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.
It depends very much on what areas of harmonic analysis you're interested in, of course. Grafakos' books are excellent and really quite advanced, and if you wish to continue in that style of harmonic analysis, then there's not much else you can do other than start reading many of the articles that he cites. On the other hand, there are interesting areas in harmonic analysis not covered by Grafakos. I'd recommend a couple of textbooks by Stein: Singular Integrals and Differentiability Properties of Functions and Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. There are probably some other interesting textbooks on singular integral operators that might be useful (though I can't think of any off the top of my head). One other interesting (and very modern) area is wavelets: Mayer's book Wavelets and Operators is probably the place to start there.
Other useful resources are lecture notes or survey articles about harmonic analysis available online. For example, Pascal Auscher taught a course at ANU on harmonic analysis using real-variable methods last year, and one of the students in the class typed up notes, which are available here. Similarly, Terry Tao taught a course a few years ago, and he has lecture notes here and here. Finally, if you want to learn about harmonic analysis with an operator-theoretic bent, there are useful lecture notes here and here.
Best Answer
Two good books for an introduction to global algebraic number theory (i.e., number fields) are:
Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger (Dover Books on Mathematics) May 19, 2008 by Pierre Samuel (less than $8 in paperback)
A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland & Michael Rosen, Springer