IWASAWA THEORY (TCC COURSE)
This is the webpage for my TCC course on Iwasawa theory.
Main reference:
Cyclotomic Fields and Zeta Values (J. Coates and R. Sujatha).
Auxiliary references:
Introduction to Cyclotomic Fields (L. C. Washington).
Iwasawa theory (lecture notes of R. Sharifi, available at https://www.math.ucla.edu/~sharifi/iwasawa.pdf).
Lectures: Tuesday mornings 10.0012.00, from 12 Oct 2021 to 30th Nov 2021.
Assessment of the course will be based on 3 problem sheets, to be distributed after lectures 2, 4 and 7.
Sheet 1 (review of algebraic number theory + structure theory of Zp[[T]] modules). Here.
Sheet 2 (lectures 3 and 4).
Sheet 3 (lectures 5, 6 and 7). Here. Deadline: December 24th.
Syllabus.

The Iwasawa main conjecture: an overview.

Review of algebraic number theory and padic fields.

Structure theory of Zp[[T]]modules.

Iwasawa's control theory on Zpextensions.

padic measures and the padic zeta function.

Coleman power series.

The Iwasawa theorem.

How to prove the Iwasawa main conjecture: a quick introduction to Euler systems and to the MazurWiles method.
Any comment regarding the organization of the course/feedback is more than welcomed!
I also appreciate if you report to me any typo or mistake you find in the slides and notes of the course.
Schedule
Lecture 1 (October 12th).
Overview of Iwasawa theory and presentation of the course. Slides here.
Brief review of padic fields. Slides here.
Lecture 2 (October 19th). Structure theory of Zp[[T]]modules. We will cover essentially Section 13.2 of Washington's book.
Lecture 3 (October 26th). Iwasawa's control theory on Zpextensions I.
You can see the notes here.
And here a summary without proofs.
For those of you not very familiar with CFT, this is a link to a rather nice review of the main statements: https://math.mit.edu/classes/18.785/2017fa/LectureNotes27.pdf
Lecture 4 (November 2nd). Iwasawa's control theory on Zpextensions II + the big picture (Chapter 1 CoatesSujatha).
You can see the notes for the first part here (continuation of Lecture 3).
Here the notes for the second part (Chapter 1 of CoatesSujatha).
Lecture 5 (November 9th). Coleman power series (Chapter 2 CoatesSujatha).
You can see the notes here. Take a look if possible at the proof of the last theorem (that we can't do time for time constraints).
Lecture 6 (November 16th). padic measures and the padic zeta function (Chapter 3 CoatesSujatha).
You can see the notes here.
Lecture 7 (November 23th). The Iwasawa theorem (Chapter 4 CoatesSujatha).
You can see the notes here.
Lecture 8 (November 30th). How to prove the Iwasawa main conjecture? discussion of some of the ideas involved in the two main approaches (Euler systems and Rubin/MazurWiles methods).
You can see the notes here.