## What is a good roadmap for learning Shimura curves?

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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.

Thanks.

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Alas, the answer to Question 1 is definitely no, and I don't know a good answer for Question 2. I just mention that the article Carayol, Henri. Sur la mauvaise réduction des courbes de Shimura. (French) [Bad reduction of Shimura curves] Compositio Math. 59 (1986), no. 2, 151--230. MR0860139 is something of a classic, and a basic reference, but it is also quite difficult. – JS Milne Jan 9 2010 at 6:53

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+1 Milne. I learnt a lot from Carayol, which is really the only place for Shimura curves over totally real fields. Over Q there are other places to look. Wait until Pete Clark wakes up and he'll probably tell you good places to start over Q. – Kevin Buzzard Jan 9 2010 at 8:34

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@Buzzard. I plus-ed your comment for the remark on Pete Clark waking up. – Anweshi Jan 9 2010 at 14:51
@Ben, I retitled your question to reflect more general second question; it includes the first one as well. Feel free to revert! – Ilya Nikokoshev Feb 4 2010 at 19:27

## 2 Answers

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://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:

1) http://math.uga.edu/~pete/thesis.pdf

is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.

2) For my (incomplete) lecture notes from 2005, go to

http://math.uga.edu/~pete/expositions.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.

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I can't add much to this fairly comprehensive answer, so I'm going to leave this as a comment. Miyake's book "Modular forms" also covers some ground related to automorphic forms on Shimura curves. It is mostly upper-half-plane stuff in a manner similar to more elementary references than you're looking for with respect to modular curves. However, it at least has the benefit of covering the Eichler-Selberg trace formula (and, correspondingly, some about CM points) for Shimura curves in some detail. – Tyler Lawson Jan 9 2010 at 12:50

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@TL: That's a useful remark. The trace formula is certainly something that should be in "The Bible of Modular Curves" and is not found in either Shimura or Katz-Mazur. – Pete L. Clark Jan 9 2010 at 13:00
@Pete: This response is really fantastic. Thanks so much! – Ben Linowitz Jan 9 2010 at 13:43

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@Pete Clark. Perhaps for the situation of moduli of elliptic curves the expose of Deligne-Rapoport ought to be more accessible than Katz-Mazur. – Anweshi Jan 9 2010 at 15:02

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[I decided my previous answer was long enough, so I'm adding this one separately and making it Community Wiki. Feel free to add to it!]

Other people's PhD theses that have nice expositions on Shimura curves include:

David Helm (Berkeley 2003)
Bruce Jordan (Harvard 1981)
David Roberts (Harvard 1989)
Victor Rotger Cerda (Universitat de Barcelona 2002)
John Voight (Berkeley 2005)

[Edit (Emerton):] Ken Ribet's Inventiones 100 article describes certain instances of Shimura curves over $\mathbb Q$, including some relations with orders in quaternion algebras, and some information about $p$-adic uniformization and their bad reduction at primes describing the discriminant. Like all of Ribet's papers, it is a masterpiece of exposition.

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