and the methods of scheme-theoretic alg. geom. are among the central tools of modern number theory. So certainly they are applied to the theory of elliptic curves


On the other hand, you won't find it so easy to synthesize your reading on the arithmetic of elliptic curves and your reading of scheme theory. For example, even Silverman's (first) book, which is quite a bit more advanced than Silverman--Tate, doesn't use schemes. Some of the arguments can be clarified by using schemes, but it takes a bit of sophistication to see how to do this, or even where such clarification is possible or useful.

Hartshorne has a discussion of elliptic curves in Ch. IV, but it doesn't touch on the number theoretic aspects of the theory; indeed, Hartshorne's book doesn't make it at all clear how scheme-theoretic techniques are to be applied in number theory.

With my own students, one exercise I give them to get them to see how make scheme-theoretic arguments and use them to study elliptic curves is the following:

Let $E$ be an elliptic curve over $\mathbb Q$ with good or multiplicative reduction at $p$; then prove that reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb Q}$ to the endomorphisms of the reduction of $E$ mod $p$ over $\overline{\mathbb F}$ is injective.

The proof isn't that difficult, but requires some amount of sophistication to discover, if you haven't seen this sort of thing before.


Finally:

None of the results on torsion on elliptic curves over $\mathbb Q$, or modularity, or Sato--Tate, or BSD, will be accessible to you in the time-frame of your masters (I would guess); any one of them takes an enormous amount of time and effort to learn (a strong Ph.D. student working on elliptic curves might typically learn some aspects of one of them over the entirety of their time as a student). I don't mean to be discouraging --- I just want to say that it will take time, patience, and also a good advisor, if you want to learn how schemes are applied to the theory of elliptic curves, or any other part of modern number theory.

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  • $\begingroup$ Thank you for an extremely comprehensive answer and also for your reference to that older post, which contained links to two papers (an expository article on $X_0 (11)$ and $X_1 (11)$ and another masters thesis) which I think look suitable for me to read when I'm more informed. I will also try your exercise when I have the tools to do so! I appreciate that a lot of this material is too advanced for me at this stage, but part of this thesis is getting ideas and a headstart for my PhD so I don't find it discouraging. $\endgroup$ – Alex Saad Jan 9 '15 at 10:26
  • $\begingroup$ I'm curious about your opinion on the following. Let's suppose that I care only about elliptic curves, and I'm willing to black box what reduction really means from an abstract perspective (i.e. minimal regular models/Neron models are a thing I don't worry about). One may then begin to wonder what, really, is the connection between $T_p E$ and $T_p \mathcal{E}_{\mathbb{F}_p}$ if $E$ is an elliptic curve over $\mathbb{Q}_p$ with good reduction, and $\mathcal{E}_{\mathbb{F}_p}$ it's reduction. How does one answer that without schemes? I mean, the 'correct' answer is that the interpolating $\endgroup$ – Alex Youcis Nov 23 '16 at 15:10
  • $\begingroup$ object is the $p$-divsible group $\varinjlim \mathcal{E}[p^n]$ which, even if you don't want to think about fppf sheaves, is still a 'colimit' (in what if not sheaves?) of non-reduced group schemes over $\mathbb{Z}_p$? I mean, I guess you can maybe get around this by saying that there is a Galois equivariant surjection $T_p E\to T_p\mathcal{E}_{\mathbb{F}_p}$ with kernel the ($\overline{\mathbb{Z}_p}$-points) the connected part of $\mathcal{E}[p^\infty]$, and this is just the 'formal group' (which is also a notion that's kind of confusing without thinking about formal schemes/fppf sheaves). $\endgroup$ – Alex Youcis Nov 23 '16 at 15:13
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    $\begingroup$ How does one formalize this to themselves without using scheme theory? I know I'm preaching to the choir, but doesn't this whole story just get very convoluted/difficult to discuss without schemes? I don't recall how Silverman sidesteps this, but I know he doesn't talk about $p$-divisible groups. Thanks for any insight! If there is a 'non-fancy' way of seeing this, I would love to know. $\endgroup$ – Alex Youcis Nov 23 '16 at 15:14

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