This year Math 620 will run in the "advanced mode" in the two-year cycle. The topic this year will be Shimura varieties and the Langlands correspondence. We will start with the some classical examples of Shimura varieties before explaining the general Shimura varieties. We will also discuss the cohomology of Shimura varieties and the applications to the Langlands correspondence. (As it is not possible to explain both the theory of automorphic representations and the theory of Shimura varieties in this course, we will only give essential ideas to most of the applications.)
The required background knowledge are basic algebraic number theory and algebraic geometry: adeles and ideles, finiteness of class numbers, the definition and basic properties of schemes and morphisms between schemes.
Remarks about the homework assignments:
These assignment will be posted irregularly. Some of them are given as exercises in class, but not all those are collected in the assignments. Some of them are routine, and some are "more interesting"; you should be able to tell which is which. Let me know if you have solved any of these less routine ones. There will be no written final exam for this course. Part of your grade is based on problems you solve, either from the exercies given in class, or from these assignments. You can just tell me your solutions orally, or give me written versions of your solutions. ONLY the less routine ones, please.