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?
 A: 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. 
