Shimura Curves [Lecture notes]

✍ Scribed by Pete L. Clark

• Year: 2010
• Tongue: English
• Category: Library

✦ Synopsis

Shimura Curves

Notes from my 2005 ISM course: (106 pages total)
These are all the notes I typed up for my 2005 ISM course on Shimura varieties. In the lectures, I presented more material on Hilbert and Siegel modular varieties, adelic double coset constructions, and strong approximation than has survived in the lecture notes. Most of the omitted material is of a rather standard sort -- it appears in many places -- which is not to say that it shouldn't appear here as well. The reader will notice that the notes are significantly more polished at the beginning and the end than in the middle. I am quite pleased with the very last lecture, which seems to put some of the pieces of the theory together in a new way. I would like to see more detail on arithmetic groups and lots more detail on quaternion orders and trace formulas. Inevitably for notes of this length, the most important results -- like the existence of rational and integral canonical models -- get stated and kicked around a bit but not proved. To remedy this will require significantly more work.
Lecture 0: Modular curves. (pdf) (6 pages)
Lecture 1: Endomorphisms of elliptic curves. (pdf) (13 pages)
Lecture 2: Fuchsian groups. (pdf) (18 pages)
Lecture 3: More Fuchsian groups. (pdf) (6 pages)
Lecture 4: Arithmetic Fuchsian groups. (pdf) (6 pages)
Lecture 4.5: A Crash Course on Linear Algebraic Groups. (pdf) (7 pages)
Lecture 5: The Adelic Perspective. (pdf) (4 pages)
Lecture 6: Special points and canonical models. (pdf) (10 pages)
Lecture 7: Real points. (pdf) (6 pages)
Lecture 8: Quaternion orders. (pdf) (10 pages)
Lecture 9: Quaternionic moduli. (pdf) (4 pages)
Lecture 10: Integral structures, genera and class numbers. (pdf) (16 pages)
(I do not have any good explanation for the bizarre numbering. In actuality there were many more than 12 lectures, and there was nothing exceptional about the lecture I gave on linear algebraic groups, except that when I defined unipotent groups one of the attendees had the guts and honesty to ask, "What is the point of all this?" The point is that you need to know about a whole lot of different things to understand the definition of a Shimura variety!)