# Elliptic curves and algebraic stacks

I am a student almost without background on algebraic geometry (but I do know basic graduate algebra and topology). Now I am trying to understand something about algebraic stacks.

This article tries to define algebraic stacks by introducing the notion of the moduli space of elliptic curves. However, I do not know anything about elliptic curves and there are just too many possible books.

Can someone give some suggestions about how to proceed to learn enough elliptic curves so as to understand something about algebraic stacks? I also welcome other suggestions. Thank you.

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Learn algebraic geometry first... –  zeb Jul 20 '10 at 16:12
The answers of zeb and Ryan Reich may seem a little harsh, but I think they are also realistic -- it is unlikely that you will be able to acquire a good working knowledge of algebraic stacks without a rather strong background in algebraic geometry. On the other hand, is it possible that you don't actually need a good working knowledge but just have some specific question or issue which is phrased in the language of stacks and you need to be "decoded"? If so, try asking that here. –  Pete L. Clark Jul 20 '10 at 16:27
To paraphrase Pete, it might help focus the answers if you told us your motivation for trying to learning stacks. The subject arose out of the needs of algebraic geometers, and that's the traditional path to it, but in principle there could be others. Perhaps for what you want, it might be sufficient to look up the definition of an orbifold, which arose independently in geometry/topology. On the other hand, elliptic curve theory is beautiful subject, and learning it would be time well spent. However, it's going to be a long road to stacks, be patient. –  Donu Arapura Jul 20 '10 at 21:24

You should really learn algebraic geometry first. After you're done with that, try reading Mumford's paper "Picard Groups of Moduli Problems", which is a fascinating window into the mind of one of the people who later invented algebraic stacks as he was himself figuring out what you are also trying to figure out. From that, you should have a sketch of how the properties of elliptic curves contribute to the properties of their moduli stack; make sure you work out or look up the various things he says about them. Then you can read a regular treatise on stacks.

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Telling someone "you should learn algebraic geometry" is like telling them "you should learn math." Obviously you don't mean all of it, but it's not clear which parts are necessary/relevant, or where the stopping point should be. –  Charles Staats Jul 20 '10 at 22:06
That's true. It's not clear which parts are necessary, but I think in fact that just about everything is relevant since so much of the theory of schemes has word-for-word generalizations to stacks. However, simply to understand the definition of an algebraic stack, you need at least to really appreciate the concept of base change (i.e. fibered product) and know what a smooth morphism is. To understand what is going on, you also have to love the functor of points. This is experiential knowledge; you can't research it, you can only assimilate it. Hence, "learn algebraic geometry first". –  Ryan Reich Jul 20 '10 at 22:20

It is rather simple to study elliptic curve themselves. But since your objective is stacks, you are going to have to be much more abstract than is usual. If you want to avoid algebraic geometry as much as possible, you could take the analytic approach. For this, you would need:

1. A sound footing in Complex Analysis(which I assume you have)

2. And you would need to know some very basic things about compact Riemann surfaces. These aren't that hard, provided you know complex variables. Compact Riemann surfaces are like manifolds; just that the local model is that of an open set in a complex plane and the transition maps are analytic.

3. Next is the fact to keep in your mind that "smooth algebraic curves over the complex numbers" and "Compact Riemann surfaces" are one and the same thing. Here you are to think of algebraic curves as curves specified by algebraic equations, and you are to define smoothness by the Jacobian criterion.

4. Elliptic curves can then be studied using the analytic approach. They are nothing but the complex plane modulo some lattice, ie 1 dimensional complex tori, which is also the same thing as genus $1$ algebraic curves with one marked point.

5. Then, there is the notion of an analytic space. The local model here is that approximately of the zero set of some finite number of analytic functions, and you glue such up using analytic transition maps.

6. An elliptic curve $E$ over an analytic space $S$ can be thought of as a morphism $E \to S$ with a section $0$, with some properties being satisfied, in order that each fiber would then be an elliptic curve.

And I suppose you could then read the AMS notices article you cite.

But I strongly suggest that you do not follow the above approach, as you would miss out on all the interesting algebraic geometry. I suggest that you first learn it well(boosted by some background of compact Riemann surfaces). This is not easy and is best done through a course given by some teacher. And then you can read stacks with the illuminating examples which would all appear trivial to you once you know the requisite background.

For elliptic curves themselves, you could try Knapp's book for an introduction.

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Since nobody has mentioned it yet, I would suggest Frances Kirwan's book "Complex Algebraic Curves" for an introduction to algebraic curves and Riemann Surfaces. –  Adrián Barquero Jul 21 '10 at 4:15

I think it is possible to go quite a few steps towards an understanding of stacks, and even that of elliptic curves, without mastering lots of Algebraic Geometry:

There is a wonderful, friendly written, undergraduate readable book which starts out introducing moduli spaces (and later gives a rough idea abut stacks as well): It is Kock/Vainsencher: An Invitation to Quantum Cohomology

Chapter 1-2, maybe 3, can give you a feeling of how you can work with moduli objects, and are a pleasant reading experience. You will get to know moduli stacks/spaces of curves, close enough to elliptic curves for a start, maybe (those latter are trickier to compactify, but first first you have to get far enough to want to to do that, and this book is wonderful for that purpose).

A very readable and short introductory source for Algebraic Stacks is then Gomez: Algebraic Stacks

You say you have no background in Algebraic Geometry - maybe you should know that the concept of stack is not limited to the world of Algebraic Geometry. If you are more comfortable with Topology or Differential Geometry, maybe this helps: Heinloth: Some notes on Differentiable Stacks

If you are categorically minded and can read french, there is a very nice master course on Stacks by Bertrand Toen. It is very abstract - you can fill in your favourite geometric context - and will leave a considerable gap to the study of the concrete moduli stack of elliptic curves, but it explains very well the other, non-moduli, motivations for introducing stacks and algebraic spaces, e.g. "bad quotients". The good thing about is that you won't feel a lack of algebro-geometric background, even "scheme" is defined - but you need categorical background.

Finally, again thinking of your non-Algebraic-Geometry background, the moduli stack of elliptic curves is used in Algebraic Topology (maybe this is your motivation to study it?) and there are some introductions (quite high-level though) aimed at people with an according background. You can look around here

Have fun exploring this nice topic!!

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The article of Richard Hain "Lectures on moduli spaces of elliptic curves", arXiv:0812.1803, is a very good introduction to the subject, which contains a lot of motivations. Here you can learn many things about orbifolds, and there is also an appendix on stacks.

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I want to second Hain's lecture notes. They're very gentle and concrete. –  Andy Putman Jul 20 '10 at 19:56

I basically agree with Anweshi on this one.In fact,one of the traditional routes for breaking slowly into algebraic geometry is to learn elliptic curves first as one the critical classes of algebraic curves. From there, you would learn thier generalization as varieties,then schemes,then stacks.

The definitive beginning text on elliptic curves is Joseph Silverman's The Arithmetic of Elliptic Curves-it only requires basic graduate algebra and complex analysis. After that,you could try Fulton's classic on algebraic curves-still probably the single best book on classical algebraic geometry.

My best recommendation for you,however,is Rick Miranda's Algebraic Curves And Riemann Surfaces,available through the AMS. Miranda's book is an attempt to introduce modern algebraic geometry to graduate students by beginning with a complete development of complex algebraic curves and then using them as concrete examples of modern algebraic geometric structures like varieties and schemes. The last part of the book develops all the machinery of AG in this context, all the way up to sheaves and deformation theory.

If you want a bridge from basic algebra and analysis into modern algebraic geometry,I can't think of a better source.

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While I do like the three books you mentioned, your description of them is bizarre (have you actually tried to read any of them?). Silverman is a much more advanced book than Fulton... –  Andy Putman Jul 20 '10 at 19:34
Andrew might be referring to earlier editions of Fulton. As far as I can tell, the earlier editions incorporated a lot more material than the current one. –  Qiaochu Yuan Jul 20 '10 at 21:29
A comment on Miranda: it's very gentle, and consequently not very concise. –  Charles Staats Jul 20 '10 at 22:09
@Qiaochu : Are you sure? Looking at the prefaces to the 1989 and 2008 versions (available on Fulton's webpage here : math.lsa.umich.edu/~wfulton/CurveBook.pdf ), it appears that he only performed minor revisions. @Charles : Yes, I agree. It's a very nice place to learn standard material about compact Riemann surfaces (Riemann-Roch, Abel's thm, etc), but the second half is verbose enough that I haven't read much of it. –  Andy Putman Jul 20 '10 at 22:35