PhD in Mathematics

An introduction to Iwasawa theory

An introduction to Iwasawa theory

Academic year 2024/2025

Some familiarity with inverse limits and with p-adic numbers will be very useful (they will be introduced from zero, but at a very fast pace). Also, some knowledge of functional analysis, while not strictly necessary, might prove helpful (a large amount of what I am going to explain can be thought of as functional analysis in a non-archimedean setting). Elementary notions from the theory of analytic functions (as studied in complex analysis) might appear at some point and will be taken for granted.

Sommario del corso

• Program
• Course delivery
• Learning assessment methods
• Suggested readings and bibliography
• Notes
• Class schedule
• Teaching tools
• Course card

Program

Contents: profinite groups and rings; (commutative) Iwasawa algebras and their interpretation in terms of measures and distributions; Mahler's theorem and the Mahler/Amice transform; structure theorem for finitely generated modules over Iwasawa algebras and applications to growth control; the Iwasawa main conjecture for cyclotomic fields.

The idea is to start without assuming too much (some basic course in commutative algebra and algebraic number theory, but, for example, no real familiarity with inverse limit constructions) and spend some time in building a framework which can encompass much current research, a bit more general than the standard textbook material (so the Iwasawa algebra to be discussed would be Z_p[ [Gamma] ], with Gamma isomorphic to Z_p^d) and also some applications beyond the standard textbooks.

Ideally I would also give a complete proof of the simplest case of the main conjecture, but if time should not suffice I might decide to be sketchy on some parts, since details can be found in textbooks.

Course delivery

The course is delivered in class, but it is also possible to follow online (interested people should contact me for the link). 

I am trying to record the lectures, for those people who cannot attend in the scheduled time. Please contact me for links to past lectures.

Learning assessment methods

The exam will consist in giving a seminar on some topic agreed with the teacher.

I plan to write some notes and upload them online. However, there might be some waiting time between the lecture and the respective notes.

Notes

Hours: 30 Period: spring 2025

Class schedule

• Teaching material
• Exams and finals
• Class schedule
• Go to Moodle
• Register for the course
• Enrolled students
• Invia email agli studenti registrati
• Export table