## 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 10.00-12.00, 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 p-adic fields.

• Structure theory of Zp[[T]]-modules.

• Iwasawa's control theory on Zp-extensions.

• p-adic measures and the p-adic zeta function.

• Coleman power series.

• The Iwasawa theorem.

• How to prove the Iwasawa main conjecture: a quick introduction to Euler systems and to the Mazur--Wiles 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 p-adic 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 Zp-extensions 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 Zp-extensions II + the big picture (Chapter 1 Coates--Sujatha).

You can see the notes for the first part here (continuation of Lecture 3).

Here the notes for the second part (Chapter 1 of Coates--Sujatha).

Lecture 5 (November 9th). Coleman power series (Chapter 2 Coates--Sujatha).

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). p-adic measures and the p-adic zeta function (Chapter 3 Coates--Sujatha).

You can see the notes here.

Lecture 7 (November 23th). The Iwasawa theorem (Chapter 4 Coates--Sujatha).

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/Mazur--Wiles methods).

You can see the notes here.