First of all, Kevin is being quite modest in his comment above: his paper
Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.
contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in the field: 22 citations on MathSciNet. The most recent is a paper of mine:
Clark, Pete L. On the Hasse principle for Shimura curves. Israel J. Math. 171 (2009), 349--365.
http://alpha.math.uga.edu/~pete/plclarkarxiv7.pdf
Section 3 of this paper spends 2-3 pages summarizing results on the structure of the canonical integral model of a Shimura curve over $\mathbb{Q}$ (with applications to the existence of local points). From the introduction to this paper:
"This result [something about local points] follows readily enough from a description of their [certain Shimura curves over Q] integral canonical models. Unfortunately I know of no unique, complete reference for this material. I have myself written first (my 2003 Harvard thesis) and second (notes from a 2005 ISM course in Montreal) approximations of such a work, and in so doing I have come to respect the difficulty of this expository problem."
I wrote that about three years ago, and I still feel that way today. Here are the documents:
- http://alpha.math.uga.edu/~pete/thesis.pdf
is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.
- For my (incomplete) lecture notes from 2005, go to
http://alpha.math.uga.edu/~pete/expositions2012.html
and scroll down to "Shimura Curves". There are 12 files there, totalling 106 pages [perhaps I should also compile them into a single file]. On the other hand, the title of the course was Shimura Varieties, and although I don't so much as attempt to give the definition of a general Shimura variety, some of the discussion includes other PEL-type Shimura varieties like Hilbert and Siegel moduli space. These notes do not entirely supercede my thesis: each contains some material that the other omits.
When I applied for an NSF grant 3 years ago, I mentioned that if I got the grant, as part of my larger impact I would write a book on Shimura curves. Three years later I have written up some new material (as yet unreleased) but am wishing that I had not said that so directly: I would need at least a full semester off to make real progress (partly, of course, to better understand much of the material).
Let me explain the scope of the problem as follows: there does not even exist a single, reasonably comprehensive reference on the arithmetic geometry of the classical modular curves (i.e., $X_0(N)$ and such). This would-be bible of modular curves ought to contain most of the material from Shimura's book (260 pages) and the book of Katz and Mazur Arithmetic Moduli of Elliptic Curves (514 pages). These two books don't mess around and have little overlap, so you get a lower bound of, say, 700 pages that way.
Conversely, I claim that there is some reasonable topology on the arithmetic geometry of modular curves whose compactification is the theory of Shimura curves. The reason is that in many cases there are several ways to establish a result about modular curves, and "the right one" generalizes to Shimura curves with little trouble. (For example, to define the rational canonical model for classical modular curves, one could use the theory of Fourier expansions at the cusps -- which won't generalize -- or the theory of moduli spaces -- which generalizes immediately. Better yet is to use Shimura's theory of special points, which nowadays you need to know anyway to study Heegner point constructions.) Most of the remainder concerns quaternion arithmetic, which, while technical, is nowadays well understood and worked out.
I'm not a big fan of full roadmaps and reading lists. Exploring mathematics is something that can be totally different depending on where and who you are. Any serious roadmap needs to be flexible and take account of the course: reading maths is a skill (one you seem well on your way to learning btw! but still...), initially you may find actual teaching easier to grasp- and your reading should work along side that. So here's my attempt at a flexible roadmap:
1) Buy some very carefully chosen books and read them cover to cover: There's a lot of baffling books out there- even some that look really UG friendly can have you weeping by page 5 in your first year- and you only want 4-5 to start with (any more will just be too expensive and you won't get round to reading all of them- top up via the library). My recommendations are: 'Naive set theory- Halmos', 'Finite dimensional vector spaces- Halmos', 'Principles of mathematical analysis- Rudin' and 'Proofs from the book- Aigner and Ziegler'.
[These, ostensibly, cover the exact same material as the book you have decided to buy- but to develop quickly I urge you to buy more mathematical texts like these: Halmos' and Rudin's writing styles are very clear but technical (in a way that the book you are interested in will not be), and will make you a better mathematician faster than any book that tries to 'bridge the gap' ever would. I also seconded Owen's call for proofs from the book: it is simultaneously inspiring and useful as a way of seeing 'advanced' topics in action- it's something you'll keep coming back to, right up to your third year!]
2) Do all of the excercises: Or as much as you can bear to- even if it looks like it's beneath you (if you're half decent- a lot of first year will!) you will be surprised as to how much it helps with your mathematical development (and the crucial high mark you'll need for a good PhD placement). This applies to classes and your 4-5 text books.
3) Ask your tutor about doing some modules from the year above: If you've read all of those textbooks and done all of the excercises, you will be ready. Get some advice from your tutor about what would be best and roll with it (most unis won't make you take the exam, so if you don't feel comfortable you're fine). Taking something like metric spaces or group theory in your first year will put you top of the pile.
4) Keep doing all of these things: Immerse yourself in maths- keep on MO, meet likeminded people and no matter how slow the course seems to be moving, no matter the allure of apathy: keep at it. Advice for later books would be pointless now, but there will be people who can give it to you there and then (use maths forums if you want). Oh, and never rule out an area- you never know where intrigue will come from...
Best of luck,
Tom
Best Answer
To understand Perelman's proof of the Poincaré Conjecture, you need a solid background in Riemannian geometry. Many books can be used for an introduction to this field. There are two books I like on this subject: Riemannian Geometry, by Gallot, Hulin and Lafontaine and Riemannian Geometry by Petersen.
After, you can try to learn about Ricci flow, a good starting point is Chow and Knopff's "The Ricci Flow: an Introduction". It covers the basics of Ricci flow including Hamilton's theorem that on a compact 3-manifold with $Ric>0$, the (normalized) flow will converge to constant curvature.
Then, if you want to go into Perelman's work, there is the book "Ricci Flow and the Poincaré Conjecture" by Morgan and Tian. However you also have to understand Thurston's Geommetrization Conjecture, so you need a solid background in 3-manifold topology, I don't know the references for this part, maybe Thurston's lecture notes?
Another interesting road is to study the proof of the differentiable sphere theorem by Brendle and Schoen, a good reference is Brendle's "Ricci Flow and the Sphere Theorem".
I hope that was helpful.