Fall 2009: Math 678 Introduction to Iwasawa theory


Outline: This course is a topics class in number theory, which will provide an introduction to the subject of Iwasawa theory. The main goal is to formulate the so-called main conjecture in the cyclotomic case and explain the proof using Euler systems. I will be providing some lecture notes below. There is nothing new or original in these notes: the material is quite well known and appears in several sources. Please see below for a list of references.
Lecture 1: (9/9/09) Introduction
Lecture 2: (9/11/09) L-functions of Dirichlet characters
Lecture 3: (9/14/09) Special values of L-functions of Dirichlet characters
Lecture 4: (9/16/09) The analytic class number formula
Lecture 5: (9/18/09) Class numbers and Bernoulli polynomials
Lecture 6: (9/21/09) Distributions and measures
Lecture 7: (9/23/09) The Kummer congruences
Lecture 8: (9/25/09) More congruences
Lecture 9: (9/28/09) Gauss sums and Stickelberger's theorem
Lecture 10: (9/30/09) Herbrand's theorem
Lecture 11: (10/02/09) The index of the Stickelberger ideal
Lecture 12: (10/05/09) Kummer theory and class groups
Lecture 13: (10/07/09) The plus part of the class group
Lecture 14: (10/09/09) Cyclotomic units and their index
Lecture 15: (10/12/09) Introduction to p-adic L-functions: Measures and power series I
Lecture 16: (10/14/09) Measures and power series II
Lecture 17: (10/16/09) Measures and power series III
Lecture 18: (10/21/09) Construction of p-adic L-functions I: p-adic analytic functions
Lecture 19: (10/23/09) Construction of p-adic L-functions II: power series
Lecture 20: (10/26/09) Applications to class numbers I
Lecture 21: (10/28/09) Applications to class numbers II
Lecture 22: (10/30/09) Z_p-extensions I
Lecture 23: (11/2/09) Z_p-extensions II
Lecture 24: (11/4/09) Z_p-extensions III




References:
Lang, Cyclotomic Fields.
Washington, Introduction to Cyclotomic fields.
Iwasawa, Lectures on p-adic L-functions.
Coates and Sujatha, Cyclotomic fields and zeta values.