|Advanced Number Theory - MA843 - Fall 2013|
|Course Home Page|
Instructor: Jared Weinstein
Lectures: TTh 12:30 pm - 2:00 pm in MCS B31
Office Hours: T 2:30 pm - 4:30 pm and Th 2:30 pm - 3:30 pm in MCS 227
This course is a survey of the theory of Shimura varieties. It is a sequel to my Fall 2011 course on automorphic forms, although that won't be a prerequisite.
In the setting of the Langlands program, Shimura varieties provide a crucial link between automorphic forms and Galois representations. We'll begin by studying the process by which a two-dimensional Galois representation is attached to a cuspidal Hecke eigenform. Then we will work through Deligne's proof of the Ramanujan conjecture for cusp forms via the Weil conjectures. Further topics may include: Abelian varieties, Shimura curves, Deligne's formalism of Shimura varieties for general groups, and the association of Galois representations to higher-rank automorphic forms.
Algebraic number theory (number fields, local fields, Galois theory) and some algebraic geometry.
Weekly assignments account for 100% of your grade in this course. Assignments will appear on this page. These are to be turned in to me during lecture, or in my mailbox, or e-mailed to me as a PDF. Working together is encouraged, but everything you turn in must be in your own words.
Assignment 1, due Sept. 12.
Assignment 2, due Sept. 24.
Assignment 3, due Oct. 17.
Abelian varieties and Jacobians
The Eichler-Shimura relation
Variations of Hodge structures
Variations of Hodge structures, 2
Hermitian symmetric domains
The Baily-Borel theorem
The formalism of Shimura varieties
Two very helpful sources are Milne's notes on elliptic modular forms and Shimura Varieties.
Some files relevent to this course are here.
|MA843 Home Page||Jared Weinstein||Department of Mathematics||Boston University|