# History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they evolved, and what the unsolved problems are in this corner of mathematics. I want to see the landscape, so to say.

The history has already been discussed to some extent here: History of elliptic curves

So... what are the big open questions? What are the lesser ones?

• I think you need to be more specific about what you're looking for. Even then the answers are probably going to turn into Wikipedia articles. Anyway, BSD is still open. Get to work!
– Hoot
May 20, 2016 at 0:07

Your question covers a vast amount of material and without specifying further it is hard to say exactly what you're looking for. Also, you did not state the level of material you were looking for. Almost any text on Algebraic Geometry (or more specifically Arithmetic Algebraic Geometry), Elliptic Curves, or Number Theory related to these areas would contain such problems and their history.

To name a few texts at various level:

Elliptic Curves: Milne : A classic introductory text to these areas.

Elliptic Tales : A nice general historical overview with some discussion of open problems.

Rational Points on Elliptic Curves : An undergraduate text relating to these areas. It does talk about some open problems.

Arithmetic of Elliptic Curves : A very dense graduate text introducing elliptic curves.

Elliptic Curves, Modular Forms, and Their L-functions : A more readable, accessible to undergraduates, text spanning elliptic curves and open problems.

Elliptic Curves : Another graduate introductory text to elliptic curves. I found this more accessible than Silverman's book for a first approach.

Elliptic Curves: Number Theory and Cryptography : Another great undergraduate book for elliptic curves. This is probably less difficult than the book of Silverman.

Arithmetic Geometry : A dense introduction to the results/research of arithmetic geometry.

Arithmetic Algebraic Geometry : Yet another dense introduction to the results/research of arithmetic geometry.

Journey through Genius : A well written text discussing several big math problems. Though this will not focus on your areas of interest for much of the book, it is a cheap and good read.

Fermat's Last Theorem : A general introductory text to these areas through Fermat's Last Theorem.

Modular Forms and Fermat's Last Theorem : Another general introductory text to these areas through Fermat's Last Theorem.

Fermat's Last Enigma : A more readable book on the history of FLT.

As for easy access papers:

Elliptic Curves: Milne : Milne's book referenced above.

Elliptic Curves : Lecture notes on elliptic curves.

Introduction to Fermat's Last Theorem : A nice history by Cox on the history of FLT.

Fermat's Last Theorem : A readable introduction to the theories going into FLT.

The ABC Conjecture : A discussion on another big (possibly still open, a proof is being verified) open problem.

From the Taniyama-Shimura Conjecture to Fermat’s Last Theorem

Wiles' Proof of Fermat's Last Theorem : A discussion of the techniques that went into FLT.

36 Unsolved Problems in Number Theory : A list of unsolved number theory problems.

More Unsolved Problems : More unsolved problems.

Introduction to Arithmetic Geometry : A nice introductory text to arithmetic geometry.

Roadmap to Learning Arithmetic Algebraic Geometry : Same as above

I do believe there are several OCW courses on the MIT website whose notes you could look through that would also give a bit more material for you to look through. Hopefully, this is what you were hoping for. Possible areas for you to look into would be the ABC conjecture (you could look here for more on this but these papers are accessible to VERY few people), BSD (Birch and Swinnerton-Dyer Conjecture), you might look into Perfectoid Spaces, ranks of elliptic curves, the Langland's Program, etc etc etc. As I said, the field is very big and vibrant. It depends on if you are looking more towards the elliptic curve end, algebraic number theory side, or algebraic geometry side of things.

• Thanks for a splendid answer. I was looking for an introductory treatment and the links seem very interesting, My level of understanding is advanced undergraduate - fresh graduate. It is hard to know what one does not know, hence the diffuse question.
– R.s
May 20, 2016 at 0:30
• @R.s Not a problem. This is my general area that I am working in for my PhD so I always have links ready. Note that many of the texts can be found for free online with searching. I did not link them here as I am unaware which are available legally and which are not. Since you seem new to the area, perhaps you should try some of the general interests texts first before trying texts which specifically try to introduce you to the actual mathematics of the field - which often require years of study because of the vast amount of mathematics the materials touches upon: algebra, analysis, geo., cetc May 20, 2016 at 0:34
• The book by Lawrence Washington, Elliptic Curves: Number Theory and Cryptography, is a really nice introduction (with proofs), although I'm only half way through it. Not familiar with the other texts, but will look at them. The third one about rational points, or one book by Silvermann (don't recall the name; starts with an introduction to algebraic curves), might be my next reads.
– R.s
May 20, 2016 at 0:41
• @R.s Many of them are good but the problem very quickly becomes that introductory texts take 'short cuts' by presenting material which can be easily presented to undergraduates. When trying to find a slightly more advanced book, one often finds the difference between the previous book and the next very steep. This is because the usual techniques of the field require commutative alg, alg. geometry, and various pieces of analytic and alg number theory. If you can follow it, Husemullers book would be a good next step or "Primes of the form $x^2+ny^2$". It all depends on your level of interest. May 20, 2016 at 0:46
• Found it: "Silverman: The arithmetic of elliptic curves" is what I was thinking about initially, but now I will try to locate the Husemullers text.
– R.s
May 20, 2016 at 0:52