Registration Deadline: January 24, 2027

Instructor: Professor Antonio Lei, University of Ottawa

Course Dates: January 11 - April 14, 2027
Mid-Semester Break: February 15-19, 2027
Lecture Time: Wednesdays, 10:00 AM - 11:20 AM | Fridays, 8:30 AM - 9:50 AM (ET)
Office Hours: by appointment

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: 

Format: 

Course Description

This course develops the arithmetic of infinite Galois extensions and the theory of $p$-adic $L$-functions, central themes in modern number theory. Students will first study towers of number fields and the structure of infinite Galois groups, with particular emphasis on cyclotomic and other natural infinite extensions. The tools of Iwasawa theory will be introduced to analyze arithmetic invariants in these towers, such as class groups, and to relate their growth to deep structural phenomena.

The second part of the course focuses on the construction and properties of $p$-adic $L$-functions. Starting from complex $L$-functions and their special values, students will see how to interpolate these values $p$-adically and how to formulate and study main conjectures linking $p$-adic $L$-functions to arithmetic modules over Iwasawa algebras.

A solid understanding of infinite extensions and $p$-adic $L$-functions is essential for contemporary research in Iwasawa theory and many aspects of the Langlands programme. It provides both the conceptual and technical foundation for current work on special values of $L$-functions and the deeper interplay between Galois representations and arithmetic geometry.

We will cover the following topics:

• Review of Galois theory and infinite groups
• Infinite Galois groups and examples
• $p$-adic numbers and local fields
• Iwasawa algebras and structure theorems
• Control theorems and growth of ideal class groups
• $p$-adic measures and Mahler transforms
• Introduction to global fields
• Locally analytic functions and distributions
• The Kubota-Leopoldt $p$-adic $L$-function
• The Iwasawa main conjecture
• The $\mu = 0$ conjecture and further topics

Prerequisites: Abstract algebra, Galois theory and number theory (undergraduate level)

Evaluation:

• Attendance: 10%
• Two Assignments: 20% each
• A final project (either a written report or a presentation): 50%

Reference Material

I will use the textbook of Washington on cyclotomic fields and the lecture notes by Joaquín Rodrigues Jacinto and Chris Williams as a basis, combined with some personal notes of mine. Students will not be required to acquire any textbooks for this course.