Do schemes help us understand elliptic curves?

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.

Solution 1:

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.