Spring 2012 Iwasawa theory

The goal of this course is to introduce Iwasawa theory from the point of view of the explicit reciprocity law.

(I)  Iwasawa theory for the rational number field Q and cyclotomic units.

Bernoulli numbers, Dirichlet L-values and Kummer congruences.

Gauss sums and Stickelberger-Herbrand's theorem.

Cyclotomic units and Ribet's theorem (I) :  Preliminaries.

Cyclotomic units and Ribet's theorem (II): Euler system argument.

Kubota-Leopodlt p-adic L-functions.

Class number formulae.

Iwasawa theory for Zp-extensions.

Local units and Coates-Wiles homomorphism.

Kolyvagin-Rubin's proof of Iwasawa main conjecture over Q.

Sinnot's proof of the vanishing of mu-invariant.

(II) Iwaswa theory for elliptic curves and control theorems.

Mazur's control theorem.

The exact control theorem for elliptic curves with non-vanishing central L-values

References:

Book: Introduction to cyclotomic fields (GTM 83), by L. Washington.

The article "Iwasawa theory for elliptic curves" by R. Greenberg in Lecture notes of mathematics 1716

Prerequisites:

Standard knowledge on p-adic numbers and Dirichlet L-functions/

Chaper I-VI in "A course in arithmetic" by J.-P. Serre.

Elliptic curves /

Rational points on elliptic curves by J. Silverman and J. Tate.

Basic algebraic number theory /

Algebraic number theory: proceedings edited by Cassels and Frohlich, Chapter I-VII.

At least Chapter 1-4 and Chapter 8 in "Number fields" by D. Marcus.