I am interested in learning about Shimura curves. Unlike most of the people who post reference requests however (see this question for example), my problem is not sorting through an abundance of books but rather dealing with what appears to be an extreme paucity of sources.
Anyway, I'm a graduate student and have spent the last year or so thinking about the arithmetic of orders in quaternion algebras (and more generally in central simple algebras). The study of orders in quaternion algebras seems to play an important role in Shimura curves, and I'd like to study these connections more carefully.
Unfortunately, it has been very difficult for me to find a good place to start. I only really know of two books that explicitly deal with Shimura curves:
- Shimura's Introduction to the arithmetic theory of automorphic functions
- Alsina and Bayer's Quaternion Orders, Quadratic Forms, and Shimura Curves
Neither book has been particularly helpful however; the first only mentions them briefly in the final section, and the second has much more of a computational focus then I'd like.
Question 1: Is there a book along the lines of Silverman's The Arithmetic of Elliptic Curves for Shimura curves?
I kind of doubt that such a book exists. Thus I've tried to read the introductory sections of a few papers & theses, but have run into a problem. There seem to be various ways of thinking about a Shimura curve, and it has been the case that every time I look at an article I'm confronted with a different one. For example, this talk by Voight and this paper by Milne. By analogy, it seems to be a lot like trying to learn class field theory by switching between articles with ideal-theoretic statements and articles taking an adelic slant without having a definitive source which tells you that both are describing the same theorems.
My second question is therefore:
Question 2: Can anyone suggest a 'roadmap' to Shimura curves? Which theses or papers have especially good expository accounts of the basic properties that one needs in order to understand the literature.
Clearly I need to say something about my background. As I mentioned above, I'm an algebraic number theorist with a particular interest in quaternion algebras. I don't have the best algebraic geometry background in the world, but have read Mumford's Red Book, the first few chapters of Hartshorne and Qing Liu's Algebraic Geometry and Arithmetic Curves. I've also read Silverman's book The Arithmetic of Elliptic Curves and Diamond and Shurman's A First Course in Modular Forms.