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)

Lecture 23: (11/2/09)

Lecture 24: (11/4/09)

References:

Lang, Cyclotomic Fields.

Washington, Introduction to Cyclotomic fields.

Iwasawa, Lectures on p-adic L-functions.

Coates and Sujatha, Cyclotomic fields and zeta values.