Lectures on Shimura Curves 0: Modular Curves

General remarks about these documents: These are lecture notes. Let us reflect separately on the meaning of both words. The first word is meant to indicate some correspondence between what is said in the lectures and what appears in these notes. This correspondence is not precise: on the one hand, in response to a question or a comment, I may say things in class which do not make it into these documents. (If I or someone else says something important that does not appear here, let me know, and I will try to incorporate it into a later draft.) On the other hand in certain places the notes will go into significantly more depth than what is said in class. It is also hoped that gaps in arguments which get pointed out in class will be filled in the notes. The meaning of the second word is that these documents are provisional and works in progress – we hope that the mathematics presented in here is clear, and is complete at least where it claims to be complete. It would be nice if every unproved fact were accompanied by a precise reference, but this may be too much to hope for. An important part of your responsibility in taking the class is to look through the notes and to suggest improvements. Perhaps some day these notes will be turned into a formal written document, but that day is not especially near at hand.